Existence and regularity results for a class of parabolic problems with double phase flux of variable growth

Abstract

We study the homogeneous Dirichlet problem for the equation

$$\begin{aligned} u_t-{\text {div}}\left( \mathcal {F}(z,\nabla u)\nabla u\right) =f, \quad z=(x,t)\in Q_T=\Omega \times (0,T), \end{aligned}$$

where $\Omega \subset \mathbb {R}^N$, is a bounded domain with $\partial \Omega \in C^2$, and $\mathcal {F}(z,\xi )=a(z)\vert \xi \vert ^{p(z)-2}+b(z)\vert \xi \vert ^{q(z)-2}$. The variable exponents p, q and the nonnegative modulating coefficients a, b are given Lipschitz-continuous functions. It is assumed that $\frac{2N}{N+2}<p(z),\ q(z)$, and that the modulating coefficients and growth exponents satisfy the balance conditions

$$\begin{aligned} a(z)+b(z)\ge \alpha >0,\quad \vert p(z)-q(z)\vert <\frac{2}{N+2}\hbox { in }\overline{Q}_T \end{aligned}$$

with $\alpha =const$. We find conditions on the source f and the initial data $u(\cdot ,0)$ that guarantee the existence of a unique strong solution u with $u_t\in L^2(Q_T)$ and $a\vert \nabla u\vert ^{p}+b\vert \nabla u\vert ^q\in L^\infty (0,T;L^1(\Omega ))$. The solution possesses the property of global higher integrability of the gradient,

$$\begin{aligned} \vert \nabla u\vert ^{\min \{p(z),q(z)\}+r}\in L^1(Q_T)\quad \text {with any }r\in \left( 0,\frac{4}{N+2}\right) , \end{aligned}$$

which is derived with the help of new interpolation inequalities in the variable Sobolev spaces. The global second-order differentiability of the strong solution is proven:

$$\begin{aligned} D_i\left( \sqrt{\mathcal {F}(z,\nabla u)}D_j u\right) \in L^{2}(Q_T),\quad i=1,2,\ldots ,N. \end{aligned}$$

The same results are obtained for the equation with the regularized flux $\mathcal {F}(z,\sqrt{\epsilon ^2+(\xi ,\xi )})\xi$.

1 Introduction

Let $\Omega \subset \mathbb {R}^N$ be a smooth bounded domain, $N \ge 2$ and $0<T< \infty$. We consider the following parabolic equation with the homogeneous Dirichlet boundary conditions:

$$\begin{aligned} {\left\{ \begin{array}{ll} &{} u_t - {{\,\textrm{div}\,}}\left( \mathcal {F}(z,\nabla u)\nabla u\right) = f(z) \quad \hbox { in}\ Q_T, \\ &{} \hbox {} u=0 \, \hbox {on}\, \Gamma _T, \quad u(x,0)=u_0(x) \text{ in } \Omega , \end{array}\right. } \end{aligned}$$

(1.1)

where $z=(x,t)$ denotes the point in the cylinder $Q_T=\Omega \times (0,T]$ and $\Gamma _T= \partial \Omega \times (0,T)$ is the lateral boundary of the cylinder. The function

$$\begin{aligned} \mathcal {F}(z,\nabla u)\nabla u=\left( a(z)\vert \nabla u\vert ^{p(z)-2} + b(z) \vert \nabla u\vert ^{q(z)-2}\right) \nabla u \end{aligned}$$

represents the flux that depends on the given coefficients a(z), b(z) and the variable exponents p(z), q(z). It is assumed that the coefficients a, b are nonnegative in $Q_T$ and satisfy the condition

$$\begin{aligned} a(z)+b(z)\ge \alpha \hbox { in }\,\overline{Q}_T\, \hbox {with a positive constant }\alpha . \end{aligned}$$

(1.2)

Equations of the type (1.1) fall into the class of double phase equations intensively studied in the last decades. This name, introduced in [23, 24], reflects the fact that the coercivity and growth conditions on the flux vary from point to point in the problem domain and depend on the relation between p(z) and q(z) and the properties of the coefficients. As special cases, equation (1.1) contains the equations

$$\begin{aligned} u_t={\text {div}}\left( \vert \nabla u\vert ^{p(z)-2}\nabla u\right) +f,\qquad a\equiv 1,\;b\equiv 0, \end{aligned}$$

(1.3)

and

$$\begin{aligned} \begin{aligned}&u_t={\text {div}}\left( \vert \nabla u\vert ^{p(z)-2}\nabla u+b(z)\vert \nabla u\vert ^{q(z)-2}\nabla u\right) +f, \\&\quad a(z)\equiv 1,\quad b(z)\ge 0,\quad q(z)\ge p(z). \end{aligned} \end{aligned}$$

(1.4)

Equations (1.3), (1.4) are prototypes of PDEs with variable growth conditions with a gap between the conditions of monotonicity and coercivity. In case of (1.3)

$$\begin{aligned} -c+\vert \xi \vert ^{p^-}\le \mathcal {F}(z,\xi )\xi \cdot \xi \le \vert \xi \vert ^{p^+}+C, \end{aligned}$$

where c, C are nonnegative constants, $p^-=\inf p(z)<\sup p(z)=p^+$ with the supremum and infimum taken over the problem domain. Much attention has been paid in the literature to the study of the special case (1.3). This interest is motivated by applications to the mathematical modelling of various real-world phenomena, such as the flows of electrorheological or thermorheological fluids, the thermistor problem, or the problem of processing of digital images. With no intention to provide an exhaustive list of references concerning equation (1.3), we confine ourselves to [1, 5, 7, 8, 13, 30, 32, 48] and references therein.

Equation (1.4) furnishes an example of the double phase operator where the growth of the flux is controlled by the first or the second term depending upon the values of the modulating coefficient b(z). More precisely, on the set $\{z \in Q_T: b(z)=0\}$ the growth is controlled by the p(z)-power of the gradient, while on the set $\{z \in Q_T: b(z) \ne 0\}$ it is given by the sum of the p(z) and q(z) powers of the gradient. This is one of the reasons why we regard (1.4) as an equation with the double phase operator. Such operators appeared for the first time in the late 1970s and 1980s in the works by Ball [11] and Zhikov [49] for the interpretation of physical processes in the nonlinear elasticity theory. In particular, Zhikov [49, 50] investigated the models of strongly anisotropic materials and introduced the following energy functional

$$\begin{aligned} u \mapsto \int _{\Omega } \left( \vert \nabla u\vert ^p + b(x) \vert \nabla u\vert ^q\right) ~dx, \quad 0 \le b(x) \le L, \quad 1<p<q, \end{aligned}$$

where the modulating coefficient b(x) dictates the geometry of the composite made of two materials with ordered hardening exponent p and q. It is easy to see that the corresponding differential form of the integral functional can be drafted as

$$\begin{aligned} u \mapsto - {{\,\textrm{div}\,}}\left( \vert \nabla u\vert ^{p-2}\nabla u+b(z)\vert \nabla u\vert ^{q-2}\nabla u \right) .\end{aligned}$$

The study of such operators was continued in the seminal works of Marcellini [40, 41] and Mingione et al. [1, 23, 24]. Later on, problems involving the double phase operators attracted attention of many researchers. Despite significant progress in the study of the double phase problems, they remain the subject of active research. On the one hand, the study of equations with the double phase operators is a challenging mathematical problem. On the other hand, such problems started appearing in a variety of physical models. We refer here to [10] for applications in transonic flows, [12] for quantum physics and [18] for reaction-diffusion systems. There is an extensive literature devoted to the study of the questions of existence and qualitative properties of solutions to the stationary and evolution double phase problems for single PDEs and systems of equations. We refer to [42] for a comprehensive review of the available literature and a presentation of the current state of the theory of equations with nonstandard growth.

The results on the existence and qualitative properties of solutions to evolution problems can be found in [4, 16, 21, 31, 36, 41, 47]. For a review of the existence results we refer to papers [16, 20, 21]. The natural analytic framework for the study of double-phase problems is the theory of Musielak–Orlicz spaces, for the p(z)-Laplace equation the natural function space is the variable Sobolev space $\mathbb {W}_{p(\cdot )}(Q_T)$. These functions space are introduced and briefly described in Sect. 2. The existence results of [16, 21] are obtained in the context of the Musielak–Orlicz spaces. When applied to the model equation (1.4), these results guarantee the existence of distributional solutions if the exponents p, q and the coefficient b satisfy certain regularity and balance conditions (see [16, Example 1.18 (E2)]) sufficient for the density of smooth functions in the Musielak–Orlicz space the solution belongs to.

The variational approach to a wide class of parabolic equations and systems with nonstandard growth is discussed in [14, 41], see also references therein for the previous results. The existence results of [41] apply to the Dirichlet problem for the equations

$$\begin{aligned} u_t={\text {div}} D_\xi f(x,u,\nabla u)-D_u f(x,u,\nabla u) \end{aligned}$$

with a convex integrand $f:\Omega \times \mathbb {R}\times \mathbb {R}^N\mapsto [0,\infty ]$ satisfying the coercivity condition $f(x,u,\xi )\ge \nu \vert \xi \vert ^p-g(x)(1+\vert u\vert )$ with some constants $\nu >0$, $p>1$ and a nonnegative function $g\in L^{\frac{p}{p-1}}(\Omega )$. The initial function is assumed to satisfy the conditions $u_0\in L^{2}(\Omega )\cap W^{1,p}(\Omega )$, $f(x,u_0,\nabla u_0)\in L^1(\Omega )$ (cf. with the assumptions on $u_0$ in Theorem 2.1). This class of equations includes (1.3) and (1.4) with constant p, q and $b\equiv b(x)\ge 0$. For the latter, the integrand f has the form $f(x,\xi ) =\frac{1}{p}\vert \xi \vert ^{p}+\frac{b(x)}{q}\vert \xi \vert ^q$.

In the present work, we address the following two questions.

• Sufficient conditions of existence of strong solutions to problem (1.1). By the strong solution we mean a solution whose time derivative belongs to $L^2(Q_T)$ and $\mathcal {F}(z,\nabla u)\vert \nabla u\vert ^2\in L^{\infty }(0,T;L^{1}(\Omega ))$, see Definition 2.1.

• The global properties of higher integrability of the gradient and the second-order differentiability with respect to the spatial variables.

Higher differentiability of solutions of equations with nonstandard growth has already been studied in the literature. It is known [30] that for the solutions of equation (1.3) with the flux $(\mu ^2+\vert \nabla u\vert ^2)^{\frac{p-2}{2}}\nabla u$, $\mu \in [0,1]$,

$$\begin{aligned} D_{x_i}((\mu ^2+\vert \nabla u\vert ^2)^{\frac{p-2}{4}}D_{x_j}u)\in L^2_{loc}(Q_T)\hbox { and }u_t\in L^{\frac{p}{p-1}}_{loc}(Q_T). \end{aligned}$$

For systems of parabolic equations with (p, q) growth and non-smooth coefficients, existence of weak solutions with $D_{x_i}((\mu ^2+\vert \nabla u\vert ^2)^{\frac{p-2}{4}}D_{x_j}u)\in L^2_{loc}(Q_T)$ and $u_t\in L^{\frac{p}{q-1}}_{loc}(Q_T)$ was recently proven in [34] in the case $2\le p\le q$, and in [27] for $p>\lambda \frac{2N}{N+2}$ with a constant $\lambda \ge 1$ depending on the coefficients in the equation. The weak solutions are constructed as the limits of sequences of solutions of regularized problems. The local higher regularity of the approximations follows from local interpolation inequalities derived with the help of Caccioppoli-type inequalities.

The global regularity of solutions of the evolution p-Laplace equation is studied in [22]. It is shown that

$$\begin{aligned} u_t\in L^2(Q_T)\hbox { and }D_{x_i}(\vert \nabla u\vert ^{p-2}D_{x_j} u)\in L^{2}(Q_T), \end{aligned}$$

provided that $f\in L^2(Q_T)$ and $u_0\in L^2(Q_T)\cap W^{1,p}_0(\Omega )$. This regularity result is sharp - see [22, Remark 2.3]. For the evolution p(z)-Laplace equation (1.3), existence of solutions with the global regularity properties $D_{x_i}(\vert \nabla u\vert ^{\frac{p(z)-2}{2}}D_{x_j}u)\in L^{2}(Q_T)$, $u_t\in L^2(Q_T)$ is proven in [8]. For the singular equation (1.3) with $p(z)\le 2$, the inclusion $D_{x_ix_j}^2u\in L^{2}(\Omega \times (\delta ,T))$, $\delta >0$, is proven in [6] under specific assumptions on the exponent p(z).

Our approach to the questions of existence and regularity of solutions is different from the methods of the cited works. We find a solution as the limit of a sequence of finite-dimensional Galerkin’s approximations. Because of the special choice of the basis and the assumptions on the regularity of the data, the finite-dimensional approximations are smooth up to the parabolic boundary of the cylinder $Q_T$. This allows us to derive a global interpolation inequality based on the Gagliardo-Nirenberg inequalities and the covering argument. Combining the interpolation inequality with the a priori estimates for the Galerkin approximations, we prove global in $Q_T$ higher integrability of the gradients of the approximations, which entails strong convergence of the gradients and the second-order differentiability. Passing to the limit we obtain a solution that inherits the following properties of global regularity:

$$\begin{aligned} u_t\in L^2(Q_T),\qquad D_{x_i}\left( \sqrt{\mathcal {F}(z,\nabla u)}D_{x_j}u\right) \in L^{2}(Q_T). \end{aligned}$$

(1.5)

Properties (1.5) hold for the solutions of the problems with the regularized flux $\mathcal {F}(z,(\epsilon ^2+(\nabla u,\nabla u))^{\frac{1}{2}})\nabla u$, $\epsilon >0$. It is worth noting that the use of Galerkin’s approximations prevents us from employing the techniques developed in [22].

A crucial element of the proofs is the property of global higher integrability of the gradient. The property of higher integrability of the gradient is interesting in itself. This is an intrinsic property of solutions of nonlinear elliptic and parabolic equations. For the solutions of evolution equations and systems of p-Laplace structure, it was established in [38] and has been studied since then in numerous works under different conditions on the nonlinear structure of the equation. We refer to [1,2,3, 7, 13, 34, 43, 44, 48, 51] for the parabolic equations with nonstandard power growth. For the solutions of equation (1.3), the local version of this property reads as follows: if u is a weak solution of equation (1.3) and $\vert \nabla u\vert ^{p(z)}\in L^{1}(Q_T)$, then for every strictly interior sub-cylinder $Q'=\Omega '\times (\epsilon ,T)$, $\Omega '\Subset \Omega$, $\epsilon >0$, there is a constant $\delta >0$ such that $\vert \nabla u\vert ^{p(z)+\delta }\in L^1(Q')$. The constant $\delta$ depends on the distance between the parabolic boundaries of $Q_T$ and $Q'$. It was recently shown in [2] for constant p and in [3] for variable p(x, t) that this property remains valid in the domains $\Omega \times (\epsilon ,T)$ under mild conditions on the regularity of $\partial \Omega$. The global higher integrability of the gradient in the whole cylinder $Q_T$ is proven in [8] for the solutions of equation (1.3), now we prove its analogue for the solutions of equation (1.1) in the absence of the order relation between the variable exponents p(z) and q(z). We show that

$$\begin{aligned} \vert \nabla u\vert ^{\min \{p(z),q(z)\}+r}\in L^1(Q_T)\quad \hbox { with any}\ r\in \left( 0,\frac{4}{N+2}\right) . \end{aligned}$$

(1.6)

In application to the solutions of equation (1.3), or equation (1.4) with the exponents subject to the additional order condition $p(z)\le q(z)$, this property reads as follows:

$$\begin{aligned} \displaystyle \vert \nabla u\vert ^{p(z)+r}\in L^1(Q_T)\hbox { with any }0<r<\frac{4}{N+2}. \end{aligned}$$

This result refines the property of global higher integrability of the gradient proven in [8] for the solutions of equation (1.3) and recovers the best order of local integrability of the gradient proven for weak solutions of equations with constant (p, q)-growth conditions in [14] in the case $2\le p\le q$, and in [46] for the case $\frac{2N}{N+2}<p<2$ and $p\le q$.

The results and novelties of the present work are summarized as follows.

• It is proven that problem (1.1) has a unique strong solution. This result does not require any assumption on the null sets of the coefficients $a(\cdot )$, $b(\cdot )$, except for condition (1.2). The constructed solution is continuous with respect to the problem data.

• The solution possesses the property of global higher integrability of the gradient (1.6).

• The solution has the global higher differentiability properties (1.5).

• The results do not require any order relation between the variable exponents p(z) and q(z). We impose instead the condition on the admissible gap between the values of p(z) and q(z):

  $$\begin{aligned} \max \{p(z),q(z)\}<\min \{p(z),q(z)\}+\frac{2}{N+2}. \end{aligned}$$

  (1.7)

  We do not distinguish between degenerate or singular equations. The exponents are only subject to condition (1.7) and may vary within the intervals $[p^-,p^+], [q^-,q^+]\subset (\frac{2N}{N+2},\infty )$ independently of each other.

2 Assumptions and results

To formulate the results we have to introduce the function spaces the data and solutions of problem (1.1) belong to.

2.1 The functions spaces

2.1.1 Variable Lebesgue spaces

We begin with a brief description of the Lebesgue and Sobolev spaces with variable exponents. A detailed insight into the theory of these spaces and a review of the bibliography can be found in [26, 28, 39]. Let $\Omega \subset \mathbb {R}^N$, $N \ge 2$, be a bounded domain with Lipschitz continuous boundary $\partial \Omega$. Define the set

$$\begin{aligned} \mathcal {P}(\Omega ):= \{\text {measurable functions on }\Omega \hbox { with values in }(1, \infty )\}. \end{aligned}$$

Given $r \in \mathcal {P}(\Omega )$, we introduce the modular

$$\begin{aligned} A_{r(\cdot )}(f)= \int _{\Omega } \vert f(x)\vert ^{r(x)} ~dx \end{aligned}$$

(2.1)

and the set

$$\begin{aligned} L^{r(\cdot )}(\Omega ) = \{f: \Omega \rightarrow \mathbb {R}\ \vert \ \text {measurable on}\ \Omega ,\ A_{r(\cdot )}(f) < \infty \}. \end{aligned}$$

The set $L^{r(\cdot )}(\Omega )$ equipped with the Luxemburg norm

$$\begin{aligned} \Vert f\Vert _{r(\cdot ), \Omega }= \inf \left\{ \lambda >0 : A_{r(\cdot )}\left( \frac{f}{\lambda }\right) \le 1\right\} \end{aligned}$$

becomes a Banach space. By convention, from now on we use the notation

$$\begin{aligned} r^- := {{\,\textrm{ess}\,}}\min _{x \in \Omega } r(x), \quad r^+ := {{\,\textrm{ess}\,}}\max _{x \in \Omega } r(x). \end{aligned}$$

If $r \in \mathcal {P}(\Omega )$ and $1< r^- \le r(x) \le r^+ < \infty$ in $\Omega$, then the following properties hold.

• (i) $L^{r(\cdot )}(\Omega )$ is a reflexive and separable Banach space.

• (ii) For every $f \in L^{r(\cdot )}(\Omega )$

  $$\begin{aligned} \min \{\Vert f\Vert ^{r^-}_{r(\cdot ), \Omega }, \Vert f\Vert ^{r^+}_{r(\cdot ), \Omega }\} \le A_{r(\cdot )}(f) \le \max \{\Vert f\Vert ^{r^-}_{r(\cdot ), \Omega }, \Vert f\Vert ^{r^+}_{r(\cdot ), \Omega }\}. \end{aligned}$$

  (2.2)

• (iii) For every $f \in L^{r(\cdot )}(\Omega )$ and $g \in L^{r'(\cdot )}(\Omega )$, the generalized Hölder inequality holds:

  $$\begin{aligned} \int _{\Omega } \vert fg\vert \le \left( \frac{1}{r^-} + \frac{1}{(r')^-} \right) \Vert f\Vert _{r(\cdot ), \Omega } \Vert g\Vert _{r'(\cdot ), \Omega } \le 2 \Vert f\Vert _{r(\cdot ), \Omega } \Vert g\Vert _{r'(\cdot ), \Omega }, \end{aligned}$$

  (2.3)

  where $r'= \frac{r}{r-1}$ is the conjugate exponent of r.

• (iv) If $p_1, p_2 \in \mathcal {P}(\Omega )$ and satisfy the inequality $p_1(x) \le p_2(x)$ a.e. in $\Omega$, then $L^{p_2(\cdot )}(\Omega )$ is continuously embedded in $L^{p_1(\cdot )}(\Omega )$ and for all $u \in L^{p_2(\cdot )}(\Omega )$

  $$\begin{aligned} \Vert u\Vert _{p_1(\cdot ), \Omega } \le C\Vert u\Vert _{p_2(\cdot ), \Omega },\qquad C=C(\vert \Omega \vert , p_1^\pm , p_2^\pm ). \end{aligned}$$

  (2.4)

• (v) For every sequence $\{f_k\} \subset L^{r(\cdot )}(\Omega )$ and $f \in L^{r(\cdot )}(\Omega )$

  $$\begin{aligned} \Vert f_k-f\Vert _{r(\cdot ),\Omega } \rightarrow 0 \quad \hbox {iff} \, A_{r(\cdot )}(f_k-f) \rightarrow 0\, \hbox {as}\, k \rightarrow \infty . \end{aligned}$$

  (2.5)

2.1.2 Variable Sobolev spaces

The variable Sobolev space $W^{1,r(\cdot )}_0(\Omega )$ is the set of functions

$$\begin{aligned} W^{1,r(\cdot )}_0(\Omega )= \left\{ u: \Omega \rightarrow \mathbb {R}\ \vert \ u \in L^{r(\cdot )}(\Omega )\cap W^{1,1}_0(\Omega ),\; \vert \nabla u\vert \in L^{r(\cdot )}(\Omega ) \right\} \end{aligned}$$

equipped with the norm

$$\begin{aligned} \Vert u\Vert _{W^{1,r(\cdot )}_0(\Omega )}= \Vert u\Vert _{r(\cdot ), \Omega } + \Vert \nabla u\Vert _{r(\cdot ), \Omega }. \end{aligned}$$

If $r \in C^0(\overline{\Omega })$, the Poincaré inequality holds: for every $u\in W_0^{1,r(\cdot )}(\Omega )$

$$\begin{aligned} \Vert u\Vert _{r(\cdot ),\Omega } \le C \Vert \nabla u\Vert _{r(\cdot ),\Omega }. \end{aligned}$$

(2.6)

Inequality (2.6) means that the equivalent norm of $W^{1,r(\cdot )}_0(\Omega )$ is given by

$$\begin{aligned} \Vert u\Vert _{W^{1,r(\cdot )}_0(\Omega )}=\Vert \nabla u\Vert _{r(\cdot ),\Omega }. \end{aligned}$$

(2.7)

Let us denote by $C_{\textrm{log}}(\overline{\Omega })$ the subset of $\mathcal {P}(\Omega )$ composed of the functions continuous on $\overline{\Omega }$ with the logarithmic modulus of continuity:

$$\begin{aligned} {p} \in C_{\textrm{log}}(\overline{\Omega })\quad \Leftrightarrow \quad \vert p(x)-p(y)\vert \le \omega (\vert x-y\vert )\quad \forall x,y\in \overline{\Omega }, \;\vert x-y\vert <\frac{1}{2}, \end{aligned}$$

(2.8)

where $\omega$ is a nonnegative function such that

$$\begin{aligned} \limsup _{s\rightarrow 0^+}\omega (s)\ln \frac{1}{s}=C,\quad C=const. \end{aligned}$$

If $r \in C_{\textrm{log}}(\overline{\Omega })$, then the set $C_{0}^\infty (\Omega )$ of smooth functions with finite support is dense in $W^{1,r(\cdot )}_0(\Omega )$. This property allows one to use the equivalent definition of the space $W^{1,r(\cdot )}_0(\Omega )$:

$$\begin{aligned} W^{1,r(\cdot )}_0(\Omega )=\left\{ \text {the closure of } C_{0}^{\infty }(\Omega )\hbox { with respect to the norm }\Vert \cdot \Vert _{W^{1,r(\cdot )}_0(\Omega )}\right\} . \end{aligned}$$

Given a function $u\in W^{1,r(\cdot )}_0(\Omega )$ with $r \in C_{\textrm{log}}(\overline{\Omega })$, the smooth approximations of u in $W^{1,r(\cdot )}_0(\Omega )$ can be obtained by mollification.

2.1.3 Musielak–Orlicz spaces

Let $a_0,b_0:\overline{\Omega }\mapsto [0,\infty )$ be given functions, $a_0,b_0\in C^{0,1}(\overline{\Omega })$. Assume that the exponents $p(\cdot ), q(\cdot )\in C^{0,1}(\overline{\Omega })$ take values in the intervals $[p^-,p^+]$, $[q^-,q^+]$. Set

$$\begin{aligned} s(x)=\max \{2,\min \{p(x),q(x)\}\},\quad r(x)=\max \{2,p(x)\},\quad \sigma (x)=\max \{2,q(x)\} \end{aligned}$$

and consider the function

$$\begin{aligned} \mathcal {H}(x,t)=t^{s(x)}+a_0(x)t^{r(x)}+b_0(x)t^{\sigma (x)},\quad t\ge 0,\quad x\in \Omega . \end{aligned}$$

The set

$$\begin{aligned} L^\mathcal {H}(\Omega )=\left\{ u:\Omega \mapsto \mathbb {R}\ \vert \ u\hbox { is measurable},\,\rho _{\mathcal {H}}(u)=\int _{\Omega }\mathcal {H}(x,\vert u\vert )\,dx<\infty \right\} \end{aligned}$$

equipped with the Luxemburg norm

$$\begin{aligned} \Vert u\Vert _{\mathcal {H}}=\inf \left\{ \lambda >0:\,\rho _{\mathcal {H}}\left( \dfrac{u}{\lambda }\right) \le 1\right\} \end{aligned}$$

becomes a Banach space. The space $L^\mathcal {H}(\Omega )$ is separable and reflexive [33]. By $W^{1,\mathcal {H}}(\Omega )$ we denote the Musielak-Sobolev space

$$\begin{aligned} W^{1,\mathcal {H}}(\Omega )=\left\{ u\in L^{\mathcal {H}}(\Omega ):\,\vert \nabla u\vert \in L^{\mathcal {H}}(\Omega )\right\} \end{aligned}$$

(2.9)

with the norm

$$\begin{aligned} \Vert u\Vert _{1,\mathcal {H}}=\Vert u\Vert _{\mathcal {H}}+\Vert \nabla u\Vert _{\mathcal {H}}. \end{aligned}$$

The space $W^{1,\mathcal {H}}_0(\Omega )$ is defined as the closure of $C_{0}^\infty (\Omega )$ with respect to the norm of $W^{1,\mathcal {H}}(\Omega )$.

For the elements of the space $W^{1,\mathcal {H}}_0(\Omega )$, the norm and modular convergence are equivalent:

$$\begin{aligned} \begin{aligned}&\Vert u\Vert _{W^{1,\mathcal {H}}_0(\Omega )}\rightarrow 0\quad \Leftrightarrow \quad \rho _{\mathcal {H}}(u)+\rho _{\mathcal {H}}(\vert \nabla u\vert )\rightarrow 0. \end{aligned} \end{aligned}$$

(2.10)

The detailed presentation of the theory of Musielak–Orlicz spaces can be found in the monograph [37] and the references therein.

2.1.4 Spaces of functions depending on x and t

For the study of parabolic problem (1.1) we use the spaces of functions depending on $z=(x,t)\in Q_T$. Assume that $\partial \Omega \in C^2$, $p \in C^{0,1}(\overline{Q}_T)$ and define the spaces

$$\begin{aligned} \begin{aligned}&\mathbb {V}_{p(\cdot ,t)}(\Omega ) = \{u: \Omega \rightarrow \mathbb {R}\ \vert \ u \in L^2(\Omega ) \cap W_0^{1,1}(\Omega ),\,\vert \nabla u\vert ^{p(x,t)}\in L^{1}(\Omega ) \},\quad t\in (0,T), \\&\mathbb {W}_{p(\cdot )}(Q_T)= \{u : (0,T) \rightarrow \mathbb {V}_{p(\cdot ,t)}(\Omega ) \ \vert \ u \in L^2(Q_T), \vert \nabla u\vert ^{p(z)}\in L^1(Q_T)\}. \end{aligned}\nonumber \\ \end{aligned}$$

(2.11)

The norm of $\mathbb {W}_{p(\cdot )}(Q_T)$ is defined by

$$\begin{aligned} \Vert u\Vert _{\mathbb {W}_{p(\cdot )}(Q_T)}=\Vert u\Vert _{2,Q_T}+\Vert \nabla u\Vert _{p(\cdot ),Q_T}. \end{aligned}$$

Since $p\in C^{0,1}(\overline{Q}_T)\subset C_{\textrm{log}}(\overline{Q}_T)$, the space $\mathbb {W}_{p(\cdot )}(Q_T)$ is the closure of $C_{0}^{\infty }(Q_T)$ with respect to this norm.

2.2 Main results

Let $p,q: Q_T \mapsto \mathbb {R}$ be functions satisfying the following conditions:

$$\begin{aligned} \begin{aligned}&\frac{2N}{N+2}< p^{-} \le p(z) \le p^{+},\quad \frac{2N}{N+2} < q^{-} \le q(z) \le q^{+} \quad \hbox { in}\ \overline{Q}_T \end{aligned} \end{aligned}$$

(2.12)

with positive constants $p^\pm$, $q^\pm$,

$$\begin{aligned} \begin{aligned} p,q \in C^{0,1}(\overline{Q}_T)\hbox { with the Lipschitz constant } L_{p,q}. \end{aligned} \end{aligned}$$

(2.13)

Let us define the functions

$$\begin{aligned} \underline{s}(z)=\min \{p(z),q(z)\},\quad \overline{s}(z)=\max \{p(z),q(z)\} \end{aligned}$$

(2.14)

and accept the notation

$$\begin{aligned} \underline{s}^-=\min _{\overline{Q}_T}\underline{s}(z),\quad \overline{s}^+=\max _{\overline{Q}_T}\overline{s}(z), \quad \dfrac{2N}{N+2}<\underline{s}^-\le \overline{s}^+\le \max \{p^+,q^+\}. \end{aligned}$$

(2.15)

We will repeatedly use the following threshold numbers:

$$\begin{aligned} \ r^\sharp =\dfrac{4}{N+2}, \qquad r_*=\dfrac{2}{N+2}. \end{aligned}$$

The modulating coefficients $a(\cdot )$ and $b(\cdot )$ satisfy the following conditions:

$$\begin{aligned} \begin{aligned}&\hbox {} a(z) \hbox {,}\, b(z)\hbox { are nonnegative in }\overline{Q}_T, \\&a,b \in C^{0,1}(\overline{Q}_T)\hbox { with the Lipschitz constants }L_{a}\hbox { and }L_b, \\&\max _{\overline{Q}_T}a(z)=a^+,\quad \max _{\overline{Q}_T}b(z)=b^+, \\&a(z) + b(z) \ge \alpha >0\hbox { in }\overline{Q}_T\hbox { with a positive constant }\alpha . \end{aligned}\nonumber \\ \end{aligned}$$

(2.16)

Definition 2.1

A function $u:Q_T\mapsto \mathbb {R}$ is called strong solution of problem (1.1) if

• 1. $u \in \mathbb {W}_{\overline{s}(\cdot )}(Q_T)$, $\vert \nabla u\vert \in L^{\infty }(0,T; L^{r(\cdot )}(\Omega ))$ with $r(z)=\max \{2,\underline{s}(z)\}$, and $u_t \in L^2(Q_T)$,

• 2. for every $\phi \in \mathbb {W}_{\overline{s}(\cdot )}(Q_T)$

  $$\begin{aligned} \int _{Q_T} u_t \phi ~dz + \int _{Q_T} \mathcal {F}(z,\nabla u)\nabla u\cdot \nabla \phi ~dz= \int _{Q_T} f \phi \,dz, \end{aligned}$$

  (2.17)

• 3. for every $\psi \in C_0^1(\Omega )$

  $$\begin{aligned} \int _{\Omega } (u(x,t)-u_0(x)) \psi ~dx \rightarrow 0\quad \text {as }t \rightarrow 0. \end{aligned}$$

The main results are given in the following theorems. We assume that the initial datum belongs to $W_0^{1,\mathcal {H}}(\Omega )$ (see (2.9)):

$$\begin{aligned} \Vert \nabla u_0\Vert _{2,\Omega }^2+\int _\Omega \mathcal {F}((x,0),\nabla u_0)\vert \nabla u_0\vert ^2\,dx=K<\infty \end{aligned}$$

(2.18)

with the coefficients a, b and exponents p, q in the flux function $\mathcal {F}$ taken at the instant $t=0$.

Theorem 2.1

(Existence and uniqueness) Let $\Omega \subset \mathbb {R}^N$, $N\ge 2$, be a bounded domain with the boundary $\partial \Omega \in C^2$. Assume that $p(\cdot )$, $q(\cdot )$ satisfy conditions (2.12), (2.13),

$$\begin{aligned} \max _{z \in \overline{Q}_T} \vert p(z)-q(z)\vert < r_*, \end{aligned}$$

(2.19)

and the coefficients $a(\cdot ), b(\cdot )$ satisfy conditions (2.16). Then, for every $u_0 \in W^{1,\mathcal {H}}_0(\Omega )$ satisfying (2.18) and for every $f_0 \in L^2(0,T; W_0^{1,2}(\Omega ))$, problem (1.1) has a unique strong solution u. The solution satisfies the estimate

$$\begin{aligned} \Vert u_t\Vert _{2,Q_T}^{2} + {\text {ess}}\sup _{(0,T)}\int _{\Omega } \left( \vert \nabla u\vert ^{2}+\mathcal {F}(z,\nabla u)\vert \nabla u\vert ^2 \right) \,dx \le C \end{aligned}$$

(2.20)

with a constant C depending upon N, $\partial \Omega$, T, $p^\pm$, $q^\pm$, the constants in conditions (2.13), (2.16), $\Vert u_0\Vert _{W_0^{1,\mathcal {H}}(\Omega )}$ and $\Vert f_0\Vert _{L^{2}(0,T;W^{1,2}_0(\Omega ))}$.

Theorem 2.2

(Continuity with respect to the data) Under the conditions of Theorem 2.1, the solution u satisfies the energy equality

$$\begin{aligned} \dfrac{1}{2}\Vert u(t)\Vert _{2,\Omega }^2+\int _{Q_T} \mathcal {F}(z,\nabla u)\vert \nabla u\vert ^2 \,dz= \dfrac{1}{2}\Vert u_0\Vert _{2,\Omega }^2+\int _{Q_T}uf\,dz. \end{aligned}$$

(2.21)

The solution is continuous with respect to the data: if u, v are two strong solutions of problem (1.1) corresponding to the initial data $u_0$, $v_0$ and the free terms f, g, then

$$\begin{aligned} \begin{aligned}&\sup _{(0,T)}\Vert u-v\Vert ^2_{2,\Omega }(t) + \int _{Q_T}\mathcal {F}(z,\nabla (u-v))\ \vert \nabla (u-v)\vert ^2\,dz \rightarrow 0, \\&\int _{Q_T}\vert \nabla (u-v)\vert ^{\underline{s}(z)}\,dz\rightarrow 0 \quad \text {as }\Vert u_0-v_0\Vert _{2,\Omega }+\Vert f-g\Vert _{2,Q_T}\rightarrow 0. \end{aligned} \end{aligned}$$

(2.22)

Theorem 2.3

(Global regularity) Let the conditions of Theorem 2.1 be fulfilled. Then,

• 1. for any $0\le r<\frac{4}{N+2}$

  $$\begin{aligned} \int _{Q_T}\vert \nabla u\vert ^{\underline{s}(z)+r }\,dz \le C \end{aligned}$$

  (2.23)

  with a constant C which depends on $N, \partial \Omega$, $T, p^\pm , q^\pm$, r, the constants in conditions (2.13), (2.16), (2.18) and $\Vert f_0\Vert _{L^{2}(0,T;W^{1,2}_0(\Omega ))}$;

• 2.
  $$\begin{aligned} \begin{aligned}&D_{x_i}\left( \sqrt{\mathcal {F}(z,\nabla u)} D_{x_j}u\right) \in L^2(Q_T), \\&D^2_{x_ix_j} u\in L^{\underline{s}(\cdot )}_{loc}(Q_T \cap \{z: \max \{p(z), q(z)\}< 2\}), \end{aligned} \end{aligned}$$

  and the corresponding norms are bounded by constants depending only on the data.

Outline of the paper. In Sect. 3, we collect several technical assertions which are repeatedly referred to in the rest of the text. These are elementary algebraic relations and some useful relations between the elements of the space $\mathbb {W}_{\underline{s}(\cdot )}(Q_T)$ and the functions with $\mathcal {F}(z,\nabla v)\vert \nabla v\vert ^2\in L^1(Q_T)$.

Section 4 is of utter importance for the rest of the work. This is where we derive the interpolation inequalities which yield the property of global higher integrability of the gradient, the key tool for the further study. These inequalities are proven for the regularized fluxes $\mathcal {F}\left( z,(\epsilon ^2+(\nabla v,\nabla v))^{\frac{1}{2}}\right)$, $\epsilon \in (0,1)$. The strategy of the proof consists in reduction to a situation where the classical Gagliardo-Nirenberg inequalities in Sobolev spaces with constant exponents become applicable. To this end, we make use of the uniform continuity of the exponents $p(\cdot )$, $q(\cdot )$ and choose a finite cover of the domain $\Omega$, or the cylinder $\Omega \times (0,T)$, by subdomains where the oscillations of p and q are suitably small. Proceeding in this way we overcome the difficulties typical for variable Sobolev spaces where, unlike the classical Sobolev spaces, the inequalities for the norms and modulars need not coincide - see, e.g., (2.2). In the result, we show that for all sufficiently smooth functions v and every $\beta >0$

$$\begin{aligned} \alpha \int _{Q_T}\vert \nabla v\vert ^{\underline{s}(z)+r}\,dz\le \beta \int _{Q_T}\mathcal {F}\left( z,(\epsilon ^2+(\nabla v,\nabla v))^{\frac{1}{2}}\right) \sum _{i,j=1}^{N}\vert D^2_{x_ix_j}v\vert ^{2}\,dz+C \end{aligned}$$

with $\underline{s}(z)=\min \{p(z),q(z)\}$, any $0<r<(0,\frac{4}{N+2})$, $\alpha >0$, $\epsilon >0$, and a constant C depending on the known constants, $\beta$, and ${\text {ess}}\sup _{(0,T)}\Vert u(t)\Vert _{2,\Omega }$.

In Sect. 5 we formulate the problem with the regularized flux and describe the scheme of construction of a solution as the limit of a sequence of finite-dimensional approximations. Special attention is paid to the choice of the basis. On the one hand, in our conditions on the data, the set of smooth functions with compact support $C_0^\infty (\Omega )$ is dense in the variable Sobolev spaces and the generalized Musielak–Orlicz space $W^{1,\mathcal {H}}_0(\Omega )$. On the other hand, it is convenient to take for the basis the set of eigenfunctions of the Dirichlet problem for the Laplace operator but this set approximates the elements of $C_0^\infty (\Omega )$ in a suitable function space only if the boundary $\partial \Omega$ possesses an extra regularity. For this reason, we proceed in two steps. In the first step, we prove the main assertions for the regularized problem in a smooth domain with $\partial \Omega \in C^{k}$, $k\ge 2+\left[ \frac{N}{2}\right]$. In the second step, we extend all these assertions to the domain with minimal required regularity $\partial \Omega \in C^2$. This is done by approximation of $\Omega$ from the interior by a family of expanding smooth domains.

Solvability of the regularized problem and the regularity of the solution are studied in Sects. 6, 7. In Sect. 6 we derive uniform a priori estimates on the sequence of Galerkin’s approximations in a smooth domain, including the estimates on the nonlinear terms that involve the second-order derivatives. In Sect. 7 we use these estimates to extract a subsequence with suitable convergence properties, and to show that its limit is a solution of the regularized problem. The second-order higher regularity of the limit follows from the a priori estimates on the regularized fluxes and the pointwise convergence of the sequence of gradients of the approximations.

Theorems 2.1, 2.2, 2.3 are proven in Sect. 8 by passing to the limit with respect to the regularization parameter.

In the special case $p(z)=q(z)$ and $a=b=const$, equation (1.1) transforms into the evolution p(z)-Laplace equation (1.3). For this equation, the questions of global higher integrability of the gradient and the second-order spatial regularity were studied in [8]. The assertions of Theorem 2.1 and 2.3 improve the corresponding results in [8] and, by the same token, complete the results of [9] for another special case (1.4).

3 Auxiliary propositions

We collect here the technical assertions used throughout the rest of the work. With certain abuse of notation, here we denote by p(x), q(x), $\underline{s}(x)$, $\overline{s}(x)$, a(x), b(x) the functions of the variables (x, t) with “frozen" $t\in [0,T]$. By continuity of these functions in $\overline{Q}_T$, conditions (2.19), (2.15) and (2.16) remain in force if we fix t and consider p(x, t), q(x, t) and a(x, t), b(x, t) as functions of the variable $x\in \Omega$. So, we write

$$\begin{aligned} \begin{aligned}&\overline{s}(x):= \max \{p(x), q(x)\},\qquad \overline{s}^+:= \max _{x \in \overline{\Omega }} \overline{s}(x), \\&\underline{s}(x):= \min \{p(x), q(x)\}, \qquad \underline{s}^-:= \min _{x \in \overline{\Omega }} \underline{s}(x). \end{aligned} \end{aligned}$$

Given $\epsilon \in (0,1)$, $\xi \in \mathbb {R}^N$, and non-negative functions $s_1, s_2$, we denote

$$\begin{aligned} \begin{aligned}&\beta _{\epsilon }(\xi )=\epsilon ^2+\vert \xi \vert ^{2}, \qquad \gamma ^{(p(x))}_{\epsilon }(\xi )=\beta ^{\frac{p(x)-2}{2}}_{\epsilon }(\xi ), \\&\mathcal {F}^{(s_1, s_2)}_{\epsilon }(x,\xi )= a(x) \beta ^{\frac{p(x)+s_1(x)-2}{2}}_{\epsilon }(\xi ) + b(x) \beta ^{\frac{q(x)+s_2(x)-2}{2}}_{\epsilon }(\xi ). \end{aligned} \end{aligned}$$

The function $\mathcal {F}_{\epsilon }^{(s_1, s_2)}(y,\xi )$ depends on y implicitly, through the exponents p, q, $s_1, s_2$ and the coefficients a, b. For this reason, we will write $\mathcal {F}_{\epsilon }^{(s_1, s_2)}(x,\xi )$ if the exponents and coefficients depend on $x\in \Omega$, or $\mathcal {F}_{\epsilon }^{(s_1, s_2)}(z,\xi )$ if at least one of these functions depends on $z=(x,t)\in Q_T$. The functions $\mathcal {F}_{\epsilon }^{(s_1,s_2)}(y,\xi )\xi$ approximate the flux: $\mathcal {F}_{0}^{(0,0)}(y,\xi )\xi \equiv \mathcal {F}(y,\xi )\xi$.

• 1. For every $\epsilon \in (0,1)$, $\mu >0$, and $\xi \in \mathbb {R}^N$

  $$\begin{aligned} \vert \xi \vert ^{2\mu }\le \beta ^{\mu }_{\epsilon }(\xi )\le {\left\{ \begin{array}{ll} (2\vert \xi \vert ^2)^{\mu }=2^\mu \vert \xi \vert ^{2\mu } &{} \text {if }\vert \xi \vert \ge \epsilon , \\ (2\epsilon ^2)^\mu &{} \hbox { if}\ \vert \xi \vert <\epsilon \end{array}\right. } \le 2^\mu \left( 1+\vert \xi \vert ^{2\mu }\right) . \end{aligned}$$

  (3.1)

• 2. For the nonnegative coefficients $a, b \in L^\infty (\Omega )$ and the exponents $s_1,s_2 \ge 0$, p, q such that $p+s_1\ge 2$, $q+s_2\ge 2$

  $$\begin{aligned} \begin{aligned} a(x)\vert \xi \vert ^{p+s_1-2}+b(x)\vert \xi \vert ^{q+s_2-2}&\le \mathcal {F}^{(s_1,s_2)}_{\epsilon }(x,\xi ) \\&\le C\left( 1+a(x)\vert \xi \vert ^{p+s_1-2}+b(x)\vert \xi \vert ^{q+s_2-2}\right) \end{aligned} \end{aligned}$$

  with a constant C independent of x and $\xi$.

• 3. For the nonnegative coefficients $a, b \in L^\infty (\Omega )$, $\epsilon \in (0,1)$, $\xi \in \mathbb {R}^N$, and the parameters $s_1,s_2 \ge 0$

  $$\begin{aligned} \begin{aligned} \mathcal {F}^{(s_1, s_2)}_{\epsilon }(x,\xi )\beta _{\epsilon }(\xi )&\le {\left\{ \begin{array}{ll} a(x)(2\epsilon ^2)^{p+s_1}+b(x)(2\epsilon ^2)^{q+s_2} &{} \text {if } \vert \xi \vert \le \epsilon , \\ 2 \mathcal {F}^{(s_1, s_2)}_\epsilon (x,\xi )\vert \xi \vert ^2 &{} \hbox { if}\ \vert \xi \vert >\epsilon , \end{array}\right. } \\&\le C+ 2\mathcal {F}_\epsilon ^{(s_1, s_2)}(x,\xi )\vert \xi \vert ^2. \end{aligned} \end{aligned}$$

  (3.2)

  In particular,

  $$\begin{aligned} \begin{aligned} \mathcal {F}^{(s_1, s_2)}_0(x,\xi )\vert \xi \vert ^2\,dx&\equiv a(x)\vert \xi \vert ^{p+s_1}+b(x)\vert \xi \vert ^{q+s_2} \\&\le \mathcal {F}^{(s_1, s_2)}_{\epsilon }(x,\xi )\beta _{\epsilon }(\xi ) \le C+2\mathcal {F}^{(s_1, s_2)}_\epsilon (x,\xi )\vert \xi \vert ^2. \end{aligned} \end{aligned}$$

  (3.3)

• 4. For every $\zeta >0$, $\xi \in \mathbb {R}^N$, and $\mu \in (0,\zeta )$

  $$\begin{aligned} \begin{aligned} \vert \xi \vert ^{\zeta }\vert \ln \vert \xi \vert \vert&={\left\{ \begin{array}{ll} \left( \vert \xi \vert ^{\zeta +\mu }\right) \left( \vert \xi \vert ^{-\mu }\vert \ln \vert \xi \vert \vert \right) &{} \text {if } \vert \xi \vert \ge 1, \\ \left( \vert \xi \vert ^{\zeta -\mu }\right) \left( \vert \xi \vert ^{\mu }\vert \ln \vert \xi \vert \vert \right) &{} \text {if }\vert \xi \vert < 1, \end{array}\right. } \\&\le C(\mu ,\zeta )\left( 1+\vert \xi \vert ^{\zeta +\mu }\right) . \end{aligned} \end{aligned}$$

  (3.4)

Proposition 3.1

(Strict monotonicity) Let $\epsilon \in [0,1)$. There exists a constant $C=C(p^\pm )$ such that for all $\xi ,\zeta \in \mathbb {R}^N {\setminus } \{0\}$ and every $z\in Q_T$

$$\begin{aligned} \begin{aligned} \mathcal {S}_p(\xi ,\eta )&:=\big (\gamma ^{(p)}_\epsilon (\xi )\xi - \gamma ^{(p)}_\epsilon (\eta )\eta ,\xi -\eta \big ) \\&\ge C{\left\{ \begin{array}{ll} \vert \xi -\zeta \vert ^{p} &{} \text {if }p\ge 2, \\ \vert \xi -\eta \vert ^2\left( \epsilon ^2 +\vert \xi \vert ^2+\vert \eta \vert ^2\right) ^{\frac{p-2}{2}} &{} \hbox { if}\ p\in (1,2). \end{array}\right. } \end{aligned} \end{aligned}$$

(3.5)

Proof

The second line of (3.5) with $\epsilon \not =0$ is proven in [8, Lemma 6.1], in the case $\epsilon =0$ the proof can be found in [19, Lemma 17.3]. For this reason we only have to consider the case $p\ge 2$. For every $\xi ,\eta \in \mathbb {R}^N$

$$\begin{aligned} \begin{aligned} \mathcal {S}_p(\xi ,\eta )= \gamma ^{(p)}_\epsilon (\xi )\vert \xi \vert ^2 + \gamma ^{(p)}_\epsilon (\eta )\vert \eta \vert ^2- \left( \gamma ^{(p)}_\epsilon (\xi )+\gamma ^{(p)}_\epsilon (\eta )\right) (\xi ,\eta ). \end{aligned} \end{aligned}$$

Expressing the inner product $(\xi ,\eta )$ from the relation

$$\begin{aligned} \vert \xi -\eta \vert ^2=(\xi -\eta ,\xi -\eta )=\vert \xi \vert ^2+\vert \eta \vert ^2-2(\xi ,\eta ) \end{aligned}$$

we transform the previous equality to the form

$$\begin{aligned} \begin{aligned} \mathcal {S}_p(\xi ,\eta )&= \frac{1}{2}\left( \gamma ^{(p)}_\epsilon (\xi ) +\gamma ^{(p)}_\epsilon (\eta )\right) \vert \xi -\eta \vert ^2 + \frac{1}{2}\left( \gamma ^{(p)}_\epsilon (\xi )-\gamma ^{(p)}_\epsilon (\eta )\right) (\vert \xi \vert ^2-\vert \eta ^2\vert ). \end{aligned} \end{aligned}$$

For $p\ge 2$ the function $(\epsilon ^2+\vert \xi \vert ^2)^{\frac{p-2}{2}}$ is monotone increasing as a function of $\vert \xi \vert$, therefore the second term is nonnegative. Dropping it, we estimate $\mathcal {S}_p(\xi ,\eta )$ from below in the following way:

$$\begin{aligned} \begin{aligned} \mathcal {S}_p(\xi ,\eta )&\ge \frac{1}{2}\left( \gamma ^{(p)}_\epsilon (\xi ) +\gamma ^{(p)}_\epsilon (\eta )\right) \vert \xi -\eta \vert ^2 \\&\ge \frac{1}{2}\left( \vert \xi \vert ^{p-2}+\vert \eta \vert ^{p-2}\right) \vert \xi -\eta \vert ^2\ge \frac{1}{2C_p}(\vert \xi \vert +\vert \eta \vert )^{p-2}\vert \xi -\eta \vert ^2, \end{aligned} \end{aligned}$$

with the constant $C_p \ge 1$ from the inequality $(a+b)^{p-2}\le C_p(a^{p-2}+b^{p-2})$, $p \ge 2$, $a,b\ge 0$. Finally,

$$\begin{aligned} \begin{aligned} \vert \xi -\eta \vert ^{p}&=\left( \vert \xi -\eta \vert ^2\right) ^{\frac{p-2}{2}}\vert \xi -\eta \vert ^2 = \left( (\xi -\eta ,\xi -\eta )\right) ^{\frac{p-2}{2}}\vert \xi -\eta \vert ^2 \\&=(\vert \xi \vert ^2 -2(\xi ,\eta )+\vert \eta \vert ^2)^{\frac{p-2}{2}}\vert \xi -\eta \vert ^2 \le (\vert \xi \vert ^2+2\vert \xi \vert \vert \eta \vert +\vert \eta \vert ^2)^{\frac{p-2}{2}}\vert \xi -\eta \vert ^2 \\&=(\vert \xi \vert +\vert \eta \vert )^{p-2}\vert \xi -\eta \vert ^2 \le 2C_p\mathcal {S}_p(\xi ,\eta ). \end{aligned} \end{aligned}$$

$\square$

Now we consider the functions defined on the cylinder $Q_T$ and depending on the variables $z=(x,t)\in \Omega \times (0,T)$. Let us denote

$$\begin{aligned} \begin{aligned}&\mathcal {N}(\nabla w)=\int _{Q_T}(a(z)\vert \nabla w\vert ^{p(z)}+b(z)\vert \nabla w\vert ^{q(z)})\,dz, \\&\mathcal {G}_{\epsilon }(\nabla u,\nabla v)= \int _{Q_T} \left( \mathcal {F}_{\epsilon }^{(0,0)}(z,\nabla u)\nabla u-\mathcal {F}_{\epsilon }^{(0,0)}(z,\nabla v)\nabla v\right) \cdot \nabla (u-v)\,dz. \end{aligned} \end{aligned}$$

(3.6)

Proposition 3.2

For every $u,v\in \mathbb {W}_{\overline{s}(\cdot )}(Q_T)$ and any $\epsilon \in [0,1)$

$$\begin{aligned} \mathcal {N}(\nabla u-\nabla v)\le C\left( \mathcal {G}_\epsilon ^{\frac{\overline{s}^+}{2}}(\nabla u,\nabla v)+\mathcal {G}_\epsilon ^{\frac{\underline{s}^-}{2}}(\nabla u,\nabla v)+\mathcal {G}_\epsilon (\nabla u,\nabla v)\right) \end{aligned}$$

(3.7)

with a constant C depending on $p^\pm$, $q^\pm$, $a^+$, $b^+$, $\Vert \nabla u\Vert _{\overline{s}(\cdot ),Q_T}$, $\Vert \nabla v\Vert _{\overline{s}(\cdot ),Q_T}$.

Proof

Because of (3.5) it is sufficient to consider in detail the terms with the coefficient a(z), the terms with the coefficient b(z) are considered likewise. Let us split the domain $Q_T$ into two parts,

$$\begin{aligned} \begin{aligned}&Q_+=Q_T\cap \{p\ge 2\}, \quad Q_-=Q_T\cap \{p< 2\}. \end{aligned} \end{aligned}$$

Given $\epsilon \in [0,1)$, we denote

$$\begin{aligned} \begin{aligned}&\mathcal {R}(\nabla u,\nabla v)=(\epsilon +\vert \nabla u\vert ^2+\vert \nabla v\vert ^2)^{\frac{2-p(z)}{2}}, \\&\left( \mathcal {R}^{\frac{p}{2}}(\nabla u,\nabla v)\right) ^{\frac{2}{2-p}}\le C(1+\vert \nabla u\vert ^p+\vert \nabla v\vert ^p)\quad \hbox { on}\ Q_-. \end{aligned} \end{aligned}$$

By the generalized Hölder inequality (2.3)

$$\begin{aligned} \begin{aligned} \int _{Q_-}&a(z)\vert \nabla (u-v)\vert ^{p(z)}\,dz \\&=\int _{Q_-} a^{1-\frac{p}{2}} \mathcal {R}^{\frac{p}{2}}(\nabla u,\nabla v)\left( a\mathcal {R}^{-1}(\nabla u,\nabla v)\vert \nabla (u-v)\vert ^2 \right) ^{\frac{p}{2}}\,dz \\&\le 2(a^+)^{1-\frac{p}{2}}\Vert \mathcal {R}^{\frac{p}{2}}(\nabla u,\nabla v)\Vert _{\frac{2}{2-p(\cdot )},Q_-} \left\| a^{\frac{p}{2}}\mathcal {R}^{-\frac{p}{2}}(\nabla u,\nabla v)\vert \nabla (u-v)\vert ^{p}\right\| _{\frac{2}{p(\cdot )},Q_-}, \end{aligned} \end{aligned}$$

where

$$\begin{aligned} \begin{aligned}&\Vert \mathcal {R}^{\frac{p}{2}}(\nabla u,\nabla v)\Vert _{\frac{2}{2-p(\cdot )},Q_-}\le C(p^\pm , \Vert \nabla u\Vert _{p(\cdot ),Q_T},\Vert \nabla v\Vert _{p(\cdot ),Q_T}), \\&\left\| a^{\frac{p}{2}}\mathcal {R}^{-\frac{p}{2}}(\nabla u,\nabla v)\vert \nabla (u-v)\vert ^{p}\right\| _{\frac{2}{p(\cdot )},Q_-} \\&\quad \le \max \left\{ \left( \int _{Q_-}\frac{a}{\mathcal {R}(\nabla u, \nabla v)}\vert \nabla (u-v)\vert ^2\,dz\right) ^{\frac{p^+}{2}}, \left( \int _{Q_-}\ldots \right) ^{\frac{p^-}{2}}\right\} . \end{aligned} \end{aligned}$$

Gathering these inequalities and using (3.5) we obtain:

$$\begin{aligned} \begin{aligned} \int _{Q_-}a\vert \nabla (u-v)\vert ^p\,dz&\le C\left( \left( \int _{Q_-}a\mathcal {S}_{p(z)}(\nabla u,\nabla v)\,dz\right) ^{\frac{p^+}{2}} + \left( \int _{Q_-} \ldots \right) ^{\frac{p^-}{2}}\right) \\&\le C\left( \mathcal {G}_{\epsilon }^{\frac{\overline{s}^+}{2}}(\nabla u,\nabla v) + \mathcal {G}_{\epsilon }^{\frac{\underline{s}^-}{2}}(\nabla u,\nabla v)\right) . \end{aligned} \end{aligned}$$

The estimate on the integral of $a\vert \nabla (u-v)\vert ^p$ over $Q_+$ is straightforward and follows directly from the first line in (3.5):

$$\begin{aligned} \int _{Q_+}a\vert \nabla (u-v)\vert ^{p}\,dz\le C \mathcal {G}_{\epsilon }(\nabla u,\nabla v). \end{aligned}$$

The proof is concluded by gathering these inequalities with the corresponding inequalities for the terms with the coefficient b(z) and the exponent q(z).$\square$

Let

$$\begin{aligned} \mathcal {W}(Q_T)=\left\{ u:Q_T\mapsto \mathbb {R}: u\in L^2(Q_T), \,\mathcal {N}(\nabla u)<\infty \right\} . \end{aligned}$$

Lemma 3.1

There is the continuous embedding $\mathcal {W}(Q_T)\subset \mathbb {W}_{\underline{s}(\cdot )}(Q_T)$.

Proof

We have to show that for every $u\in \mathcal {W}(Q_T)$ the norm $\Vert u\Vert _{\mathbb {W}_{\underline{s}(\cdot )}(Q_T)}$ is bounded and

$$\begin{aligned} \Vert u\Vert _{\mathbb {W}_{\underline{s}(\cdot )}(Q_T)}=\Vert u\Vert _{2,Q_T}+\Vert \nabla u\Vert _{\underline{s}(\cdot ),Q_T}\rightarrow 0\quad \hbox { as}\ \Vert u\Vert ^2_{2,Q_T}+\mathcal {N}(\nabla u)\rightarrow 0. \end{aligned}$$

Let $u\in \mathcal {W}(Q_T)$. The following inequality holds:

$$\begin{aligned} \begin{aligned} \alpha \int _{Q_T} \vert \nabla u\vert ^{\underline{s}(z)}\,dz&\le \int _{Q_T}(a(z)+b(z))\vert \nabla u\vert ^{\underline{s}(z)}\,dz \\&\equiv \int _{Q_T}a^{1-\frac{\underline{s}}{p}}\left( a\vert \nabla u\vert ^{p}\right) ^{\frac{\underline{s}}{p}}\,dz + \int _{Q_T}b^{1-\frac{\underline{s}}{q}}\left( b\vert \nabla u\vert ^{q}\right) ^{\frac{\underline{s}}{q}}\,dz \\&\le A^+\int _{Q_T}\left( a\vert \nabla u\vert ^{p}\right) ^{\frac{\underline{s}}{p}}\,dz + B^+ \int _{Q_T}\left( b\vert \nabla u\vert ^{q}\right) ^{\frac{\underline{s}}{q}}\,dz \equiv A^+\mathcal {J}_a+B^+\mathcal {J}_b \end{aligned} \end{aligned}$$

with the constants $A^+=\sup _{Q_T}a^{1-\frac{\underline{s}(z)}{p(z)}}(z)$, $B^+=\sup _{Q_T}b^{1-\frac{\underline{s}(z)}{q(z)}}(z)$. The estimates on $\mathcal {J}_a$ and $\mathcal {J}_b$ are similar, for this reason we provide the details only for the first one. By the generalized Hölder inequality

$$\begin{aligned} \begin{aligned} \mathcal {J}_a&\le 2C_a\Vert (a\vert \nabla u\vert ^{p})^{\frac{\underline{s}}{p}} \Vert _{\frac{p(\cdot )}{\underline{s}(\cdot )},Q_T} \\&\le 2C_a \max \left\{ \left( \int _{Q_T}a\vert \nabla u\vert ^{p}\,dz\right) ^{\frac{\underline{s}^-}{p^+}}, \left( \int _{Q_T}a\vert \nabla u\vert ^{p}\,dz\right) ^{\frac{\overline{s}^+}{p^-}}\right\} \\&\le 2C_a\left( \mathcal {N}^{\frac{\underline{s}^-}{p^+}}(\nabla u)+\mathcal {N}^{\frac{\overline{s}^+}{p^-}}(\nabla u)\right) \end{aligned} \end{aligned}$$

with the constant $C_a=\Vert 1\Vert _{({p}(\cdot )/\underline{s}(\cdot ))',Q_T}$. Gathering this estimate with the similar estimate for $\mathcal {J}_b$ with the constant $C_b=\Vert 1\Vert _{(q(\cdot )/\underline{s}(\cdot ))',Q_T}$ we obtain

$$\begin{aligned} \alpha \int _{Q_T}\vert \nabla u\vert ^{\underline{s}(z)}\,dz\le 4(C_a+C_b)(A^++B^+)\left( \mathcal {N}^{\frac{\underline{s}^-}{\overline{s}^+}}(\nabla u)+\mathcal {N}^{\frac{\overline{s}^+}{\underline{s}^-}}(\nabla u)\right) . \end{aligned}$$

$\square$

The next assertion is an immediate byproduct of Lemma 3.1.

Lemma 3.2

Let $\epsilon \in [0,1)$. If $u,v\in \mathbb {W}_{\overline{s}(\cdot )}(Q_T)$, then

$$\begin{aligned} \int _{Q_T}\vert \nabla (u-v)\vert ^{\underline{s}(z)}\,dz\rightarrow 0 \quad \text {when}\quad \mathcal {G}_{\epsilon }(\nabla u,\nabla v)\rightarrow 0. \end{aligned}$$

4 Interpolation inequalities

The bulk of this section is devoted to deriving interpolation inequalities for functions of the variable $x\in \Omega$. By agreement, we will write $\displaystyle \vert u_{xx}\vert ^2=\sum _{i,j=1}^{N}\left| D^2_{x_ix_j}u\right| ^2.$

4.1 Global inequalities

Lemma 4.1

Let $\partial \Omega \in C^2$, $u\in C^1(\overline{\Omega })\cap H^{2}_0(\Omega )$. Assume that p, q, a, b are functions of $x\in \Omega$ and satisfy conditions (2.15) and (2.16) in $\overline{\Omega }$. If

$$\begin{aligned} \int _\Omega \mathcal {F}^{(0,0)}_{\epsilon }(x,\nabla u)\vert u_{xx}\vert ^2\,dx<\infty ,\qquad \int _{\Omega }u^2\,dx= M_0, \end{aligned}$$

then for every pair of nonnegative functions $r_1, r_2: \overline{\Omega } \rightarrow \mathbb {R}$, $r_1, r_2 \in C^{0,1}(\overline{\Omega })$ with the Lipschitz constants $L_{r_1}$, $L_{r_2}$, and arbitrary parameters $\delta , \nu \in (0,1)$

$$\begin{aligned}&\int _{\Omega } \mathcal {F}_{\epsilon }^{(r_1, r_2)}(x,\nabla u)\vert \nabla u\vert ^2\,dx \le \delta \int _{\Omega }\mathcal {F}^{(0,0)}_{\epsilon }(x,\nabla u)\vert u_{xx}\vert ^{2}\,dx \nonumber \\&\quad \le C_0 + C_1 \int _{\Omega }u^2\,dx \nonumber \\&\qquad + C_2 \int _{\Omega } \left( \vert u\vert ^{\frac{p(x)+r_1(x)}{1-\nu }}+ \vert u\vert ^{\frac{q(x)+r_2(x)}{1-\nu }}\right) \,dx \\&\qquad + C_3 \int _\Omega \vert u\vert \ \left( \beta _{\epsilon }^{\frac{p(x)+r_1(x)-1}{2}}(\nabla u) + \beta _{\epsilon }^{\frac{q(x)+r_2(x)-1}{2}}(\nabla u)\right) \,dx\nonumber \\&\qquad + C_4\int _\Omega \vert u\vert ^2 \ \left( \beta _{\epsilon }^{\frac{p(x)+2r_1(x)-2}{2}}(\nabla u)+ \beta _{\epsilon }^{\frac{q(x)+2r_2(x)-2}{2}}(\nabla u)\right) \,dx\nonumber \end{aligned}$$

(4.1)

with independent of u constants $C_i=C_i(\partial \Omega ,\delta ,\alpha , \nu , s^\pm ,N,r_i, L_{r_i}, M_0, a^+, b^+)$.

Proof

Let $\textbf{n}$ denote the outer normal to $\partial \Omega$. By employing Green’s formula

$$\begin{aligned} \begin{aligned} \int _{\Omega }a(x) \beta _{\epsilon }^{\frac{p(x)+r_1(x)-2}{2}}&(\nabla u) \vert \nabla u\vert ^2\,dx =\int _{\Omega } a(x)\ \beta _{\epsilon }^{\frac{p(x)+r_1(x)-2}{2}}(\nabla u) \ \nabla u\cdot \nabla u\,dx \\&= \int _{\partial \Omega } a(x) u \beta _{\epsilon }^{\frac{p(x)+r_1(x)-2}{2}}(\nabla u) \ \nabla u \cdot \textbf{n} \,dS \\&\quad -\int _{\Omega }u\ {\text {div}}\left( a(x)\beta _{\epsilon }^{\frac{p(x)+r_1(x)-2}{2}}(\nabla u) \nabla u\right) \,dx\\&= -\int _{\Omega }u\ {\text {div}}\left( a(x)\beta _{\epsilon }^{\frac{p(x)+r_1(x)-2}{2}}(\nabla u) \nabla u\right) \,dx := -J. \end{aligned} \end{aligned}$$

A straightforward computation leads to the following representation:

$$\begin{aligned} \begin{aligned} J&= \int _\Omega a(x) u\ \beta _{\epsilon }^{\frac{p(x)+r_1(x)-2}{2}}(\nabla u)\ \Delta u\,dx + \int _{\Omega } u\ \beta _{\epsilon }^{\frac{p(x)+r_1(x)-2}{2}} (\nabla u) \ \nabla u \cdot \nabla a \,dx \\&\quad + \int _\Omega (p(x)+r_1(x)-2)\ a(x)u\ \beta _{\epsilon }^{\frac{p(x)+r_1(x)-2}{2}-1}(\nabla u) \sum _{i=1}^n \left( u_{x_i}\sum _{j=1}^{n}u_{x_j}u_{x_ix_j}\right) \,dx \\&\quad + \int _{\Omega }a(x)u\ \beta _{\epsilon }^{\frac{p(x)+r_1(x)-2}{2}} (\nabla u) \ln (\beta _{\epsilon }(\nabla u)) \nabla u \cdot \nabla (p+ r_1) \,dx \\&:= J_1 + J_2 + J_3 + J_4. \end{aligned} \end{aligned}$$

The terms $J_i$ are estimated as follows. Using Young’s inequality, for any $\lambda _1 \in (0,1)$ we have

$$\begin{aligned} \begin{aligned}&\vert J_1\vert \le \int _\Omega \left( \vert u\vert \ (a(x))^{\frac{1}{2}} \beta _{\epsilon }^{\frac{p(x)+r_1(x)-2}{2}- \frac{p(x)-2}{4}}(\nabla u) \right) \\&\qquad \times \left( (a(x))^{\frac{1}{2}} \ \beta _{\epsilon }^{\frac{p(x)-2}{4}}(\nabla u)\ \vert \Delta u\vert \right) \,dx\\&\quad \le \lambda _1 \int _\Omega a(x) \ \beta _{\epsilon }^{\frac{p(x)-2}{2}}(\nabla u)\ \vert \Delta u\vert ^2\,dx + C' \int _\Omega \vert u\vert ^2 \ a(x)\ \beta _{\epsilon }^{\frac{p(x)+2r_1(x)-2}{2}}(\nabla u)\,dx\\&\quad \le \lambda _1 \int _\Omega a(x) \ \beta _{\epsilon }^{\frac{p(x)-2}{2}}(\nabla u)\ \vert u_{xx}\vert ^2\,dx + C \int _\Omega \vert u\vert ^2 \beta _{\epsilon }^{\frac{p(x)+2r_1(x)-2}{2}}(\nabla u)\,dx \end{aligned}\nonumber \\ \end{aligned}$$

(4.2)

with $C=C(\lambda _1, a^+)$ and

$$\begin{aligned} \begin{aligned} \vert J_2\vert&\le \int _\Omega \vert u\vert \ \vert \nabla a\vert \ \beta _{\epsilon }^{\frac{p(x)+r_1(x)-2}{2}}(\nabla u)\ \vert \nabla u\vert \,dx \\&\le L_{a} \int _\Omega \vert u\vert \ \beta _{\epsilon }^{\frac{p(x)+r_1(x)-1}{2}}(\nabla u)\,dx. \end{aligned}\nonumber \\ \end{aligned}$$

(4.3)

By Young’s inequality, for any $\lambda _2 \in (0,1)$ we obtain

$$\begin{aligned} \begin{aligned} \vert J_3\vert&\le \lambda _2 \int _\Omega a(x) \ \beta _{\epsilon }^{\frac{p(x)-2}{2}}(\nabla u)\ \vert u_{xx}\vert ^2\,dx + C \int _\Omega \vert u\vert ^2 \beta _{\epsilon }^{\frac{p(x)+2r_1(x)-2}{2}}(\nabla u)\,dx \end{aligned} \end{aligned}$$

(4.4)

with $C= C(p^\pm ,r_1, \lambda _2, a^+)$, $a^+=\max _{\overline{\Omega }}a(x)$, and

$$\begin{aligned} \begin{aligned} \vert J_4\vert&\le (L_{p} + L_{r_1}) \int _{\Omega }\vert u\vert \ a(x) \ \beta _{\epsilon }^{\frac{p(x)+r_1(x)-1}{2}}(\nabla u) \vert \ln (\beta _{\epsilon }(\nabla u))\vert \,dx. \end{aligned} \end{aligned}$$

(4.5)

Applying (3.4) with $\vert \xi \vert =\beta _\epsilon (\nabla u)$, $\zeta =\frac{p(x)+r_1(x)-1}{2}$ and $\mu =\nu /2\in (0,1/2)$, we estimate

$$\begin{aligned} \begin{aligned} \beta _{\epsilon }^{\frac{p(x)+r_1(x)-1}{2}}(\nabla u)\vert \ln ( \beta _{\epsilon }(\nabla u))\vert&\le C+C(\nu )\beta ^{\frac{p(x)+r_1(x)-1+\nu }{2}}_{\epsilon }(\nabla u). \end{aligned} \end{aligned}$$

(4.6)

By virtue of (3.3)

$$\begin{aligned} \vert \nabla u\vert ^{{\frac{p(x)+r_1(x)}{2}}} \le \beta _{\epsilon }^{\frac{p(x)+r_1(x)}{2}}(\nabla u) \le C+2 \beta _{\epsilon }^{\frac{p(x)+r_1(x)-2}{2}}(\nabla u)\vert \nabla u\vert ^2. \end{aligned}$$

(4.7)

By using (4.6) and (4.7) in (4.5), and applying then Young’s inequality with $\lambda _3 \in (0,1)$, we obtain

$$\begin{aligned} \begin{aligned}&\vert J_4\vert \le C \int _{\Omega }a(x) \vert u\vert \,dx + C\int _{\Omega } \left( \vert u\vert \ (a(x))^{\frac{1-\nu }{p(x)+r_1(x)}}\right) \\&\qquad \times \left( (a(x))^{\frac{p(x)+r_1(x)-1+\nu }{p(x)+r_1(x)}} \beta ^{\frac{p(x)+r_1(x)-1+\nu }{2}}_{\epsilon }(\nabla u) \right) \,dx \\&\quad \le C (1 + M_0^{\frac{1}{2}}) + \frac{\lambda _3}{2} \int _{\Omega } a(x) \beta _{\epsilon }^{\frac{p(x)+r_1(x)}{2}}(\nabla u)\,dx+ C_{\lambda _3}' \int _{\Omega }a(x) \vert u\vert ^{\frac{p(x)+r_1(x)}{1-\nu }}\,dx\\&\quad \le C_1 + C_2 \int _{\Omega } \vert u\vert ^{\frac{p(x)+r_1(x)}{1-\nu }}\,dx + \lambda _3 \int _{\Omega } a(x) \beta _{\epsilon }^{\frac{p(x)+r_1(x)-2}{2}}(\nabla u) \vert \nabla u\vert ^2\,dx, \end{aligned}\nonumber \\ \end{aligned}$$

(4.8)

where $C_1= C_1(a^+, p^\pm ,r_1, \lambda _3, M_0)$ and $C_2= C_2(\lambda _3, p^\pm ,\nu , a^+)$. By combining (4.2)–(4.4) and (4.8), we get the following estimate:

$$\begin{aligned} \begin{aligned}&\int _{\Omega }a(x) \beta _{\epsilon }^{\frac{p(x)+r_1(x)-2}{2}} (\nabla u) \vert \nabla u\vert ^2\,dx \le C_1 + C_2 \int _{\Omega } \vert u\vert ^{\frac{p(x)+r_1(x)}{1-\nu }}\,dx \\&\quad + C_3 \int _\Omega \vert u\vert \ \beta _{\epsilon }^{\frac{p(x)+r_1(x)-1}{2}}(\nabla u)\,dx + C_4 \int _\Omega \vert u\vert ^2 \ \beta _{\epsilon }^{\frac{p(x)+2r_1(x)-2}{2}}(\nabla u)\,dx \\&\quad + (\lambda _1 + \lambda _2) \int _\Omega a(x) \ \beta _{\epsilon }^{\frac{p(x)-2}{2}}(\nabla u)\ \vert u_{xx}\vert ^2\,dx. \end{aligned} \end{aligned}$$

We get the claim by repeating the same arguments for the term with the exponent q on the left-hand side of (4.1) and gathering the results.$\square$

Let us rewrite inequality (4.1) in the form

$$\begin{aligned} \begin{aligned} \int _\Omega \mathcal {F}_\epsilon ^{(r_1, r_2)}(x,\nabla u)\vert \nabla u\vert ^2\,dx&\le \delta \int _{\Omega }\mathcal {F}^{(0,0)}_{\epsilon }(x,\nabla u)\vert u_{xx}\vert ^{2}\,dx \\&\quad + C_0 + C_1 \int _{\Omega }u^2\,dx + C_2 \mathcal {Q}_1^{(r_1, r_2)} \\&\quad + C_3 \mathcal {Q}_2^{(r_1, r_2)} + C_4 \mathcal {Q}_3^{(r_1, r_2)}, \end{aligned} \end{aligned}$$

(4.9)

where

$$\begin{aligned} \begin{aligned}&\mathcal {Q}_1^{(r_1, r_2)}:= \int _{\Omega } \left( \vert u\vert ^{\frac{p(x)+r_1(x)}{1-\nu }}+ \vert u\vert ^{\frac{q(x)+r_2(x)}{1-\nu }}\right) \,dx \quad \ \text {for some} \ \nu \in (0,1), \\&\mathcal {Q}_2^{(r_1, r_2)}:= \int _\Omega \vert u\vert \ \left( \beta _{\epsilon }^{\frac{p(x)+r_1(x)-1}{2}}(\nabla u) + \beta _{\epsilon }^{\frac{q(x)+r_2(x)-1}{2}}(\nabla u)\right) \,dx, \\&\mathcal {Q}_3^{(r_1, r_2)}:= \int _\Omega \vert u\vert ^2 \ \left( \beta _{\epsilon }^{\frac{p(x)+2r_1(x)-2}{2}}(\nabla u)+ \beta _{\epsilon }^{\frac{q(x)+2r_2(x)-2}{2}}(\nabla u)\right) \,dx. \end{aligned} \end{aligned}$$

The integrals $\mathcal {Q}_i^{(r_1, r_2)}$ will be estimated separately and under a special choice of the functions $r_1$, $r_2$. We assume that the exponents p, q are subject to the balance condition (2.19), which for the independent of t functions p, q reads

$$\begin{aligned} \max _{x \in \overline{\Omega }} \vert p(x) -q(x)\vert < r_*. \end{aligned}$$

(4.10)

Given p, q, we introduce the Lipschitz-continuous functions

$$\begin{aligned} r_1(x):= r^\sharp + \underline{s}(x) - p(x), \quad r_2(x):= r^\sharp + \underline{s}(x) - q(x). \end{aligned}$$

(4.11)

Let $\{\Omega _i\}_{i=1}^{K}$ be a finite cover of $\Omega$, $\Omega _i\subset \Omega$, $\partial \Omega _i\in C^2$, $\Omega \subseteq \bigcup _{i=1}^K\Omega _i$. Set

$$\begin{aligned} \begin{aligned} \underline{s}_i^+=\max _{\overline{\Omega }_i}\underline{s}(x), \quad \underline{s}_i^-=\min _{\overline{\Omega }_i}\underline{s}(x). \end{aligned} \end{aligned}$$

Fix an arbitrary number $\varsigma \in (0, r^\sharp )$. By continuity of the map $x \mapsto \underline{s}(x)$, the cover $\{\Omega _i\}_{i=1}^{K}$ can be chosen in such a way that for every $i=1,2,\ldots ,K$

$$\begin{aligned} 0\le \underline{s}_i^+-\underline{s}_i^- < \varsigma . \end{aligned}$$

(4.12)

The number of elements in the cover $\{\Omega _i\}$ depends on $\varsigma$ and the modules of continuity $\omega _p$, $\omega _q$ of functions p and q in $\Omega$: $K=K(\omega _p,\omega _q,\varsigma )$. Under the choice (4.11) we have $p(x)+r_1(x)=q(x)+r_2(x)=\underline{s}(x)+r^\sharp$, where the value of the constant $r^\sharp$ is prompted by the classical embedding inequalities used in the proofs of the forthcoming estimates on the integrals $\mathcal {Q}^{(r_1,r_2)}_{i}$.

Lemma 4.2

Let the conditions of Lemma 4.1 be fulfilled. Then for every $\varsigma \in \left( 0,r^\sharp \right)$ with $r^\sharp \equiv \dfrac{4}{N+2}$, every $\sigma >0$, and $r_1(x)$, $r_2(x)$ defined in (4.11)

$$\begin{aligned} \mathcal {Q}_{1}^{(r_1-\varsigma , r_2-\varsigma )}\le C+\sigma \int _{\Omega }\mathcal {F}_{\epsilon }^{(r_1-\varsigma , r_2-\varsigma )}(x,\nabla u)\vert \nabla u\vert ^2\,dx \end{aligned}$$

with a constant C depending on $\sigma$, N, $M_0$, $r_1$, $r_2$, $p^\pm$, $q^\pm$, $\varsigma$, and the modules of continuity $\omega _p$, $\omega _q$ of the exponents p, q in $\Omega$.

Proof

For a fixed $\varsigma \in (0, r^\sharp )$, it is sufficient to prove the required inequality for each of the sets $\Omega _i$ from the chosen cover of $\Omega$. Gathering these estimates for all $i=1,2,\ldots ,K$, we obtain the needed assertion. Let $\mathcal {Q}_{1,i}^{(r_1-\varsigma , r_2-\varsigma )}$ denote the integrals $\mathcal {Q}_{1}^{(r_1-\varsigma , r_2-\varsigma )}$ over the domains $\Omega _i$ instead of $\Omega$. By Young’s inequality, for every $i=1,2,\ldots ,K$

$$\begin{aligned} \mathcal {Q}_{1,i}^{(r_1-\varsigma , r_2-\varsigma )}= & {} \int _{\Omega _i}\left( \vert u\vert ^{\frac{p+r_1-\varsigma }{1-\nu }} + \vert u\vert ^{\frac{q+r_2- \varsigma }{1-\nu }}\right) \,dx = 2 \int _{\Omega _i} \vert u\vert ^{\frac{\underline{s}_i(x)+r^{\sharp }-\varsigma }{1-\nu }}\,dx\nonumber \\= & {} \left( \int _{\Omega _i \cap \{\vert u\vert \le 1\}} \ldots + \int _{\Omega _i \cap \{\vert u\vert > 1\}}\ldots \right) \le C + \int _{\Omega _i} \vert u\vert ^{\frac{\underline{s}^+_i+r^{\sharp }-\varsigma }{1-\nu }}\,dx\qquad \end{aligned}$$

(4.13)

with a finite constant C independent of u. To estimate the second term we apply the Gagliardo–Nirenberg inequality ([5, Lemma 1.29, Chapter 1]):

$$\begin{aligned} \Vert u\Vert _{\sigma ,\Omega _i}^{\sigma }\le C_1\Vert \nabla u\Vert _{\underline{s}^-_i,\Omega _i}^{\sigma \theta }\Vert u\Vert _{2,\Omega _i}^{\sigma (1-\theta )} +C_2\Vert u\Vert ^{\sigma }_{2,\Omega _i}\le C_1'\Vert \nabla u\Vert _{\underline{s}^-_i,\Omega _i}^{\sigma \theta }+C_2M_0^{\frac{\sigma }{2}}, \end{aligned}$$

(4.14)

with $C'_1=C_1M_0^{\frac{\sigma }{2}(1-\theta )}$ and

$$\begin{aligned} \sigma =\dfrac{\underline{s}^+_i+r^{\sharp }-\varsigma }{1-\nu }>\underline{s}^+_i+r^{\sharp }-\varsigma > \underline{s}^-_i,\quad \theta =\dfrac{\underline{s}^-_i}{\sigma }\in (0,1),\quad \dfrac{1}{\sigma }=\left( \frac{1}{\underline{s}^-_i} -\frac{1}{N}\right) \theta +\frac{1-\theta }{2}. \end{aligned}$$

Such a choice of the parameters $\sigma$, $\theta$ is possible, provided that

$$\begin{aligned} \nu =1-\dfrac{\underline{s}^+_i+r^{\sharp }-\varsigma }{\underline{s}^-_i}\dfrac{N}{N+2}>0\quad \Leftrightarrow \quad \underline{s}^+_i-\underline{s}^-_i+r^{\sharp }-\varsigma <\underline{s}^-_i\dfrac{2}{N}. \end{aligned}$$

The last inequality is true due to the assumption $\underline{s}^-_i>\frac{2N}{N+2}$, the choice of cover $\{\Omega _i\}$ with condition (4.12) and the inequality

$$\begin{aligned} \begin{aligned} \underline{s}_i^+-\underline{s}_i^- + r^{\sharp }-\varsigma< \frac{4}{N+2} =\dfrac{2}{N}\dfrac{2N}{N+2} <\frac{2}{N}\underline{s}^-_i. \end{aligned} \end{aligned}$$

By the definition of functions $r_1$, $r_2$, and by virtue of (3.2) and Lemma 3.1, we obtain

$$\begin{aligned} \alpha \int _{\Omega _i}\vert \nabla u\vert ^{\underline{s}(x)+r^{\sharp }-\varsigma } \,dx \le \int _{\Omega _i} \mathcal {F}_{\epsilon }^{(r_1-\varsigma , r_2-\varsigma )}(x,\nabla u)\beta _\epsilon (\nabla u)\,dx. \end{aligned}$$

(4.15)

Applying Young’s inequality and using (4.15) and (3.3), we conclude that for every $\lambda >0$

$$\begin{aligned} \begin{aligned} \mathcal {Q}_{1,i}^{(r_1-\varsigma , r_2-\varsigma )}\le C\left( 1+\Vert \nabla u\Vert ^{\underline{s}_i^-}_{\underline{s}_i^-,\Omega _i}\right)&\le C''+\lambda \alpha \int _{\Omega _i}\vert \nabla u\vert ^{\underline{s}(x)+r^{\sharp }-\varsigma } \,dx\\&\le C+\lambda \int _{\Omega _i} \mathcal {F}_{\epsilon }^{(r_1-\varsigma , r_2-\varsigma )}(x,\nabla u)\vert \nabla u\vert ^2\,dx. \end{aligned} \end{aligned}$$

$\square$

Lemma 4.3

Under the conditions of Lemma 4.2, for every $\varsigma \in (0,r_*)$

$$\begin{aligned} \begin{aligned} \mathcal {Q}_2^{(r_1-\varsigma , r_2-\varsigma )}&\le C +\lambda \int _{\Omega }\mathcal {F}_\epsilon ^{(r_1-\varsigma , r_2-\varsigma )}(x,\nabla u)\vert \nabla u\vert ^2\,dx \end{aligned} \end{aligned}$$

with an arbitrary $\lambda >0$ and a constant $C=C(\lambda , \omega _p,\omega _q,N,M_0,\varsigma ,\alpha , p^\pm , q^\pm )$.

Proof

As in the proof of Lemma 4.2, it is sufficient to consider the integrals $\mathcal {Q}_{2,i}^{(r_1-\varsigma , r_2-\varsigma )}$ over the elements of the cover $\{\Omega _i\}$ satisfying condition (4.12). For $\varsigma \in (0, r_*)$ we have $r^{\sharp }-\varsigma > r_*$. Set $b=\underline{s}_i^-+r_*$. By Young’s inequality

$$\begin{aligned} \begin{aligned} \mathcal {Q}_{2,i}^{(r_1-\varsigma , r_2-\varsigma )}&\le C M_0^{\frac{1}{2}} + \int _{\Omega _i} \vert u\vert \ \beta _{\epsilon }^{\frac{\underline{s}_i(x)+r^{\sharp }-\varsigma -1}{2}}(\nabla u) \,dx \\&\le C + C_\lambda \int _{\Omega _i}\vert u\vert ^{b\frac{N+2}{N}}\,dx+\lambda \int _{\Omega _i} \beta _\epsilon ^{\frac{\kappa }{2}}(\nabla u)\,dx \end{aligned} \end{aligned}$$

(4.16)

with an arbitrary $\lambda >0$ and the exponent

$$\begin{aligned} \kappa =(\underline{s}_i(x)+r^{\sharp }-\varsigma -1) \frac{b\frac{N+2}{N}}{b\frac{N+2}{N}-1}>0. \end{aligned}$$

The second term on the right-hand side of (4.16) is estimated by the Gagliardo-Nirenberg inequality

$$\begin{aligned} \int _{\Omega _i}\vert u\vert ^{b\frac{N+2}{N}}\,dx\le C\left( 1+\Vert \nabla u\Vert _{b,\Omega _i}^{\theta b \frac{N+2}{N}}\right) \quad \text {with}\quad \theta =\frac{\frac{1}{2}- \frac{N}{b(N+2)}}{\frac{N+2}{2N}-\frac{1}{b}} =\frac{N}{N+2}\in (0,1), \end{aligned}$$

whence

$$\begin{aligned} \int _{\Omega _i}\vert u\vert ^{\underline{s}_i^-\frac{N+2}{N}}\,dx\le C\left( 1+\int _{\Omega _i}\vert \nabla u\vert ^{\underline{s}_i^-+r_*}\,dx\right) ,\qquad C=C(\underline{s}_i^-,\overline{s}_i^+,N,M_0). \end{aligned}$$

To estimate the last term of (4.16) we claim that $\kappa <\underline{s}_i^-+r^\sharp -\varsigma$:

$$\begin{aligned} \begin{aligned} \underline{s}_i^++r^{\sharp }-\varsigma -1&<(\underline{s}_i^-+r^{\sharp }-\varsigma )\left( 1-\frac{N}{b(N+2)}\right) \\&\Leftrightarrow \quad \underline{s}_i^+- \underline{s}_i^- <1-\frac{N}{N+2}\frac{\underline{s}_i^-+r^{\sharp }-\varsigma }{\underline{s}_i^-+r_*}. \end{aligned}\nonumber \\ \end{aligned}$$

(4.17)

The last inequality is equivalent to

$$\begin{aligned} \begin{aligned} \underline{s}_i^+-\underline{s}_i^- +( r^\sharp -\varsigma -r_*)\frac{N}{b(N+2)}<\frac{2}{N+2}, \end{aligned} \end{aligned}$$

which is fulfilled due to (4.12) and the choice of $\varsigma$:

$$\begin{aligned} \begin{aligned}&\underline{s}_i^+-\underline{s}_i^- +(r^{\sharp }-\varsigma -r_*)\frac{N}{b(N+2)} \\&\quad< \frac{4}{N+2} - r^{\sharp } + \varsigma +(r^{\sharp }-\varsigma -r_*)\frac{N}{b(N+2)} < \frac{2}{N+2} \\&\quad \Leftrightarrow \quad (r^{\sharp }-\varsigma -r_*) \left( 1-\frac{N}{b(N+2)} \right) >0. \end{aligned}\nonumber \\ \end{aligned}$$

(4.18)

Gathering these estimates and applying once again the Young inequality we arrive at the estimate

$$\begin{aligned} \begin{aligned} \mathcal {Q}_{2,i}^{(r_1-\varsigma , r_2-\varsigma )}&\le C+\int _{\Omega }\vert \nabla u\vert ^{\underline{s}_i^-+r_*}\,dx+\lambda \int _{\Omega _i}\vert \nabla u\vert ^{\underline{s}_i^-+r^{\sharp }-\varsigma }\,dx \\&\le C+\delta \int _{\Omega _i}\vert \nabla u\vert ^{\underline{s}(x)+r^{\sharp }-\varsigma }\,dx \end{aligned} \end{aligned}$$

with any $\delta >0$ and $\varsigma \in (0, r_*)$. By collecting the estimates for all $i=1,2,\ldots ,K$ and using (4.15), we obtain the needed inequality. $\square$

Lemma 4.4

Let the conditions of Lemma 4.2 be fulfilled. Then, for every $\varsigma \in (0, r_*/2)$ and every $\mu >0$

$$\begin{aligned} \begin{aligned} \mathcal {Q}_{3}^{(r_1-\varsigma , r_2-\varsigma )}&\le C+\mu \int _{\Omega }\mathcal {F}_\epsilon ^{(r_1-\varsigma , r_2-\varsigma )}(x,\nabla u)\vert \nabla u\vert ^2\,dx \end{aligned} \end{aligned}$$

with a constant $C=C(\mu , \omega _p,\omega _q,N,M_0,K,\alpha ,\varsigma , p^\pm , q^\pm )$

Proof

Fix a number $\varsigma \in (0, r_*/2)$ and take a finite cover $\{\Omega _i\}$ of $\Omega$ such that, instead of (4.12),

$$\begin{aligned} \underline{s}_i^+-\underline{s}_i^-<\dfrac{\varsigma }{4}\quad \hbox { on each}\ \Omega _i. \end{aligned}$$

Set $h=\underline{s}_i^-+r^{\sharp }-(\varsigma + \sigma _{i, \varsigma })>0$, where $\sigma _{i, \varsigma }>0$ is a number to be defined. By Young’s inequality, for every $\lambda >0$

$$\begin{aligned} \mathcal {Q}_{3,i}^{(r_1-\varsigma , r_2-\varsigma )}\le C_\lambda \int _{\Omega _i}\vert u\vert ^{h\frac{N+2}{N}}\,dx+\lambda \left( \int _{\Omega _i}\beta _{\epsilon }^{\frac{\kappa _1}{2}}(\nabla u)\,dx + \int _{\Omega _i}\beta _{\epsilon }^{\frac{\kappa _2}{2}}(\nabla u)\,dx \right) \nonumber \\ \end{aligned}$$

(4.19)

with

$$\begin{aligned} \kappa _1=(p(x)+2(r_1-\varsigma -1))\frac{h\frac{N+2}{2N}}{h\frac{N+2}{2N}-1}, \qquad \kappa _2=(q(x)+2(r_2-\varsigma -1))\frac{h\frac{N+2}{2N}}{h\frac{N+2}{2N}-1}. \end{aligned}$$

The first term on the right-hand side of (4.19) is estimated by the Gagliardo-Nirenberg inequality:

$$\begin{aligned} \int _{\Omega _i}\vert u\vert ^{h\frac{N+2}{N}}\,dx\le C\left( 1+\Vert \nabla u\Vert _{h,\Omega _i}^{\theta h\frac{N+2}{N}}\right) ,\qquad \theta =\dfrac{\frac{1}{2} -\frac{N}{h(N+2)}}{\frac{N+2}{2N}-\frac{1}{h}}=\frac{N}{N+2}\in (0,1), \end{aligned}$$

with a constant C depending on $\underline{s}_i^+$, $\underline{s}_i^-$, N, $M_0$ and $\varsigma$. Thus, for every $\nu >0$

$$\begin{aligned} \int _{\Omega _i}\vert u\vert ^{h\frac{N+2}{N}}\,dx\le C\left( 1+\int _{\Omega _i}\vert \nabla u\vert ^{\underline{s}_i^-+r^{\sharp }-(\varsigma + \sigma _{i, \varsigma })}\,dx\right) \le C'+\nu \int _{\Omega _i}\vert \nabla u\vert ^{\underline{s}(x)+r^{\sharp }-\varsigma }\,dx. \end{aligned}$$

To estimate the second term we claim $0<\kappa _1<\underline{s}(x)+r^{\sharp }-\varsigma$ on $\Omega _i$. The first inequality is satisfied due to the assumption $\underline{s}^- > \frac{2N}{N+2}$, the choice $\varsigma \in (0, r_*/2)$, and (4.10):

$$\begin{aligned} \begin{aligned}&p(x) +2(\underline{s}(x)+r^\sharp -p(x)-\varsigma -1) \ge \underline{s}(x)-p(x)+\frac{8}{N+2}+\frac{2N}{N+2}-2\varsigma -2 \\&\;\;= {\left\{ \begin{array}{ll} \frac{4}{N+2}-2\varsigma &{} \hbox { if}\ p(x)\le q(x) \\ q(x)-p(x)+\frac{4}{N+2} -2\varsigma &{} \hbox { if}\ p(x)>q(x) \end{array}\right. } > {\left\{ \begin{array}{ll} 0 &{} \hbox { if}\ p(x)\le q(x) \\ \frac{4}{N+2}-2r_*&{} \text {otherwise} \end{array}\right. } =0. \end{aligned} \end{aligned}$$

Since $h>\dfrac{2N}{N+2}$ by definition, $\kappa _1>0$. To fulfill the second inequality for $\kappa _1$, we claim that the stronger condition holds:

$$\begin{aligned} p(x)+2(r_1-\varsigma -1) <(\underline{s}_i^-+r^{\sharp }-\varsigma ) \left( 1-\dfrac{2N}{h(N+2)}\right) . \end{aligned}$$

Due to the choice of h and $r_1$, and because of the inequalities $\underline{s}_i^-\le \underline{s}(x)\le p(x)$ on $\Omega _i$, this is true if

$$\begin{aligned} \begin{aligned} 2\underline{s}(x)-p(x)&+r^\sharp -\varsigma \le 2\underline{s}_i^+- \underline{s}_i^-+\frac{4}{N+2}-\varsigma <2+\underline{s}_i^- -\frac{2N}{N+2} -\frac{2N\sigma _{i,\varsigma }}{h(N+2)}. \end{aligned} \end{aligned}$$

The second inequality holds if $\sigma _{i,\varsigma }$ is chosen from the condition

$$\begin{aligned} 2(\underline{s}_i^+-\underline{s}_i^-)<\varsigma -\frac{2N\sigma _{i,\varsigma }}{(N+2)(\underline{s}_i^- +r^\sharp -\varsigma -\sigma _{i,\varsigma })}\quad \Leftarrow \quad 0<\sigma _{i,\varsigma }<\dfrac{(N+2)(2-\varsigma )\varsigma }{4N+(N+2)\varsigma }. \end{aligned}$$

The third term on the right-hand side of (4.19) is estimated in the same way. Gathering these estimates and applying the Young inequality we conclude that for every $\mu >0$ and $\varsigma \in (0,r_*/2)$

$$\begin{aligned} \begin{aligned} \mathcal {Q}_{3,i}^{(r_1-\varsigma , r_2-\varsigma )}&\le C + \mu \int _{\Omega _i}\mathcal {F}_\epsilon ^{(r_1-\varsigma , r_2-\varsigma )}(x,\nabla u)\vert \nabla u\vert ^2\,dx. \end{aligned} \end{aligned}$$

The proof is completed by gathering the estimates for all $\Omega _i$. $\square$

Lemma 4.5

Let the conditions of Lemma 4.1 and condition (4.10) be fulfilled. For every $u\in C^1(\overline{\Omega })\cap H^2_0(\Omega )$, any $\varsigma \in (0,r^\sharp )$, and an arbitrary $\delta >0$

$$\begin{aligned} \int _{\Omega }\mathcal {F}_{\epsilon }^{(r_1-\varsigma , r_2-\varsigma )}(x,\nabla u)\vert \nabla u\vert ^2\,dx\le \delta \int _{\Omega }\mathcal {F}_{\epsilon }^{(0,0)}(x,\nabla u)\vert u_{xx}\vert ^2\,dx +C \end{aligned}$$

(4.20)

with a constant $C=C(\omega _p,\omega _q,N,M_0,\alpha ,\delta ,\varsigma )$.

Proof

For $\varsigma \in (0,r_*/2)$ inequality (4.9) follows from the estimates of Lemmas 4.2, 4.3, 4.4 with $\sigma +\lambda +\mu <1$. If $\varsigma \in (r_*/2,r^\sharp )$, it is sufficient to observe that by virtue of Young’s inequality and (3.3)

$$\begin{aligned} \begin{aligned} \mathcal {F}_{\epsilon }^{(r_1-\varsigma , r_2-\varsigma )}(x,\nabla u)\vert \nabla u\vert ^2\,dx \le&\, \mathcal {F}_{\epsilon }^{(r_1-\varsigma , r_2-\varsigma )}(x,\nabla u) \beta _\epsilon (\nabla u) \\ \le&\,C_1 +\mathcal {F}_{\epsilon }^{(r_1-\widetilde{\varsigma }, r_2-\widetilde{\varsigma })}(x,\nabla u) \beta _\epsilon (\nabla u) \\ \le&\, C_2 + 2 \mathcal {F}_{\epsilon }^{(r_1-\widetilde{\varsigma }, r_2-\widetilde{\varsigma })}(x,\nabla u)\vert \nabla u\vert ^2\,dx \end{aligned} \end{aligned}$$

with $\widetilde{\varsigma }< \varsigma$, $\widetilde{\varsigma }\in (0, r_*/2)$ and independent of u constants $C_1$, $C_2$. $\square$

The estimate of Lemma 4.5 can be extended to the functions defined on the cylinder $Q_T$, provided that for a.e. $t\in (0,T)$ the exponents p, q and the coefficients a, b satisfy the conditions of Lemma 4.1 and (4.10) with $r^\sharp = \frac{4}{N+2}$.

Theorem 4.1

Let $\partial \Omega \in C^2$, $u\in C^{0}([0,T];C^1(\overline{\Omega })\cap H^2_0(\Omega ))$. Assume that the exponents $p(\cdot )$, $q(\cdot )$ satisfy conditions (2.12), (2.13), (2.19), and the coefficients $a(\cdot )$, $b(\cdot )$ satisfy conditions (2.16). If

$$\begin{aligned} \int _{Q_T}\mathcal {F}^{(0,0)}_{\epsilon }(z,\nabla u)\vert u_{xx}\vert ^2\,dz<\infty ,\qquad \sup _{(0,T)}\Vert u(t)\Vert _{2,\Omega }^2= M_0, \end{aligned}$$

(4.21)

then for every $\varsigma \in (0,r^\sharp )$ and every $\beta \in (0,1)$

$$\begin{aligned} \begin{aligned} \int _{Q_T} \mathcal {F}_{\epsilon }^{(r_1-\varsigma , r_2-\varsigma )}(z,\nabla u)\vert \nabla u\vert ^2\,dz\le \beta \int _{Q_T}\mathcal {F}^{(0,0)}_{\epsilon }(z,\nabla u)\vert u_{xx}\vert ^{2}\,dx+C \end{aligned} \end{aligned}$$

(4.22)

with

$$\begin{aligned} r_1(z):=\underline{s}(z)+r^\sharp - p(z), \quad \text {and} \quad r_2(z):=\underline{s}(z)+r^\sharp - q(z) \end{aligned}$$

and a constant $C=C(\partial \Omega ,\beta ,\alpha , \omega _p,\omega _q,N,M_0, L, \varsigma )$, where $\omega _p$, $\omega _q$ denote the modules of continuity of p and q in $Q_T$. Moreover, for every $r\in (0,r^\sharp )$

$$\begin{aligned} \alpha \int _{Q_T}\vert \nabla u\vert ^{\underline{s}(z)+r} \,dz\le \beta \int _{Q_T}\mathcal {F}^{(0,0)}_{\epsilon }(x,\nabla u)\vert u_{xx}\vert ^{2}\,dx+C \end{aligned}$$

(4.23)

with a constant C depending on the same quantities as the constant in (4.22).

Proof

Since the exponents p, q are Lipschitz-continuous in $\overline{Q}_T$, for every $0< \varsigma < r_*/2$ there exists a finite cover of $Q_T$ composed of the cylinders $Q^{(i)}=\Omega _i\times (t_{i-1},t_i)$, $i=1,2,\ldots ,K$, such that

$$\begin{aligned} \begin{aligned}&t_0=0,\quad t_K=T,\quad t_i-t_{i-1}=\rho ,\quad Q_T\subset \bigcup _{i=1}^{K}Q^{(i)},\quad \partial \Omega _i\in C^2, \\&\quad \underline{s}_i^+=\max _{\overline{Q}^{(i)}} \underline{s}(z),\quad \underline{s}_i^{-}=\min _{\overline{Q}^{(i)}}\underline{s}(z), \quad \underline{s}_i^+-\underline{s}_i^- <\dfrac{\varsigma }{4},\quad i=1,\ldots ,K,\;\;K=K(\omega _p,\omega _q,\varsigma ). \end{aligned} \end{aligned}$$

For a.e. $t\in (0,T)$ the function $u(\cdot ,t):\Omega \mapsto \mathbb {R}$ satisfies inequality (4.20) on each of $\Omega _i$. Integrating these inequalities over the intervals $(t_{i-1},t_i)$ and summing the results we obtain (4.22) with $\varsigma <r_*/2$. The case $\varsigma \in [r_*/2,r^\sharp )$ follows as in the proof of Lemma 4.5. Due to the choice of $r_1(z)$, $r_2(z)$, inequality (4.23) follows from (4.22) with the help of (3.3) and Lemma 3.1.$\square$

4.2 Estimates on the traces

We will need estimates on the traces on the lateral boundary of the cylinder $Q_T$. In the assertions formulated below the exponents p, q and coefficients a, b are considered as functions of $x\in \Omega$ and the variable t is regarded as a parameter.

Lemma 4.6

[Lemma 4.3, [9]] Let $\Omega \subset \mathbb {R}^N$, $N\ge 2$ be a bounded domain with the boundary $\partial \Omega \in C^{2}$, and ${a,b \in W^{1,\infty }(\Omega )}$ be given nonnegative functions. Assume that $v\in H^{3}_0(\Omega )$ and denote

$$\begin{aligned} \mathcal {K}=\int _{\partial \Omega } \mathcal {F}^{(0,0)}_\epsilon (x, \nabla v) \left( \Delta v \,(\nabla v\cdot \textbf{n})-\nabla (\nabla v\cdot \textbf{n})\cdot \nabla v\right) \,dS, \end{aligned}$$

(4.24)

where $\textbf{n}$ stands for the exterior normal to ${\partial \Omega }$. There exists a constant $\kappa =\kappa (\partial \Omega )$ such that

$$\begin{aligned} \mathcal {K}\le \kappa \int _{\partial \Omega }\mathcal {F}^{(0,0)}_\epsilon (x, \nabla v) \vert \nabla v\vert ^2\,dS. \end{aligned}$$

Theorem 4.2

Let $\partial \Omega \in C^2$, $u\in C^1(\overline{\Omega })\cap H^2_0(\Omega )$. Assume that a(x), b(x), p(x), q(x) satisfy the conditions of Lemma 4.1 and (4.10). Then, for every $\lambda \in (0,1)$

$$\begin{aligned} \int _{\partial \Omega } \mathcal {F}^{(0,0)}_{\epsilon }(x,\nabla u)\vert \nabla u\vert ^{2}\,dS\le \lambda \int _{\Omega } \mathcal {F}^{(0,0)}_{\epsilon }(x,\nabla u) \vert u_{xx}\vert ^2\,dx+ C \end{aligned}$$

(4.25)

with a constant C depending on $\lambda$, $\underline{s}^-$, $\underline{s}^+$, N, L, $\alpha$, and $\Vert u\Vert _{2,\Omega }$.

Proof

Applying [9, Corollary 4.1] to $\vert \nabla u\vert$ we obtain

$$\begin{aligned} \begin{aligned} \int _{\partial \Omega } \mathcal {F}^{(0,0)}_{\epsilon }(x,\nabla u) \vert \nabla u\vert ^2\,dS&\le \delta \int _{\Omega } \mathcal {F}^{(0,0)}_{\epsilon }(x,\nabla u) \vert u_{xx}\vert ^2\,dx \\&\quad + C' \int _{\Omega } \left( \vert \nabla u\vert ^{p(x)} + \vert \nabla u\vert ^{q(x)}\right) \,dx \\&\quad + C'' \int _{\Omega } (\vert \nabla u\vert ^{p(x)} + \vert \nabla u\vert ^{q(x)})\vert \ln \vert \nabla u\vert \vert \,dx +C''' \end{aligned}\nonumber \\ \end{aligned}$$

(4.26)

with independent of u constants $C', C'', C'''$. Choose $\ell _0 \in (0,1)$ and $\ell _{00} \in [r_*, r^{\sharp })$ such that

$$\begin{aligned} \overline{s}(x)+\ell _0< \underline{s}(x) + \ell _{00} < \underline{s}(x) + r^{\sharp }. \end{aligned}$$

Using (3.4) and Young’s inequality we estimate

$$\begin{aligned} (\vert \nabla u\vert ^{p(x)} + \vert \nabla u\vert ^{q(x)})\vert \ln \vert \nabla u\vert \vert \le C(\ell _0,p^\pm , q^\pm ) (1+ \vert \nabla u\vert ^{\underline{s}(x)+ \ell _{00}}) \end{aligned}$$

with a constant C independent of u. Using the above inequality and then applying Lemma 4.5 with $\varsigma = r^\sharp -\ell _{00}$ we continue (4.26) as follows:

$$\begin{aligned} \begin{aligned} \int _{\partial \Omega } \mathcal {F}^{(0,0)}_{\epsilon }(x,\nabla u) \vert \nabla u\vert ^2\,dS&\le \delta \int _{\Omega } \mathcal {F}^{(0,0)}_{\epsilon }(x,\nabla u) \vert u_{xx}\vert ^2\,dx \\&\qquad +C \left( 1+\alpha \int _{\Omega } \vert \nabla u\vert ^{\underline{s}(x)+ \ell _{00}} \,dx\right) \\&\le (\delta + \beta ) \int _{\Omega }\mathcal {F}^{(0,0)}_{\epsilon }(x,\nabla u)\vert u_{xx}\vert ^2\,dx +C. \end{aligned} \end{aligned}$$

$\square$

5 Approximation of the regularized problem

Let us consider the family of regularized unordered double phase parabolic equations:

$$\begin{aligned} \left\{ \begin{aligned} {} \partial _t u - {{\,\textrm{div}\,}}(\mathcal {F}^{(0,0)}_{\epsilon }(z,\nabla u) \nabla u)&{}= f(z){} & {} \quad \text{ in } \, Q_T, \\ u&{}= 0{} & {} \quad \text{ on } \, \Gamma _T, \\ u(\cdot ,0)&{}= u_0{} & {} \quad \ \text{ in }\, \ \Omega , \ \epsilon \in (0,1), \end{aligned} \right. \end{aligned}$$

(5.1)

where $\mathcal {F}^{(0,0)}_{\epsilon }(z,\xi )= a(z) \beta ^{\frac{p(z)-2}{2}}_{\epsilon }(\xi ) + b(z) \beta ^{\frac{q(z)-2}{2}}_{\epsilon }(\xi )$.

Definition 5.1

A function $u_\epsilon :Q_T\mapsto \mathbb {R}$ is called strong solution of problem (5.1) if

• 1. $u_\epsilon \in \mathbb {W}_{\overline{s}(\cdot )}(Q_T)$, $u_{\epsilon t} \in L^2(Q_T)$ and $\vert \nabla u_\epsilon \vert \in L^{\infty }(0,T; L^{r(\cdot )}(\Omega ))$ with $r(z) = \max \{2,\overline{s}(z)\}$,

• 2. for every $\phi \in \mathbb {W}_{\overline{s}(\cdot )}(Q_T)$

  $$\begin{aligned} \int _{Q_T} u_{\epsilon t} \phi ~dz + \int _{Q_T} \mathcal {F}_{\epsilon }^{(0,0)}(z,\nabla u_\epsilon )\nabla u_\epsilon \cdot \nabla \phi ~dz = \int _{Q_T} f \phi \,dz, \end{aligned}$$

  (5.2)

• 3. for every $\psi \in C_0^1(\Omega )$

  $$\begin{aligned} \int _{\Omega } (u_\epsilon (x,t)-u_0(x)) \psi ~dx \rightarrow 0\quad \text {as } t \rightarrow 0. \end{aligned}$$

5.1 Dense sets in $W^{1,p(\cdot )}_0(\Omega )$

Let $\{\phi _i\}$ and $\{\lambda _i\}$ be the eigenfunctions and the corresponding eigenvalues of the Dirichlet problem for the Laplacian:

$$\begin{aligned} (\nabla \phi _i,\nabla \psi )_{2,\Omega }=\lambda _i(\phi _i,\psi )\qquad \forall \psi \in H^{1}_0(\Omega ). \end{aligned}$$

(5.3)

The functions $\phi _i$ form an orthogonal basis of $L^2(\Omega )$ and are mutually orthogonal in $H^1_0(\Omega )$. If $\partial \Omega \in C^k$, $k\ge 1$, then $\phi _i\in C^{\infty }(\Omega )\cap H^{k}(\Omega )$. Let us denote by $H^k_{\mathcal {D}}(\Omega )$ the subspace of the Hilbert space $H^{k}(\Omega )$ composed of the functions f for which

$$\begin{aligned} f=0,\quad \Delta f=0,\quad \ldots , \Delta ^{[\frac{k-1}{2}]} f=0\quad \hbox { on}\ \partial \Omega ,\qquad H^{0}_{\mathcal {D}}(\Omega )=L^2(\Omega ). \end{aligned}$$

The relations

$$\begin{aligned} {[}f,g]_{k}={\left\{ \begin{array}{ll} (\Delta ^{\frac{k}{2}}f,\Delta ^{\frac{k}{2}}g)_{2,\Omega } &{} \text {if } k\hbox { is even}, \\ (\Delta ^{\frac{k-1}{2}}f,\Delta ^{\frac{k-1}{2}}g)_{H^1(\Omega )} &{} \text {if }k\hbox { is odd} \end{array}\right. } \end{aligned}$$

define an equivalent scalar product on ${H}^{k}_{\mathcal {D}}(\Omega )$: $[f,g]_k=\displaystyle \sum _{i=1}^\infty \lambda _i^k f_ig_i$, where $f_i$, $g_i$ are the Fourier coefficients of f, g in the basis $\{\phi _i\}$ of $L^2(\Omega )$. The corresponding equivalent norm of $H^k_{\mathcal {D}(\Omega )}$ is defined by $\Vert f\Vert ^2_{{H}^{k}_{\mathcal {D}}(\Omega )}=[f,f]_k$. Let $f^{(m)}=\displaystyle \sum _{i=1}^{m}f_i\phi _i$ be the partial sum of the Fourier series of $f\in L^{2}(\Omega )$. We will rely on the following known assertions.

Proposition 5.1

Let $\partial \Omega \in C^k$, $k\ge 1$. A function f can be represented by the Fourier series in the system $\{\phi _i\}$, convergent in the norm of $H^k(\Omega )$, if and only if $f\in H^{k}_{\mathcal {D}}(\Omega )$. If $f\in H^{k}_{\mathcal {D}}(\Omega )$, then the Fourier series is convergent, its sum is bounded by $C\Vert f\Vert _{H^k(\Omega )}$ with an independent of f constant C, and $\Vert f^{(m)}-f\Vert _{H^k(\Omega )}\rightarrow 0$ as $m\rightarrow \infty$. If $k\ge [\frac{N}{2}]+1$, then the Fourier series in the system $\{\phi _i\}$ of every function $f\in H^{k}_{\mathcal {D}}(\Omega )$ converges to f in $C^{k-[\frac{N}{2}]-1}(\overline{\Omega })$.

Proposition 5.2

([29], Theorem 4.7, Proposition 4.10) Let $\partial \Omega \in Lip$ and $p \in C_\textrm{log}(\overline{\Omega })$. Then, the set $C^\infty _{0}(\Omega )$ is dense in $W_0^{1,p(\cdot )}(\Omega )$.

Let $k\in \mathbb {N}$ be so large that

$$\begin{aligned} k\ge N\left( \frac{1}{2}+\frac{1}{N}-\frac{1}{q^+}\right) , \quad q(x)=\max \{2,p(x)\},\quad q^+=\sup _{\Omega }q(x). \end{aligned}$$

(5.4)

Set $\mathcal {P}_m={\text {span}} \{\phi _1,\ldots ,\phi _m\}$

Lemma 5.1

Assume k, N, q satisfy (5.4). Then, the set $\bigcup \nolimits _{m=1}^\infty \mathcal {P}_m$ is dense in the space $W^{1,q(\cdot )}_0(\Omega )$.

Proof

Take an arbitrary $u\in W_0^{1,q(\cdot )}(\Omega )$. We want to show that for every $\epsilon >0$ there exists $m(\epsilon )\in \mathbb {N}$ such that for every $m\ge m(\epsilon )$ there exists $v_m\in \mathcal {P}_m$ satisfying $\Vert v_m-u\Vert _{W^{1,q(\cdot )}_0(\Omega )}<\epsilon$. By Proposition 5.2 there exists $v_{\epsilon }\in C_{0}^{\infty }(\Omega )\subset H_{\mathcal {D}}^k(\Omega )$ such that

$$\begin{aligned} \Vert u-v_\epsilon \Vert _{W^{1,q(\cdot )}_0(\Omega )} <\frac{\epsilon }{2},\qquad v_\epsilon \in C_{0}^{\infty }(\Omega )\cap H_{\mathcal {D}}^k(\Omega ). \end{aligned}$$

By Proposition 5.1$\displaystyle v_\epsilon =\sum \nolimits _{i=1}^{\infty }v_{\epsilon ,i}\phi _i\in H_{\mathcal {D}}^k(\Omega ) \subset H^k(\Omega )$ and

$$\begin{aligned} v_{\epsilon }^{(k)}=\sum _{i=1}^k v_{\epsilon ,i}\phi _i\in \mathcal {P}_k, \qquad v_\epsilon ^{(k)}\rightarrow v_\epsilon \hbox { in } H_{\mathcal {D}}^k(\Omega ) \ \hbox { as }\ k \rightarrow \infty . \end{aligned}$$

Since k, N, q satisfy condition (5.4), the embeddings $H_{\mathcal {D}}^k(\Omega ) \subset W_0^{1, q^+}(\Omega ) \subseteq W^{1,q(\cdot )}_0(\Omega )$ are continuous:

$$\begin{aligned} \Vert w\Vert _{W^{1,q(\cdot )}_0(\Omega )}\le C\Vert w\Vert _{W_0^{1, q^+}(\Omega )}\le C'\Vert w\Vert _{H^k(\Omega )}\qquad \forall w\in H_{\mathcal {D}}^k(\Omega ) \end{aligned}$$

with independent of w constants C, $C'$. Given $\epsilon$, we may find $k(\epsilon )\in \mathbb {N}$ such that for all $k\ge k(\epsilon )$

$$\begin{aligned} \Vert v_\epsilon -v_{\epsilon }^{(k)}\Vert _{W_0^{1, q(\cdot )}(\Omega )} \le C'\Vert v_\epsilon -v_{\epsilon }^{(k)}\Vert _{H^{k}(\Omega )} =C' \left\| v_\epsilon -\sum _{i=1}^k v_{\epsilon ,i}\phi _i\right\| _{H^k(\Omega )}<\frac{\epsilon }{2}. \end{aligned}$$

It follows that for every $k\ge k(\epsilon )$

$$\begin{aligned} C''\Vert u-v_\epsilon ^{(k)}\Vert _{W^{1,q(\cdot )}_0(\Omega )} <\frac{\epsilon }{2}+ \frac{\epsilon }{2}=\epsilon \end{aligned}$$

with a constant $C''>0$ independent of u, $\epsilon$, and k. $\square$

Corollary 5.1

Let $q(\cdot )\in C_{\textrm{log}}(\overline{Q}_T)$ and $\partial \Omega \in C^k$ with k satisfying condition (5.4). Denote

$$\begin{aligned} \mathcal {N}_m=\left\{ w(x,t):\;w(x,t)= \sum _{i=1}^m \theta _i(t)\phi _i(x),\;\theta _i \in C^{0,1}[0,T]\right\} . \end{aligned}$$

(5.5)

Then, the set $\bigcup _{m=1}^\infty \mathcal {N}_m$ is dense in $\mathbb {W}_{q(\cdot )}(Q_T)$.

5.2 Dense sets in $W^{1,\mathcal {H}}_0(\Omega )$

Let $W^{1,\mathcal {H}}_0(\Omega )$ be the Musielak–Orlicz space defined in (2.9) where $p_0$, $q_0$, $a_0$, $b_0$ are the exponents and coefficients from equation (1.1) taken at the initial moment $t=0$.

Proposition 5.3

Let $\partial \Omega \in Lip$ and $a_0,b_0,p_0,q_0\in C^{0,1}(\overline{\Omega })$. If

$$\begin{aligned} \frac{\max \{r(x),\sigma (x)\}}{s(x)}< 1+\frac{1}{N}\quad \text {in } \Omega , \end{aligned}$$

then $C^\infty _0(\Omega )\cap W^{1,\mathcal {H}}(\Omega )$ is dense in $W^{1,\mathcal {H}}_0(\Omega )$.

The question of density of smooth functions in the Musielak-Sobolev spaces was studied in several works. The assertion of Proposition 5.3 follows from [21, Th.3.1] or [37, Th.6.4.7]. To check the fulfillment of all conditions listed in [37], one may literally repeat the proof given in [25, Theorem 2.28] for the special case of the space $W^{1,\mathcal {H}}(\Omega )$ generated by the function $\mathcal {H}$ with $b_0\equiv 0$. Proposition 5.3 follows as well from [17, Theorem 2.3].

Lemma 5.2

Let $p,q,a_0,b_0$ satisfy the conditions of Proposition 5.3. If $\partial \Omega \in C^k$ with

$$\begin{aligned} k\ge N\left( \frac{1}{2}+\frac{1}{N}-\frac{1}{\,\max \{2,p^+,q^+\}}\right) , \end{aligned}$$

(5.6)

then the set $\bigcup \nolimits _{m=1}^\infty \mathcal {P}_m$ is dense in $W^{1,\mathcal {H}}_0(\Omega )$.

Proof

The assertion follows by imitating the proof of Proposition 5.2. Given a function $v\in W^{1,\mathcal {H}}_0(\Omega )$ and an arbitrary $\epsilon >0$, by Proposition 5.3 we may find $v_\epsilon \in C_{0}^\infty (\Omega )\subset C_{0}^{\infty }(\Omega ) \cap H^{k}_{\mathcal {D}}(\Omega )$ such that $\Vert v-v_\epsilon \Vert _{W^{1,\mathcal {H}}}<\frac{\epsilon }{2}$, and then use Proposition 5.1 to approximate $v_\epsilon \in H^{k}_{\mathcal {D}}(\Omega )$ by the partial sums $v_\epsilon ^{(m)}$. $\square$

5.3 Galerkin’s method

Let $\epsilon >0$ be a fixed parameter. The sequence $\{u^{(m)}_\epsilon \}$ of finite-dimensional Galerkin’s approximations for the solutions of the regularized problem (5.1) is sought in the form

$$\begin{aligned} u^{(m)}_\epsilon (x,t)= \sum _{j=1}^m u_{j}^{(m)}(t) \phi _j(x) \end{aligned}$$

(5.7)

where $\{\phi _j\}$ and $\{\lambda _j\}$ are the eigenfunctions and the corresponding eigenvalues of problem (5.3). The coefficients $u_j^{(m)}(t)$ are characterized as the solutions of the Cauchy problem for the system of m ordinary differential equations

$$\begin{aligned} \left\{ \begin{aligned} {} (u^{(m)}_j)'(t)&{}= - \int _{\Omega } \mathcal {F}^{(0,0)}_{\epsilon }(z, \nabla u_\epsilon ^{(m)}) \nabla u^{(m)}_\epsilon \cdot \nabla \phi _j ~dx + \int _{\Omega } f \phi _j ~dx{} & {} , \\ u^{(m)}_j(0)&{}= (u_0^{(m)}, \phi _j)_{2, \Omega }, \quad j=1,2,\dots ,m,{} & {} \end{aligned} \right. \end{aligned}$$

(5.8)

where the functions $u_0^{(m)}$ are chosen in such a way that

$$\begin{aligned} \begin{aligned}&u_0^{(m)}= \sum _{j=1}^m (u_0, \phi _j)_{2, \Omega } \phi _j \in {\text {span}}\{\phi _1, \phi _2, \dots , \phi _m\}, \\&\hbox {} u_0^{(m)}\rightarrow u_0\, \hbox {in}\, W^{1,\mathcal {H}}_0(\Omega ), \quad \Vert u_0^{(m)}\Vert _{W^{1,\mathcal {H}}_0(\Omega )}\le C, \end{aligned} \end{aligned}$$

(5.9)

where C is independent of m. The existence of such a sequence follows from Lemma 5.2. By the Carathéodory existence theorem, for every m system (5.8) has an absolute continuous solution $u_1^{(m)},\ldots u_m^{(m)}$, defined on an interval $(0,T_m)$. The possibility of continuation of this solution to the whole interval (0, T) will follow from the a priori estimates derived in the next section.

6 A priori estimates

Throughout this section, when deriving the estimates for the approximations $u_\epsilon ^{(m)}$ we always assume that p, q, a, b and $\partial \Omega$ satisfy the conditions of Lemma 5.2.

Lemma 6.1

Let $\Omega$ be a bounded domain with the Lipschitz boundary. Assume that $p(\cdot ),q(\cdot )$ satisfy (2.12), $a(\cdot ), b(\cdot )$ satisfy (2.16), $u_0 \in L^2(\Omega )$ and $f \in L^2(Q_T)$. Then $u^{(m)}_\epsilon$ satisfies the estimates

$$\begin{aligned} \begin{aligned}&\sup _{t \in (0,T)} \Vert u^{(m)}_\epsilon (\cdot ,t)\Vert ^2_{2,\Omega } + \int _{Q_T} \mathcal {F}^{(0,0)}_{\epsilon }(z, \nabla u_\epsilon ^{(m)}) \vert \nabla u^{(m)}_\epsilon \vert ^2 ~dz \\&\qquad \qquad \le C_1 \textrm{e}^{ T} (\Vert f\Vert ^2_{2,Q_T} + \Vert u_0\Vert ^2_{2,\Omega }) \end{aligned} \end{aligned}$$

(6.1)

and

$$\begin{aligned} \begin{aligned}&\int _{Q_{T}} \mathcal {F}_0^{(0,0)}(z,\nabla u_{\epsilon }^{(m)})\vert \nabla u_{\epsilon }^{(m)}\vert ^2\,dz \\&\qquad \qquad \le C_2 \int _{Q_T} \mathcal {F}^{(0,0)}_{\epsilon }(z, \nabla u_\epsilon ^{(m)}) \vert \nabla u^{(m)}_\epsilon \vert ^2 ~dz + C_3 \end{aligned} \end{aligned}$$

(6.2)

with independent of m and $\epsilon$ constants $C_i.$

Proof

Let us multiply each of equations in (5.8) by $u_j^{(m)}(t)$ and sum up the results for $j=1,2,\dots ,m$:

$$\begin{aligned} \begin{aligned} \frac{1}{2} \frac{d}{dt} \Vert u^{(m)}_\epsilon {(\cdot , t)}\Vert ^2_{2,\Omega }&=\sum _{j=1}^m u_j^{(m)}(t) (u^{(m)}_j)'(t) \\&= - \sum _{j=1}^m u_j^{(m)}(t) \int _{\Omega } \mathcal {F}^{(0,0)}_{\epsilon }(z, \nabla u_\epsilon ^{(m)}) \nabla u^{(m)}_\epsilon . \nabla \phi _j \,~dx \\&\quad + \sum _{j=1}^m \int _{\Omega } f(x,t) \phi _j(x) u_j^{(m)}(t) ~dx \\&= - \int _{\Omega } \mathcal {F}^{(0,0)}_{\epsilon }(z, \nabla u_\epsilon ^{(m)}) \vert \nabla u^{(m)}_\epsilon \vert ^2 ~dx + \int _{\Omega } f(x,t) u^{(m)}_\epsilon ~dx. \end{aligned} \end{aligned}$$

(6.3)

By employing the Cauchy inequality, we obtain

$$\begin{aligned} \begin{aligned}&\frac{1}{2} \frac{d}{dt} \Vert u^{(m)}_\epsilon {(\cdot , t)}\Vert ^2_{2,\Omega } + \int _{\Omega } \mathcal {F}^{(0,0)}_{\epsilon }(z, \nabla u_\epsilon ^{(m)}) \vert \nabla u^{(m)}_\epsilon \vert ^2 ~dx \\&\qquad \le \frac{1}{2} \Vert f{(\cdot , t)}\Vert ^2_{2,\Omega } + \frac{1}{2} \Vert u^{(m)}_\epsilon {(\cdot , t)}\Vert ^2_{2,\Omega }. \end{aligned} \end{aligned}$$

(6.4)

Now, by rephrasing the inequality in (6.4) as

$$\begin{aligned} \begin{aligned}&\frac{1}{2} \frac{d}{dt} \left( \textrm{e}^{- t}\Vert u^{(m)}_\epsilon {(\cdot , t)} \Vert ^2_{L^2(\Omega )}\right) + \textrm{e}^{-t} \int _{\Omega }\mathcal {F}^{(0,0)}_{\epsilon }(z, \nabla u_\epsilon ^{(m)}) \vert \nabla u^{(m)}_\epsilon \vert ^2 ~dx \\&\qquad \le \frac{\textrm{e}^{-t}}{2} \Vert f{(\cdot , t)}\Vert ^2_{2,\Omega } \end{aligned} \end{aligned}$$

and integrating with respect to $0<t<T_m$, we arrive at the inequality

$$\begin{aligned} \begin{aligned}&\sup _{t \in (0,T_m)} \Vert u^{(m)}_\epsilon (\cdot ,t)\Vert ^2_{L^2(\Omega )} + \int _{Q_{T_m}} \mathcal {F}^{(0,0)}_{\epsilon }(z, \nabla u_\epsilon ^{(m)}) \vert \nabla u^{(m)}_\epsilon \vert ^2 ~dx ~dt \\&\qquad \le C \textrm{e}^{ T} \left( \Vert f\Vert ^2_{2,Q_T} + \Vert u_0\Vert ^2_{2,\Omega }\right) , \end{aligned} \end{aligned}$$

(6.5)

where the constant C is independent of $\epsilon$ and m. It follows that $u_{\epsilon }^{(m)}(\cdot ,T_m)\in L^2(\Omega )$ and, thus, system (5.8) can be solved on an interval $(T_m,T_m+h)$ with some $h>0$. In the result we obtain (6.5) with $T_m$ substituted by $T_m+h$. This process can be continued until we exhaust the interval (0, T). Moreover, (6.2) follows from (3.3) and (6.1). $\square$

Lemma 6.2

Let $\Omega$ be a bounded domain with the boundary $\partial \Omega \in C^k$ with $k\ge 2+\left[ \frac{N}{2}\right]$. Assume that $p(\cdot ),q(\cdot )$ satisfy conditions (2.12), (2.13), (2.19) and $a(\cdot ), b(\cdot )$ satisfy conditions (2.16). If $u_0 \in W^{1,\mathcal {H}}_0(\Omega )$, $f\in L^2(0,T;W_0^{1,2}(\Omega ))$, then for a.e. $t\in (0,T)$ the following inequality holds:

$$\begin{aligned} \begin{aligned}&\frac{1}{2} \frac{d}{dt} \Vert \nabla u^{(m)}_\epsilon {(\cdot , t)}\Vert ^2_{2,\Omega } + C_0 \int _{\Omega } \mathcal {F}^{(0,0)}_{\epsilon }(z, \nabla u_\epsilon ^{(m)}) \vert (u^{(m)}_\epsilon )_{xx}\vert ^2 \,dx \\&\quad \le C_1 \left( 1 + \int _{\Omega } \mathcal {F}^{(0,0)}_{\epsilon }(z, \nabla u_\epsilon ^{(m)})\vert \nabla u_\epsilon ^{(m)}\vert ^2\,dx \right. \\&\qquad \left. + \Vert \nabla u_{\epsilon }^{(m)}{(\cdot , t)}\Vert _{2,\Omega }^{2} +\Vert f{(\cdot , t)}\Vert _{W^{1,2}_0(\Omega )}^2\right) \end{aligned} \end{aligned}$$

(6.6)

where the constants $C_0$, $C_1$ are independent of $\epsilon$ and m.

Proof

Multiplying $j{\text {th}}$ equation of (5.8) by $\lambda _j u_j^{(m)}$ and then summing up the results for $j=1,2, \dots , m$, we obtain the equality

$$\begin{aligned}{} & {} \frac{1}{2} \frac{d}{dt} \Vert \nabla u^{(m)}_\epsilon {(\cdot , t)}\Vert ^2_{2,\Omega } =\sum _{j=1}^m \lambda _j (u_j^{(m)})'(t) u_j^{(m)}(t) \nonumber \\{} & {} \qquad = \sum _{j=1}^m \lambda _j u_j^{(m)} \int _{\Omega } {{\,\textrm{div}\,}}(\mathcal {F}^{(0,0)}_{\epsilon }(z, \nabla u_\epsilon ^{(m)}) \nabla u_\epsilon ^{(m)})\ \phi _j \,dx \nonumber \\{} & {} \qquad \quad + \sum _{j=1}^m \lambda _j u_{j}^{(m)} \int _{\Omega } f(x,t) \phi _j \,dx\nonumber \\{} & {} \qquad = - \int _{\Omega } {{\,\textrm{div}\,}}(\mathcal {F}^{(0,0)}_{\epsilon }(z, \nabla u_\epsilon ^{(m)}) \nabla u_\epsilon ^{(m)}) \ \Delta u^{(m)}_\epsilon \,dx + \int _{\Omega } f(x,t) \Delta u^{(m)}_\epsilon \,dx. \end{aligned}$$

(6.7)

The assumption $\partial \Omega \in C^{k}$ with $k\ge 2+[\frac{N}{2}]$ yields fulfillment of condition (5.6) and the inclusion $u_\epsilon ^{(m)}(\cdot ,t)\in C^{1}(\overline{\Omega })\cap H^k_{\mathcal {D}}(\Omega )$ for every fixed $t\in [0,T]$. It follows that $\left( \mathcal {F}^{(0,0)}_{\epsilon }(z, \nabla u_\epsilon ^{(m)}) ( u^{(m)}_\epsilon )_{x_i}\right) _{x_i} \in L^{2}(\Omega )$, therefore we may transform the first term on the right-hand side of (6.7) using the Green formula two times (see [9, Lemma 6.2] or [8, Lemma 3.2] for the details):

$$\begin{aligned} \begin{aligned}&- \int _{\Omega } \textrm{div}\left( \mathcal {F}^{(0,0)}_{\epsilon }(z, \nabla u_\epsilon ^{(m)}) \nabla u_\epsilon ^{(m)}\right) \,\Delta u_\epsilon ^{(m)}\,dx \\&\qquad = - \int _{\Omega } \mathcal {F}^{(0,0)}_{\epsilon }(z, \nabla u_\epsilon ^{(m)}) \vert (u_{\epsilon }^{(m)})_{xx}\vert ^2\,dx +P_{1}+P_2+P_{\partial \Omega } + P_{a,b}, \end{aligned} \end{aligned}$$

where $\textbf{n}=(n_1,\ldots ,n_N)$ is the outer normal vector to $\partial \Omega$. Here

$$\begin{aligned} \begin{aligned} P_1:=&\int _{\Omega } (2-p(z)) a(z) (\epsilon ^2 + \vert \nabla u^{(m)}_\epsilon \vert ^2)^{\frac{p(z)-2}{2}-1} \left( \sum _{k=1}^N \left( \nabla u^{(m)}_\epsilon \cdot \nabla (u^{(m)}_\epsilon )_{x_k} \right) ^2 \right) \,dx\\&+ \int _{\Omega } (2-q(z)) b(z) (\epsilon ^2 + \vert \nabla u^{(m)}_\epsilon \vert ^2)^{\frac{q(z)-2}{2}-1} \left( \sum _{k=1}^N \left( \nabla u^{(m)}_\epsilon \cdot \nabla (u^{(m)}_\epsilon )_{x_k} \right) ^2 \right) \,dx, \\ P_2 :=&- \sum _{k,i=1}^N \int _{\Omega } \left( a(z) (\epsilon ^2 + \vert \nabla u^{(m)}_\epsilon \vert ^2)^{\frac{p(z)-2}{2}} \frac{p_{x_k}}{2} + b(z) (\epsilon ^2 + \vert \nabla u^{(m)}_\epsilon \vert ^2)^{\frac{q(z)-2}{2}} \frac{q_{x_k}}{2}\right) \\&\times (u^{(m)}_\epsilon )_{x_k x_i } (u^{(m)}_\epsilon )_{x_i} \ln (\epsilon ^2 + \vert \nabla u^{(m)}_\epsilon \vert ^2)\,dx, \\ P_{\partial \Omega }=&- \int _{\partial \Omega } \mathcal {F}^{(0,0)}_{\epsilon }(z, \nabla u_\epsilon ^{(m)}) \left( \Delta u^{(m)}_\epsilon (\nabla u^{(m)}_\epsilon \cdot \textbf{n})- \nabla u^{(m)}_\epsilon \cdot \nabla (\nabla u^{(m)}_\epsilon \cdot \textbf{n})\right) \,dS, \\ P_{a,b}=&- \int _{\Omega } \sum _{i,k=1}^N a_{x_k} (u^{(m)}_\epsilon )_{x_i} (\epsilon ^2 + \vert \nabla u^{(m)}_\epsilon \vert ^2)^{\frac{p(z)-2}{2}} {(u_\epsilon ^{(m)})_{x_k x_i}}\,dx \\&- \int _{\Omega } \sum _{i,k=1}^N b_{x_k} (u^{(m)}_\epsilon )_{x_i} (\epsilon ^2 + \vert \nabla u^{(m)}_\epsilon \vert ^2)^{\frac{q(z)-2}{2}} {(u_\epsilon ^{(m)})_{x_k x_i}}\,dx. \end{aligned} \end{aligned}$$

By substituting these equalities into (6.7), we obtain the following inequality:

$$\begin{aligned} \begin{aligned}&\frac{1}{2} \frac{d}{dt} \Vert \nabla u^{(m)}_\epsilon {(\cdot , t)}\Vert ^2_{2,\Omega } + \int _{\Omega } \mathcal {F}^{(0,0)}_{\epsilon }(z, \nabla u_\epsilon ^{(m)}) \vert (u^{(m)}_\epsilon )_{xx}\vert ^2\,dx \\&\quad = P_{1}+P_2+P_{\partial \Omega } + P_{a,b} - \int _{\Omega }\nabla f \cdot \nabla u_{\epsilon }^{(m)}\,dx \\&\quad \le P_{1}+P_2+P_{\partial \Omega } + P_{a,b} +\frac{1}{2}\Vert \nabla u_{\epsilon }^{(m)}{(\cdot , t)}\Vert _{2,\Omega }^{2}+\frac{1}{2}\Vert f (\cdot , t)\Vert _{W^{1,2}_{0}(\Omega )}^{2}. \end{aligned}\nonumber \\ \end{aligned}$$

(6.8)

The terms on the right-hand side of (6.8) are estimated in four steps.

Step 1: estimate on $P_1$. Since $a(\cdot )$ and $b(\cdot )$ are non-negative functions, the term $P_1$ can be merged into the left-hand side of (6.8). Indeed:

$$\begin{aligned} \begin{aligned}&P_1 = \int _{\{ x\in \Omega :\ p(z) \ge 2\}}(2-p(z))\ldots + \int _{\{ x\in \Omega :\ p(z)< 2\}}(2-p(z))\ldots \\&\qquad + \int _{\{ x\in \Omega :\ q(z) \ge 2\}}(2-q(z))\ldots + \int _{\{ x\in \Omega :\ q(z)< 2\}}(2-q(z))\ldots \\&\quad \le \int _{\{p(z)< 2\}} (2-p(z)) a(z) (\epsilon ^2 + \vert \nabla u^{(m)}_\epsilon \vert ^2)^{\frac{p(z)-4}{2}} \left( \sum _{k=1}^N \left( \nabla u^{(m)}_\epsilon \cdot \nabla (u^{(m)}_\epsilon )_{x_k} \right) ^2 \right) \,dx\\&\qquad + \int _{\{q(z) < 2\}} (2-q(z)) b(z) (\epsilon ^2 + \vert \nabla u^{(m)}_\epsilon \vert ^2)^{\frac{q(z)-4}{2}} \left( \sum _{k=1}^N \left( \nabla u^{(m)}_\epsilon \cdot \nabla (u^{(m)}_\epsilon )_{x_k} \right) ^2 \right) \,dx, \end{aligned} \end{aligned}$$

whence

$$\begin{aligned} \begin{aligned} \vert P_1\vert&\le \max \{0,2-p^-\}\int _{\Omega } a(z) (\epsilon ^2+\vert \nabla u_\epsilon ^{(m)}\vert ^2)^{\frac{p(z)-2}{2}} \vert (u^{(m)}_\epsilon )_{xx}\vert ^2\,dx \\&\qquad + \max \{0,2-q^-\}\int _{\Omega } b(z)(\epsilon ^2+\vert \nabla u_\epsilon ^{(m)}\vert ^2)^{\frac{q(z)-2}{2}} \vert (u^{(m)}_\epsilon )_{xx}\vert ^2\,dx. \end{aligned} \end{aligned}$$

Step 2: estimate on $P_2$. By the Cauchy inequality, for every $\beta _0>0$

$$\begin{aligned} \begin{aligned} \vert P_2\vert&\le \frac{1}{2} \Vert \nabla p \Vert _{\infty ,\Omega } \int _{\Omega } \left( (a(z))^{\frac{1}{2}} (\epsilon ^2 + \vert \nabla u^{(m)}_\epsilon \vert ^2)^{\frac{p(z)-2}{4}} \sum _{k,i=1}^N \vert (u^{(m)}_\epsilon )_{x_k x_i }\vert \right) \\&\quad \times \left( (a(z))^{\frac{1}{2}}\vert (u^{(m)}_\epsilon )_{x_i}\vert \vert \ln (\epsilon ^2 + \vert \nabla u^{(m)}_\epsilon \vert ^2)\vert (\epsilon ^2 + \vert \nabla u^{(m)}_\epsilon \vert ^2)^{\frac{p(z)-2}{4}} \right) ~dx\\&\quad + \frac{1}{2} \Vert \nabla q \Vert _{\infty ,\Omega } \int _{\Omega } \left( (b(z))^{\frac{1}{2}} (\epsilon ^2 + \vert \nabla u^{(m)}_\epsilon \vert ^2)^{\frac{q(z)-2}{4}} \sum _{k,i=1}^N \vert (u^{(m)}_\epsilon )_{x_k x_i }\vert \right) \\&\quad \times \left( (b(z))^{\frac{1}{2}} \vert (u^{(m)}_\epsilon )_{x_i}\vert \vert \ln (\epsilon ^2 + \vert \nabla u^{(m)}_\epsilon \vert ^2)\vert (\epsilon ^2 + \vert \nabla u^{(m)}_\epsilon \vert ^2)^{\frac{q(z)-2}{4}} \right) ~dx\\&\le \beta _0 \int _{\Omega } \mathcal {F}^{(0,0)}_{\epsilon }(z, \nabla u_\epsilon ^{(m)}) \sum _{k,i=1}^N \vert (u^{(m)}_\epsilon )_{x_k x_i }\vert ^2 ~dx + \mathcal {L} \end{aligned} \end{aligned}$$

(6.9)

where

$$\begin{aligned} \mathcal {L}:= C_1 \int _{\Omega } \ln ^2(\epsilon ^2 + \vert \nabla u^{(m)}_\epsilon \vert ^2) \mathcal {F}^{(0,0)}_{\epsilon }(z,\nabla u^{(m)}_\epsilon ) \vert \nabla u^{(m)}_\epsilon \vert ^2 \,dx,\;\; C_1=C_1(a^+,b^+,N, \beta _0). \end{aligned}$$

Now we apply (3.4) in the following form: for $\vartheta \in (0,1)$ and $y>0$

$$\begin{aligned} y^{\frac{p}{2}} \ln ^2 y \le \left\{ \begin{aligned} {} y^{\frac{p + \vartheta }{2}} (y^{\frac{-\vartheta }{2}} \ln ^2(y)) \le C(\vartheta ,p^+) (y^{\frac{p + \vartheta }{2}}){} & {} {}&\quad \text{ if }\ y \ge 1, \\ y^{\frac{p^-}{2}} \ln ^2(y) \le C(p^-) \quad \quad \quad \quad \quad \quad \quad{} & {} {}&\quad \text{ if }\ y \in (0,1). \end{aligned} \right. \end{aligned}$$

(6.10)

To estimate $\mathcal {L}$ we fix $t\in (0,T)$, take some $\varsigma \in (0,r^\sharp )$, apply (6.10) with $\vartheta =r_1-\varsigma$ to the first term and $\vartheta = r_2-\varsigma$ to the second term with $r_1$, $r_2$ defined in (4.11), and then use the interpolation inequality (4.20):

$$\begin{aligned} \begin{aligned}&\mathcal {L} \le C\int _\Omega \mathcal {F}_\epsilon ^{(\underline{s}(z)+r^\sharp -\varsigma ,\ \underline{s}(z)+r^\sharp -\varsigma )}(z,\nabla u_\epsilon ^{(m)})\vert \nabla u_\epsilon ^{(m)}\vert ^2\,dx \\&\quad \le \beta _1 \int _\Omega \mathcal {F}^{(0,0)}_\epsilon (z,\nabla u_\epsilon ^{(m)})\vert (u_\epsilon ^{(m)})_{xx}\vert ^2\,dx +C \end{aligned} \end{aligned}$$

(6.11)

with any $\beta _1 \in (0,1)$ and a constant $C=C(C_1,\vartheta , \beta _1)$. Gathering (6.9) and (6.11), we finally obtain:

$$\begin{aligned} \vert P_2\vert \le (\beta _0 + \beta _1)\int _{\Omega }\mathcal {F}^{(0,0)}_{\epsilon }(x,\nabla u_\epsilon ^{(m)}) \vert (u_\epsilon ^{(m)})_{xx}\vert ^{2}\,dx+ C \end{aligned}$$

with a constant C depending on $\beta _0$, $\beta _1$, $r_*$, $r^{\sharp }$, $\overline{s}^+$, $\underline{s}_-$ and $\Vert f\Vert _{2,Q_T}$, $\Vert u_0\Vert _{2,\Omega }$, but independent of $\epsilon$ and m.

Step 3: estimates on $P_{a,b}$.

$$\begin{aligned} \begin{aligned} \vert P_{a,b}\vert&\le C\int _{\Omega }\left( (\epsilon ^2+\vert \nabla u_\epsilon ^{(m)}\vert ^2)^{\frac{p-1}{2}}\vert (u_{\epsilon }^{(m)})_{xx}\vert +(\epsilon ^2+\vert \nabla u_\epsilon ^{(m)}\vert ^2)^{\frac{q-1}{2}}\vert (u_{\epsilon }^{(m)})_{xx}\vert \right) \,dx \\&\equiv C(\mathcal {I}_1+\mathcal {I}_2) \end{aligned} \end{aligned}$$

with the constant $C=\sup _{\Omega }\vert \nabla a\vert +\sup _{\Omega }\vert \nabla b\vert$. Since $\alpha \le a+b$ by assumption, $\sqrt{\alpha }\le \sqrt{a+b}\le \sqrt{a}+\sqrt{b}$.

$$\begin{aligned} \sqrt{\alpha } \mathcal {I}_1 \le \int _{\Omega }(\sqrt{a}+\sqrt{b})(\epsilon ^2+\vert \nabla u_\epsilon ^{(m)}\vert ^2)^{\frac{p-1}{2}}\vert (u_{\epsilon }^{(m)})_{xx}\vert \,dx\equiv \mathcal {K}^{(1)}_a+\mathcal {K}^{(1)}_b. \end{aligned}$$

Applying the Cauchy inequality we estimate

$$\begin{aligned} \begin{aligned} \mathcal {K}^{(1)}_a&=\int _{\Omega }\sqrt{a}(\epsilon ^2+\vert \nabla u_\epsilon ^{(m)}\vert ^2)^{\frac{p-1}{2}}\vert (u_{\epsilon }^{(m)})_{xx}\vert \,dx \\&\equiv \int _{\Omega }\left( a(\epsilon ^2+\vert \nabla u_\epsilon ^{(m)}\vert ^2)^{\frac{p-2}{2}} \vert (u_{\epsilon }^{(m)})_{xx}\vert ^2\right) ^{\frac{1}{2}} (\epsilon ^2+\vert \nabla u_\epsilon ^{(m)}\vert ^2)^{\frac{p}{4}}\,dx \\&\le \beta _2 \int _{\Omega } a(\epsilon ^2+\vert \nabla u_\epsilon ^{(m)}\vert ^2)^{\frac{p-2}{2}} \vert (u_{\epsilon }^{(m)})_{xx}\vert ^2\,dx + C\int _{\Omega }(\epsilon ^2+\vert \nabla u_\epsilon ^{(m)}\vert ^2)^{\frac{p}{2}}\,dx, \end{aligned} \end{aligned}$$

with an arbitrary $\beta _2>0$ and $C=C(\beta _2)$. By the same token

$$\begin{aligned} \begin{aligned} \mathcal {K}^{(1)}_b&=\int _{\Omega }\sqrt{b}(\epsilon ^2+\vert \nabla u_\epsilon ^{(m)}\vert ^2)^{\frac{p-1}{2}}\vert (u_{\epsilon }^{(m)})_{xx}\vert \,dx \\&\equiv \int _{\Omega }\left( b(\epsilon ^2+\vert \nabla u_\epsilon ^{(m)}\vert ^2)^{\frac{q-2}{2}} \vert (u_{\epsilon }^{(m)})_{xx}\vert ^2\right) ^{\frac{1}{2}} (\epsilon ^2+\vert \nabla u_\epsilon ^{(m)}\vert ^2)^{\frac{p-1}{2}-\frac{q-2}{4}}\,dx \\&\le \beta _2 \int _{\Omega } b(\epsilon ^2+\vert \nabla u_\epsilon ^{(m)}\vert ^2)^{\frac{q-2}{2}} \vert (u_{\epsilon }^{(m)})_{xx}\vert ^2\,dx + C\int _{\Omega }(\epsilon ^2+\vert \nabla u_\epsilon ^{(m)}\vert ^2)^{\frac{2p-q}{2}}\,dx, \end{aligned} \end{aligned}$$

with an arbitrary $\beta _2>0$. The last integrals in the estimates for $\mathcal {K}^{(1)}_a$ and $\mathcal {K}^{(1)}_b$ are estimated by virtue of Lemma 4.5, provided that

$$\begin{aligned} p(z)< \underline{s}(z)+\rho , \quad 2p(z)-q(z)<\underline{s}(z)+\rho \end{aligned}$$

(6.12)

for some $\rho \in (r_*,r^\sharp )$, which is true by virtue of (2.19). Indeed: it is sufficient to claim that

$$\begin{aligned} \begin{aligned}&p(z)\le \overline{s}(z)<\underline{s}(z)+r^\sharp \quad \Leftrightarrow \qquad \overline{s}(z)-\underline{s}(z)<r^\sharp , \\&2p(z)-q(z)\le 2\overline{s}(z)-\underline{s}(z)<\underline{s}(z)+r^\sharp \qquad \Leftrightarrow \qquad 2(\overline{s}(z)-\underline{s}(z))<r^\sharp =2r_*. \end{aligned} \end{aligned}$$

Since $\rho \in (r_*,r^\sharp )$, there exists $\varsigma \in (0,r_*)$ such that $\rho = r^\sharp -\varsigma$ and (6.12) holds. Splitting $\Omega$ into two parts,

$$\begin{aligned} \Omega _-=\{x\in : \vert \nabla u_\epsilon ^{(m)}\vert <\epsilon \},\qquad \Omega ^+=\Omega {\setminus } \Omega _-, \end{aligned}$$

we estimate the last integral in the estimate for $\mathcal {K}^{(1)}_a$ with the use of Lemma 4.5 and (4.15):

$$\begin{aligned} \begin{aligned} \int _{\Omega }(\epsilon ^2+\vert \nabla u_\epsilon ^{(m)}\vert ^{2})^{\frac{p}{2}}\,dx&\le C+ \int _{\Omega }(\epsilon ^2+\vert \nabla u_\epsilon ^{(m)}\vert ^{2})^{\frac{\underline{s}+r^\sharp -\varsigma }{2}}\,dx\\&\le C+\frac{2}{\alpha }\int _{\Omega }\mathcal {F}_{\epsilon }^{(r_1-\varsigma , r_2-\varsigma )}(z,\nabla u_{\epsilon }^{(m)})\vert \nabla u_{\epsilon }^{(m)}\vert ^2\,dx \\&\le C'+\beta _3\int _{\Omega }\mathcal {F}_{\epsilon }^{(0,0)}(z,\nabla u_{\epsilon }^{(m)})\vert (u_{\epsilon }^{(m)})_{xx}\vert ^2\,dx \end{aligned} \end{aligned}$$

with an arbitrary $\beta _3>0$ and C, $C'$ independent of m and $\epsilon$. The corresponding term in the estimate on $\mathcal {K}^{(1)}_b$ is estimated similarly because

$$\begin{aligned} \int _{\Omega }(\epsilon ^2+\vert \nabla u_\epsilon ^{(m)}\vert ^{2})^{\frac{2p-q}{2}}\,dx \le C+ \int _{\Omega }(\epsilon ^2+\vert \nabla u_\epsilon ^{(m)}\vert ^{2})^{\frac{\underline{s}+r^\sharp -\varsigma }{2}}\,dx. \end{aligned}$$

Estimating of $\mathcal {I}_2$ is practically identical to estimating $\mathcal {I}_1$, the only difference is that the exponents p and q should be replaced. Gathering the above estimates we conclude that for every $\beta _{3}\in (0,1)$

$$\begin{aligned} \vert P_{a,b}\vert \le \beta _{3}\int _{\Omega }\mathcal {F}_{\epsilon }^{(0,0)}(z,\nabla u_\epsilon ^{(m)})\vert (u_{\epsilon }^{(m)})_{xx}\vert ^2\,dx +C \end{aligned}$$

with a constant $C=C(\beta _3)$ independent of m and $\epsilon$.

Step 4: estimate on $P_{\partial \Omega }$. To estimate $P_{\partial \Omega }$ we use Lemma 4.6 and Theorem 4.2:

$$\begin{aligned} \begin{aligned} \vert P_{\partial \Omega }\vert&\le \left| \int _{\partial \Omega } \mathcal {F}^{(0,0)}_{\epsilon }(z,\nabla u_{\epsilon }^{(m)})\left( \Delta u^{(m)}_\epsilon (\nabla u^{(m)}_\epsilon \cdot \textbf{n})- \nabla u^{(m)}_\epsilon \cdot \nabla (\nabla u^{(m)}_\epsilon \cdot \textbf{n})\right) \,dS \right| \\&\le C \int _{\partial \Omega } \mathcal {F}^{(0,0)}_{\epsilon }(z,\nabla u_{\epsilon }^{(m)})\vert \nabla u_\epsilon ^{(m)}\vert ^2 \,dS\\&\le \beta _4 \int _{\Omega } \mathcal {F}^{(0,0)}_{\epsilon }(z, \nabla u_\epsilon ^{(m)}) \vert (u_\epsilon ^{(m)})_{xx}\vert ^2 \,dx \\&\quad + C\left( 1 +\int _{\Omega } \mathcal {F}_0^{(0,0)}(z,\nabla u_\epsilon ^{(m)})\vert \nabla u_\epsilon ^{(m)}\vert ^2\,dx \right) \end{aligned} \end{aligned}$$

with an arbitrary $\beta _4 \in (0,1)$ and C depending upon $\beta _4$, p, q, a, b, $\partial \Omega$ and their differential properties, but not on $\epsilon$ and m. To complete the proof and obtain (6.6), we gather the estimates on $P_1$, $P_2$, $P_{a,b}$, $P_{\partial \Omega }$ and choose $\beta _i$ so small that

$$\begin{aligned} \min \{1,s^--1\}-\sum _{i=0}^4\beta _i>0. \end{aligned}$$

$\square$

Lemma 6.3

Under the conditions of Lemma 6.2

$$\begin{aligned} \begin{aligned}&\sup _{(0,T)}\Vert \nabla u^{(m)}_\epsilon {(\cdot , t)}\Vert ^2_{2,\Omega } + \int _{Q_T} \mathcal {F}^{(0,0)}_{\epsilon }(z, \nabla u_\epsilon ^{(m)}) \vert (u^{(m)}_\epsilon )_{xx}\vert ^2 \,dz \\&\qquad \le C \left( 1+\Vert \nabla u_{0}\Vert _{2,\Omega }^{2}+ \Vert f\Vert _{L^{2}(0,T;W^{1,2}_0(\Omega ))}^2\right) \end{aligned} \end{aligned}$$

(6.13)

and for any $r\in (0,r^\sharp )$

$$\begin{aligned} \int _{Q_T}\vert \nabla u_{\epsilon }^{(m)}\vert ^{\underline{s}(z)+r} \,dz\le C', \end{aligned}$$

(6.14)

with independent of m and $\epsilon$ constants C, $C'$.

Proof

Let us multiply (6.6) by $\textrm{e}^{-2C_1t}$, integrate the result over the interval $(0,\tau )$ with $\tau \le T$, and apply (6.1) to estimate the right-hand side:

$$\begin{aligned} \begin{aligned}&\Vert \nabla u^{(m)}_\epsilon {(\cdot , \tau )}\Vert ^2_{2,\Omega } + C_0 \int _{Q_\tau } \mathcal {F}^{(0,0)}_{\epsilon }(z, \nabla u_\epsilon ^{(m)}) \vert (u^{(m)}_\epsilon )_{xx}\vert ^2 \,dz \\&\quad \le C\left( 1+\Vert u_0\Vert ^2_{W^{1,2}_0(\Omega )}+\Vert f\Vert ^2_{L^{2}(0,\tau ;W^{1,2}_0(\Omega ))}\right) . \end{aligned} \end{aligned}$$

Since $\tau \in (0,T)$ is arbitrary, inequality (6.13) follows. Estimate (6.14) is an immediate consequence of Theorem 4.1 and inequality (3.3): for every $\varsigma \in (0,r_*)$ and $r_1(z)$, $r_2(z)$ from the conditions of Theorem 4.1

$$\begin{aligned} \begin{aligned}&\int _{Q_T}\mathcal {F}_{0}^{(r_1-\varsigma , r_2-\varsigma )} (z,\nabla u_{\epsilon }^{(m)})\vert \nabla u_{\epsilon }^{(m)}\vert ^2\,dz \\&\qquad \le C+\int _{Q_T}\mathcal {F}_{\epsilon }^{(r_1-\varsigma , r_2-\varsigma )}(z,\nabla u_{\epsilon }^{(m)})\vert \nabla u_{\epsilon }^{(m)}\vert ^2\,dz \\&\qquad \le \delta \int _{Q_T} \mathcal {F}^{(0,0)}_{\epsilon }(z, \nabla u_\epsilon ^{(m)}) \vert (u^{(m)}_\epsilon )_{xx}\vert ^2 \,dz +C. \end{aligned} \end{aligned}$$

The case $\varsigma \in [r_*, r^\sharp )$, is reduced to the considered one as in Lemma 4.5. $\square$

Corollary 6.1

Under the conditions of Lemma 6.3

$$\begin{aligned} \begin{aligned}&\Vert (\epsilon ^2+\vert \nabla u_\epsilon ^{(m)}\vert ^2)^{\frac{p(z)-2}{2}}\nabla u_\epsilon ^{(m)}\Vert _{(\overline{s}(\cdot ))',Q_T} \\&\qquad + \Vert (\epsilon ^2+\vert \nabla u_\epsilon ^{(m)}\vert ^2)^{\frac{q(z)-2}{2}}\nabla u_\epsilon ^{(m)}\Vert _{(\overline{s}(\cdot ))^{'},Q_T}\le C \end{aligned} \end{aligned}$$

(6.15)

with a constant C independent of m and $\epsilon$.

Proof

Condition (2.19) entails the inequality

$$\begin{aligned} \max \left\{ \frac{\overline{s}(z)(p(z)-1)}{\overline{s}(z)-1}, \frac{\overline{s}(z)(q(z)-1)}{\overline{s}(z)-1}\right\} \le \overline{s}(z)\le \underline{s}(z)+\varsigma \quad \text {for some} \ \varsigma \in (0, r^\sharp ). \end{aligned}$$

By Young’s inequality, the assertion follows then from (6.14), (3.1) and (3.3):

$$\begin{aligned} \begin{aligned}&\int _{Q_T}(\epsilon ^2+\vert \nabla u_\epsilon ^{(m)}\vert ^2)^{\frac{\overline{s}(z)(p(z)-1)}{2(\overline{s}(z)-1)}}\,dz + \int _{Q_T}(\epsilon ^2+\vert \nabla u_\epsilon ^{(m)}\vert ^2)^{\frac{\overline{s}(z)(q(z)-1)}{2(\overline{s}(z)-1})}\,dz \\&\qquad \le C\left( 1+\int _{Q_T}\vert \nabla u_\epsilon ^{(m)}\vert ^{\underline{s}(z)+\varsigma }\,dz\right) \le C. \end{aligned} \end{aligned}$$

$\square$

Lemma 6.4

Assume the conditions of Lemma 6.2. If (5.9) holds, then

$$\begin{aligned} \begin{aligned} \Vert (u^{(m)}_\epsilon )_t\Vert ^2_{2,Q_T} +&\,\sup _{(0,T)}\int _{\Omega } \left( a(z)(\epsilon ^2+\vert \nabla u_{\epsilon }^{(m)}\vert ^2)^{\frac{p(z)}{2}} + b(z)(\epsilon ^2+\vert \nabla u_{\epsilon }^{(m)}\vert ^2)^{\frac{q(z)}{2}}\right) \,dx \\&\le C\left( 1+ \int _\Omega \mathcal {F}_0^{(0,0)}((x,0),\nabla u_0)\vert \nabla u_0\vert ^2\,dx\right) +\Vert f\Vert _{2,Q_T}^2 \end{aligned} \end{aligned}$$

(6.16)

with an independent of m and $\epsilon$ constant C.

Proof

Multiplying (5.8) by $(u^{(m)}_j)_t$ and summing over $j=1,2,\dots ,m$ we obtain the equality

$$\begin{aligned} \int _{\Omega } (u^{(m)}_\epsilon )^2_t \,dx + \int _{\Omega } \mathcal {F}^{(0,0)}_{\epsilon }(z, \nabla u_\epsilon ^{(m)}) \nabla u^{(m)}_\epsilon \cdot \nabla (u^{(m)}_\epsilon )_t \,dx = \int _{\Omega } f (u^{(m)}_\epsilon )_t \,dx.\qquad \end{aligned}$$

(6.17)

The straighforward computation leads to the equality

$$\begin{aligned} \begin{aligned}&\mathcal {F}^{(0,0)}_{\epsilon }(z, \nabla u_\epsilon ^{(m)}) \nabla u^{(m)}_\epsilon \cdot \nabla (u^{(m)}_\epsilon )_t \\&\quad = \frac{d}{dt} \left( \frac{ a(\epsilon ^2 + \vert \nabla u^{(m)}_\epsilon \vert ^2)^{\frac{p}{2}}}{p} + \frac{ b(\epsilon ^2 + \vert \nabla u^{(m)}_\epsilon \vert ^2)^{\frac{q}{2}}}{q}\right) \\&\qquad + \frac{a p_t (\epsilon ^2 + \vert \nabla u^{(m)}_\epsilon \vert ^2)^{\frac{p}{2}}}{{p^2}} \left( 1- \frac{p}{2} \ln ((\epsilon ^2 + \vert \nabla u^{(m)}_\epsilon \vert ^2))\right) - \frac{ a_t (\epsilon ^2 + \vert \nabla u_\epsilon ^{(m)}\vert ^2)^\frac{p}{2}}{p}\\&\qquad + \frac{b q_t (\epsilon ^2 + \vert \nabla u^{(m)}_\epsilon \vert ^2)^{\frac{q}{2}}}{{q^2}} \left( 1- \frac{q}{2} \ln ((\epsilon ^2 + \vert \nabla u^{(m)}_\epsilon \vert ^2))\right) - \frac{ b_t (\epsilon ^2 + \vert \nabla u_\epsilon ^{(m)}\vert ^2)^\frac{q}{2}}{q}. \end{aligned} \end{aligned}$$

Using this equality we rewrite (6.17) as

$$\begin{aligned}{} & {} \Vert (u^{(m)}_\epsilon )_t(\cdot ,t)\Vert ^2_{2,\Omega } + \frac{d}{dt}\int _{\Omega } \left( \frac{a(z)(\epsilon ^2 + \vert \nabla u^{(m)}_\epsilon \vert ^2)^{\frac{p(z)}{2}}}{p(z)} + \frac{b(z) (\epsilon ^2 + \vert \nabla u^{(m)}_\epsilon \vert ^2)^{\frac{q(z)}{2}}}{q(z)}\right) \,dx\nonumber \\{} & {} \quad =\int _{\Omega } f (u^{(m)}_\epsilon )_t \,dx - \int _{\Omega } \frac{a(z) p_t (\epsilon ^2 + \vert \nabla u^{(m)}_\epsilon \vert ^2)^{\frac{p(z)}{2}}}{p^2(z)} \left( 1- \frac{p(z)}{2} \ln (\epsilon ^2 + \vert \nabla u^{(m)}_\epsilon \vert ^2)\right) \,dx\nonumber \\{} & {} \qquad - \int _{\Omega } \frac{b(z) q_t(z) (\epsilon ^2 + \vert \nabla u^{(m)}_\epsilon \vert ^2)^{\frac{q(z)}{2}}}{q^2(z)} \left( 1- \frac{q(z)}{2} \ln ((\epsilon ^2 + \vert \nabla u^{(m)}_\epsilon \vert ^2))\right) \,dx \nonumber \\{} & {} \qquad + \int _{\Omega } \frac{a_t (\epsilon ^2 + \vert \nabla u_\epsilon ^{(m)}\vert ^2)^\frac{p(z)}{2}}{p(z)} \,dx + \int _{\Omega } \frac{b_t (\epsilon ^2 + \vert \nabla u_\epsilon ^{(m)}\vert ^2)^\frac{q(z)}{2}}{q(z)} \,dx \nonumber \\{} & {} \quad \equiv \int _{\Omega } f (u^{(m)}_\epsilon )_t \,dx +\mathcal {I}_1+\mathcal {I}_2+\mathcal {I}_3 +\mathcal {I}_4. \end{aligned}$$

(6.18)

The first term on the right-hand side of (6.18) is estimated by the Cauchy inequality:

$$\begin{aligned} \left| \int _{\Omega } f (u^{(m)}_\epsilon )_t \,dx\right| \le \frac{1}{2} \Vert (u^{(m)}_\epsilon )_t {(\cdot ,t)}\Vert ^2_{2,\Omega } + \frac{1}{2}\Vert f{(\cdot ,t)}\Vert ^2_{2,\Omega }. \end{aligned}$$

(6.19)

To estimate $\mathcal {I}_i$, fix a number $\ell \in (r_*, r^{\sharp })$ such that

$$\begin{aligned} \overline{s}(z)< \underline{s}(z) + \ell < \underline{s}(z) + r^{\sharp }. \end{aligned}$$

Applying the Young inequality and using (3.4) in the terms with the logarithmic growth we obtain:

$$\begin{aligned} \begin{aligned} \sum _{i=1}^{4}\vert \mathcal {I}_i\vert&\le C\left( 1+ \int _{\Omega } \vert \nabla u_\epsilon ^{(m)}\vert ^{\underline{s}(z) + \ell } \,dx + \int _{\Omega } \mathcal {F}_{\epsilon }^{(r_1-\varsigma , r_2-\varsigma )}(z,\nabla u_{\epsilon }^{(m)})\vert \nabla u_{\epsilon }^{(m)}\vert ^2 \,dx \right) \end{aligned} \end{aligned}$$

with $r_i$ defined in (4.11). The required inequality (6.16) follows after gathering the above estimates, integrating the result in t, and using (5.9) and (6.14).$\square$

7 The regularized problem

We are now in position to prove the existence and uniqueness of strong solutions of the regularized problem (5.1) via passing to the limit as $m \rightarrow \infty$ in the equation satisfied by the functions $u^{(m)}_\epsilon$. We proceed in two steps. In the first step we prove the existence and establish the regularity properties of solutions to the problems in a smooth domain $\Omega$, in the second step we will extend these results to the domains with $\partial \Omega \in C^2$.

Let us assume that $\partial \Omega \in C^{k}$ with $k\ge 2+[\frac{N}{2}]$ and $\epsilon \in (0,1)$ is a fixed parameter. Under the assumptions of Theorem 2.1 on the rest of the data, there exists a sequence of Galerkin approximations $\{u^{(m)}_\epsilon \}$ given by (5.7) with the coefficients $u_j^{(m)}(t)$ defined from the system of ordinary differential equations (5.8). The functions $u^{(m)}(z)$ satisfy estimates (6.1), (6.2), (6.13), (6.14), (6.15) and (6.16). The finite-dimensional approximations $u_{\epsilon }^{(m)}$ are constructed in the cylinders $\Omega \times (0,T_m)$, but the uniform estimates (6.1) and (6.16) mean that $u_{\epsilon }^{(m)}(\cdot ,T_m)\in W^{1,2}_0(\Omega )\cap W^{1,\mathcal {H}}_0(\Omega )$. This inclusion allows one to continue $u_{\epsilon }^{(m)}$ to an interval $(T_m,T_m+h)$. Continuing this procedure we exhaust the interval (0, T) and obtain the approximate solution $u_\epsilon ^{(m)}$ on the whole of the cylinder $Q_T$.

7.1 Existence and uniqueness of strong solution

Theorem 7.1

Let $u_0$, f, p, q, a satisfy the conditions of Theorem 2.1 with $\partial \Omega \in C^k$, $k\ge 2+[\frac{N}{2}]$. Then for every $\epsilon \in (0,1)$ problem (5.1) has a unique solution $u_\epsilon$ which satisfies the estimates

$$\begin{aligned} \begin{aligned}&\Vert u_\epsilon \Vert _{{ \mathbb {W}_{\overline{s}(\cdot )}(Q_T)}}\le C_0, \\&{\text {ess}}\sup _{(0,T)}\Vert u_\epsilon {(\cdot , t)}\Vert _{2,\Omega }^{2} + \Vert u_{\epsilon t}\Vert _{2,Q_T}^{2}+ {\text {ess}}\sup _{(0,T)}\Vert \nabla u_\epsilon {(\cdot , t)}\Vert _{2,\Omega }^{2} \\&+ {\text {ess}}\sup _{(0,T)}\int _{\Omega } \left( a(z)(\epsilon ^2+\vert \nabla u_\epsilon \vert ^2)^{\frac{p(z)}{2}}+b(z)(\epsilon ^2+\vert \nabla u_\epsilon \vert ^2)^{\frac{q(z)}{2}}\right) dx\le C_1. \end{aligned}\nonumber \\ \end{aligned}$$

(7.1)

Moreover, the solution $u_\epsilon$ has the following property of global higher integrability of the gradient: for every $r\in (0,r^\sharp )$, $r^\sharp =\frac{4}{N+2}$,

$$\begin{aligned} \int _{Q_T} \vert \nabla u_{\epsilon }\vert ^{\underline{s}(z)+r} \, dz \le C_2. \end{aligned}$$

(7.2)

The constants $C_0$, $C_1$, $C_2$ depend on the data but are independent of $\epsilon$.

Proof

The uniform with respect to m and $\epsilon$ estimates (6.1), (6.2), (6.13), (6.14), (6.15) and (6.16) enable one to extract a subsequence $u^{(m)}_{\epsilon }$ (for which we keep the same notation), and functions $u_\epsilon$, $\mathcal {A}_{1, \epsilon }$, $\mathcal {A}_{2, \epsilon }$ such that

$$\begin{aligned} \begin{aligned}&u^{(m)}_\epsilon \rightarrow u_\epsilon \quad \star \text {-weakly in}\, L^\infty (0,T;L^2(\Omega )), \quad (u^{(m)}_{\epsilon })_t \rightharpoonup (u_{\epsilon })_t\text { in }L^2(Q_T), \\&\nabla u^{(m)}_\epsilon \rightharpoonup \nabla u_\epsilon \hbox {in }(L^{r(\cdot )}(Q_T))^N, \quad r(z)= \max \{2, \overline{s}(z)\}, \\&\beta _\epsilon ^{\frac{p(z)-2}{2}}(\nabla u^{(m)}_\epsilon ) \nabla u^{(m)}_\epsilon \rightharpoonup \mathcal {A}_{1, \epsilon }\hbox { in } (L^{\overline{s}'(\cdot )}(Q_T))^N, \\&\beta _\epsilon ^{\frac{q(z)-2}{2}}(\nabla u^{(m)}_\epsilon ) \nabla u^{(m)}_\epsilon \rightharpoonup \mathcal {A}_{2, \epsilon }\hbox { in } (L^{\overline{s}'(\cdot )}(Q_T))^N, \end{aligned}\nonumber \\ \end{aligned}$$

(7.3)

where the third and fourth lines follow from the uniform estimate (6.15). The continuous Sobolev’s embedding implies uniform boundedness of the functions $u^{(m)}_\epsilon$ and $(u^{(m)}_\epsilon )_t$ in $L^{\infty }(0,T;W_0^{1,\underline{s}^-}(\Omega ))$ and $L^2(0,T; L^2(\Omega ))$ respectively, while $W_0^{1,r(\cdot ,t)}(\Omega ) \subseteq W^{1,\underline{s}^-}_0(\Omega )\hookrightarrow L^2(\Omega )$. By [45, Sec.8, Corollary 4] the sequence $\{u^{(m)}_\epsilon \}$ is relatively compact in $C^0([0,T];L^2(\Omega ))$ and there exists a subsequence $\{u^{(m_k)}_\epsilon \}$, which we also assume coinciding with $\{u^{(m)}_{\epsilon }\}$, such that $u^{(m)}_\epsilon \rightarrow u_\epsilon$ in $C([0,T];L^2(\Omega ))$ and a.e. in $Q_T.$ Let $\mathcal {N}_m$ be the finite-dimensional sets defined in (5.5). Fix some $m\in \mathbb {N}$. By the method of construction $u^{(m)}_\epsilon \in \mathcal {N}_m$. Since $\mathcal {N}_{k}\subset \mathcal {N}_{m}$ for $k<m$, then for every $\xi _k\in \mathcal {N}_k$ with $k\le m$

$$\begin{aligned} \int _{Q_T} u^{(m)}_{\epsilon t} \xi _k \,dz + \int _{Q_T} \mathcal {F}^{(0,0)}_{\epsilon }(z, \nabla u_\epsilon ^{(m)}) \nabla u^{(m)}_\epsilon \cdot \nabla \xi _k \,dz = \int _{Q_T} f \xi _k \,dz. \end{aligned}$$

(7.4)

Since $\bigcup \limits _{k \in \mathbb {N}} \mathcal {N}_k$ is dense in $\mathbb {W}_{\overline{s}(\cdot )}(Q_T)$, for every $\xi \in \mathbb {W}_{\overline{s}(\cdot )}(Q_T)$ there exists a sequence such that $\xi _k\in \mathcal {N}_k$ and $\mathcal {N}_k\ni \xi _k\rightarrow \xi \in \mathbb {W}_{\overline{s}(\cdot )}(Q_T)$. If $\zeta _m\rightharpoonup \zeta$ in $L^{\overline{s}'(\cdot )}(Q_T)$, then for every $\eta \in L^{\overline{s}(\cdot )}(Q_T)$ we have

$$\begin{aligned} a(z) \eta \in L^{\overline{s}(\cdot )}(Q_T) \quad \text {and}\quad \displaystyle \int _{Q_T}a \zeta _m \eta \,dz\rightarrow \int _{Q_T}a \zeta \eta \,dz. \end{aligned}$$

Using this fact we pass to the limit as $m\rightarrow \infty$ in (7.4) with a fixed k, and then letting $k\rightarrow \infty$, we conclude that

$$\begin{aligned} \int _{Q_T} u_{\epsilon t} \xi \,dz + \int _{Q_T} a(z) \ \mathcal {A}_{1, \epsilon } \cdot \nabla \xi \,dz + \int _{Q_T} b(z) \ \mathcal {A}_{2, \epsilon } \cdot \nabla \xi \,dz = \int _{Q_T} f \xi \,dz \end{aligned}$$

(7.5)

for all $\xi \in {\mathbb {W}_{\overline{s}(\cdot )}(Q_T)}$. The classical argument based on monotonicity of the flux function (see Proposition 3.1), the uniform estimates (6.14), and the density of $\bigcup \limits _{k \in \mathbb {N}} \mathcal {N}_k$ in ${\mathbb {W}_{\overline{s}(\cdot )}(Q_T)}$, allow one to identify the limit vectors $\mathcal {A}_{1, \epsilon }$ and $\mathcal {A}_{2, \epsilon }$ as follows (see, e.g., [8, Theorem 6.1] for the details): for every $\psi \in \mathbb {W}_{\overline{s}(\cdot )}(Q_T)$

$$\begin{aligned} \int _{Q_T}\left( a(z) \mathcal {A}_{1, \epsilon } + b(z) \mathcal {A}_{2, \epsilon }\right) \cdot \nabla \psi \,dz= \int _{Q_T}\mathcal {F}^{(0,0)}_{\epsilon }(z, \nabla u_\epsilon ) \nabla u_{\epsilon } \cdot \nabla \psi \,dz. \end{aligned}$$

The initial condition for $u_\epsilon$ is fulfilled by continuity because $u_\epsilon \in C^0([0,T];L^{2}(\Omega ))$.

To prove the uniqueness we argue by contradiction. Let v, w be two solutions of problem (5.1). Take an arbitrary $\rho \in (0,T]$. The uniform estimates (6.14) allow us to take $v-w \in \mathbb {W}_{\overline{s}(\cdot )}(Q_T)$ for the test function in equalities (5.2) for v and w in the cylinder $Q_\rho =\Omega \times (0,\rho )$. Subtracting these equalities and using the monotonicity of the flux (see Proposition 3.1) we arrive at the inequality

$$\begin{aligned} \frac{1}{2}\Vert v-w\Vert _{2,\Omega }^{2}(\rho ) =\int _{Q_\tau }(v-w)(v-w)_t\,dz\le 0. \end{aligned}$$

It follows that $v(x,\rho )=w(x,\rho )$ a.e. in $\Omega$ for every $\rho \in [0,T]$.

Estimates (7.1) follow from the uniform in m estimates on the functions $u_{\epsilon }^{(m)}$ and their derivatives, the properties of weak convergence (7.3), and the lower semicontinuity of the modular. Inequality (6.14) yields that for every $\ell \in (0,r^\sharp )$ the sequence $\{\nabla u_{\epsilon }^{(m)}\}$ contains a subsequence which converges to $\nabla u_\epsilon$ weakly in $(L^{\underline{s}(\cdot )+\ell }(Q_T))^N$, whence (7.2).$\square$

7.2 Strong and almost everywhere convergence of the gradients

Lemma 7.1

[Strong convergence of $\{\nabla u_{\epsilon }^{(m)}\}$] Assume the conditions of Theorem 7.1. There is a subsequence of the sequence $\{u_{\epsilon }^{(m)}\}$ such that

$$\begin{aligned} \nabla u_{\epsilon }^{(m)}\rightarrow \nabla u_\epsilon \quad \text {in } (L^{\underline{s}(\cdot )}(Q_T))^N\hbox { and a.e. in }Q_T\hbox { as }m\rightarrow \infty . \end{aligned}$$

Proof

The sequence $\{u_\epsilon ^{(m)}\}$ possesses the convergence properties (7.3). It follows from the weak convergence $\nabla u_\epsilon ^{(m)}\rightharpoonup \nabla u_\epsilon$ in $L^{\overline{s}(\cdot )}(Q_T)$, the strong convergence $u_\epsilon ^{(m)}\rightarrow u_{\epsilon }$ in $L^{2}(Q_T)$, and the Mazur Lemma, see [15, Corollary 3.8, Chapter 3], that there exists a sequence $\{v_\epsilon ^{(m)}\}$ such that $v_\epsilon ^{(m)}\in \mathcal {N}_m$, each $v_\epsilon ^{(m)}$ is a convex combination of $\{u_\epsilon ^{(1)},u_\epsilon ^{(2)},\ldots ,u_\epsilon ^{(m)}\}$, and

$$\begin{aligned} v_\epsilon ^{(m)}\rightarrow u_{\epsilon }\hbox { in } \mathbb {W}_{\overline{s}(\cdot )}(Q_T). \end{aligned}$$

(7.6)

Let us define $w_m\in \mathcal {N}_m$ as follows:

$$\begin{aligned} \Vert u_\epsilon -w_m\Vert _{\mathbb {W}_{\overline{s}(\cdot )}(Q_T)}= {\text {dist}}(u_\epsilon ,\mathcal {N}_m)\equiv \min \{\Vert u_\epsilon -w\Vert _{\mathbb {W}_{\overline{s}(\cdot )}(Q_T)}: \,w\in \mathcal {N}_m\}. \end{aligned}$$

Because of (7.6) such $w_m$ exists and

$$\begin{aligned} {\text {dist}}(u_\epsilon ,\mathcal {N}_m)=\Vert u_\epsilon -w_m\Vert _{\mathbb {W}_{\overline{s}(\cdot )}(Q_T)}\le \Vert u_\epsilon -v_\epsilon ^{(m)}\Vert _{\mathbb {W}_{\overline{s}(\cdot )}(Q_T)}\rightarrow 0\quad \text {as }m\rightarrow \infty . \end{aligned}$$

(7.7)

By the properties of the modular and the strong convergence $u_\epsilon ^{(m)}\rightarrow u_{\epsilon }$ in $L^{2}(Q_T)$ we also have

$$\begin{aligned} \begin{aligned}&\Vert u_\epsilon ^{(m)}-w_m\Vert _{2,Q_T}^2 +\int _{Q_T}\vert \nabla (u_\epsilon - w_m)\vert ^{\overline{s}(z)}\,dz \\&\qquad \le 2\Vert u_\epsilon ^{(m)}-u_\epsilon \Vert ^2_{2,Q_T}+2\Vert u_{\epsilon }-w_m\Vert ^2_{2,Q_T}+ \int _{Q_T}\vert \nabla (u_\epsilon - w_m)\vert ^{\overline{s}(z)}\,dz \rightarrow 0 \end{aligned} \end{aligned}$$

(7.8)

as $m\rightarrow \infty$. Gathering (5.2) and (5.8) with the test function $u_\epsilon ^{(m)} \in \mathcal {N}_m$ we obtain the equality

$$\begin{aligned} \begin{aligned}&\int _{Q_T} \left( \mathcal {F}_{\epsilon }^{(0,0)}(z,\nabla u_\epsilon ^{(m)})\nabla u_\epsilon ^{(m)}-\mathcal {F}_{\epsilon }^{(0,0)}(z,\nabla u_\epsilon )\nabla u_{\epsilon }\right) \nabla u_\epsilon ^{(m)}\,dz \\&\qquad =-\int _{Q_T}((u_\epsilon ^{(m)})_t-u_{\epsilon t})u_\epsilon ^{(m)}\,dz, \end{aligned} \end{aligned}$$

which can be written in the form

$$\begin{aligned} \begin{aligned}&\int _{Q_T} \left( \mathcal {F}_{\epsilon }^{(0,0)}(z,\nabla u_\epsilon ^{(m)})\nabla u_\epsilon ^{(m)}-\mathcal {F}_{\epsilon }^{(0,0)}(z,\nabla u_\epsilon )\nabla u_{\epsilon }\right) \nabla (u_\epsilon ^{(m)}-u_\epsilon )\,dz \\&\qquad =-\int _{Q_T}((u_\epsilon ^{(m)})_t-u_{\epsilon t})u_\epsilon ^{(m)}\,dz \\&\qquad \,\quad - \int _{Q_T} \left( \mathcal {F}^{(0,0)}_{\epsilon }(z,\nabla u_\epsilon ^{(m)})\nabla u_\epsilon ^{(m)}-\mathcal {F}^{(0,0)}_{\epsilon }(z,\nabla u_\epsilon )\nabla u_{\epsilon }\right) \nabla u_\epsilon \,dz. \end{aligned}\nonumber \\ \end{aligned}$$

(7.9)

Taking $w_m\in \mathcal {N}_m$ for the test-function in (5.2), (5.8), we find that

$$\begin{aligned} \begin{aligned} \int _{Q_T}&\left( \mathcal {F}^{(0,0)}_{\epsilon }(z,\nabla u_\epsilon ^{(m)})\nabla u_\epsilon ^{(m)}-\mathcal {F}^{(0,0)}_{\epsilon }(z,\nabla u_\epsilon )\nabla u_{\epsilon }\right) \nabla w_m\,dz \\&\quad +\int _{Q_T} ((u_\epsilon ^{(m)})_t-u_{\epsilon t})w_m\,dz=0. \end{aligned} \end{aligned}$$

(7.10)

Adding (7.10) to the right-hand side of (7.9) and using notation (3.6), we obtain the equality

$$\begin{aligned} \mathcal {G}_\epsilon (\nabla u_\epsilon ^{(m)},\nabla u_\epsilon )= & {} \, -\int _{Q_T} ((u_\epsilon ^{(m)})_t-u_{\epsilon t})(u_\epsilon ^{(m)}-w_m)\,dz\nonumber \\{} & {} + \int _{Q_T}\! \!\left( \mathcal {F}^{(0,0)}_{\epsilon }(z,\nabla u_\epsilon ^{(m)})\nabla u_\epsilon ^{(m)}\!-\!\mathcal {F}^{(0,0)}_{\epsilon }(z,\nabla u_\epsilon )\nabla u_{\epsilon }\right) \nabla (u_\epsilon -w_m)\,dz.\nonumber \\ \end{aligned}$$

(7.11)

The first term on the right-hand side of (7.11) tends to zero as $m\rightarrow \infty$ because $(u_{\epsilon }-u_{\epsilon }^{(m)})_t$ are uniformly bounded in $L^2(Q_T)$ and $\Vert u_\epsilon ^{(m)}-w_m\Vert _{2,Q_T}\rightarrow 0$ by the choice of $w_m$. By (6.15) and due to the choice of $w_m$, the second term of (7.11) is bounded by $C\Vert \nabla u_{\epsilon }-\nabla w_m\Vert _{\overline{s}(\cdot ),Q_T}$ and also tends to zero as $m\rightarrow \infty$. Hence, $\mathcal {G}_{\epsilon }(\nabla u_\epsilon ^{(m)},\nabla u_\epsilon )\rightarrow 0$ as $m\rightarrow \infty$. It follows now from Lemma 3.2 that

$$\begin{aligned} \int _{Q_T}\vert \nabla (u_{\epsilon }^{(m)}-u_\epsilon )\vert ^{\underline{s}(z)}\,dz\rightarrow 0\quad \text {when }m\rightarrow \infty . \end{aligned}$$

By the Riesz-Fischer Theorem $\nabla u^{(m)}_\epsilon \rightarrow \nabla u_{\epsilon }$ a.e. in $Q_T$ (up to a subsequence). $\square$

7.3 Second-order regularity

Theorem 7.2

Let the conditions of Theorem 7.1 hold. Then:

• (i) $(\mathcal {F}_{\epsilon }^{(0,0)})^\frac{1}{2}(z,\nabla u_{\epsilon })D_iu_{\epsilon }\in L^2(0,T;W^{1,2}(\Omega ))$, $i=1,2,\ldots ,N$, and

  $$\begin{aligned} \Vert (\mathcal {F}_{\epsilon }^{(0,0)})^{\frac{1}{2}}(z,\nabla u_{\epsilon }) D_iu_{\epsilon }\Vert _{L^2(0,T;W^{1,2}(\Omega ))}\le M,\quad i=1,2,\ldots ,N, \end{aligned}$$

  with an independent of $\epsilon$ constant M;

• (ii) $D_{ij}^2u_\epsilon \in L^{\underline{s}(\cdot )}_{loc}(Q_T\cap \{z:\,\max \{p(z), q(z)\}< 2\})$, $i,j=1,2,\ldots ,N$, and

  $$\begin{aligned} \sum _{i,j=1}^{N}\Vert D_{ij}^2u_\epsilon \Vert _{\underline{s}(\cdot ),Q_T\cap \{z:\,\max \{p(z), q(z)\}< 2\}}\le M' \end{aligned}$$

  with an independent of $\epsilon$ constant $M'$.

Proof

(i) The almost everywhere convergence $\nabla u_{\epsilon }^{(m)}\rightarrow \nabla u_{\epsilon }$ in $Q_T$ implies

$$\begin{aligned} (\mathcal {F}_\epsilon ^{(0,0)})^{\frac{1}{2}}(z,\nabla u_{\epsilon }^{(m)})\nabla u_{\epsilon }^{(m)}\rightarrow (\mathcal {F}_\epsilon ^{(0,0)})^{\frac{1}{2}}(z,\nabla u_{\epsilon })\nabla u_{\epsilon }\quad \hbox { a.e. in}\ Q_T. \end{aligned}$$

(7.12)

Due to the uniform estimates (6.13), (6.14), and inequality (3.4), for every $i,j=1,2,\ldots ,N$

$$\begin{aligned} \begin{aligned}&\left\| D_i \left( (\mathcal {F}_{\epsilon }^{(0,0)})^{\frac{1}{2}}(z,\nabla u_{\epsilon }^{(m)}) D_ju_{\epsilon }^{(m)}\right) \right\| _{2,Q_T}^{2} \le C\int _{Q_T}\mathcal {F}^{(0,0)}_{\epsilon }(z,\nabla u_{\epsilon }^{(m)})\vert u_{\epsilon xx}^{(m)}\vert ^2\,dz \\&\qquad \qquad +C'\int _{Q_T}\left( (\epsilon ^2+\vert \nabla u_{\epsilon }^{(m)}\vert ^2)^{\frac{p(z)}{2}}+(\epsilon ^2+\vert \nabla u_{\epsilon }^{(m)}\vert ^2)^{\frac{q(z)}{2}}\right) \vert \ln (\epsilon ^2+\vert \nabla u_{\epsilon }^{(m)}\vert ^2)\vert \,dz \\&\qquad \le C''\left( 1+ \int _{Q_T}\vert \nabla u_{\epsilon }^{(m)}\vert ^{\overline{s}(z)+\mu }\,dz\right) \le C'''\left( 1+ \int _{Q_T}\vert \nabla u_{\epsilon }^{(m)}\vert ^{\underline{s}(z)+r}\,dz\right) \le M \end{aligned} \end{aligned}$$

with some $r \in (r_*, r^{\sharp })$ and a constant M depending on the norm of $u_0$ in $W^{1,\mathcal {H}}_0(\Omega )$, the norm of f in $L^{2}(0,T;W^{1,2}_0(\Omega ))$, the constants in conditions (2.16), and N, $p^\pm$, $q^\pm$. When estimating the terms with the logarithmic growth, we used (3.4) with $\mu$ and r chosen according to the inequalities $0<\mu <r-r_*/2$, $r_*<r<r^\sharp$ and

$$\begin{aligned} \overline{s}(z) - \underline{s}(z) + \mu< r_*+ \mu \le r < r^\sharp . \end{aligned}$$

Therefore, there is a subsequence $\{u_\epsilon ^{(m_k)}\}$ (we may assume that it coincides with the whole sequence) and functions $\Theta _{ij}\in L^{2}(Q_T)$ such that

$$\begin{aligned} D_i \left( (\mathcal {F}_\epsilon ^{(0,0)})^{\frac{1}{2}}(z,\nabla u_{\epsilon }^{(m)})D_ju_\epsilon ^{(m)}\right) \rightharpoonup \Theta _{ij}\in L^{2}(Q_T)\quad \hbox { as}\ m\rightarrow \infty . \end{aligned}$$

The uniform global higher integrability of the gradients (7.2) implies the existence of $\delta >0$ such that

$$\begin{aligned} (\mathcal {F}_{\epsilon }^{(0,0)})^{\frac{1}{2}}(z,\nabla u_{\epsilon }^{(m)}) D_ju_{\epsilon }^{(m)}\quad \text {are uniformly bounded in }L^{2+\delta }(Q_T). \end{aligned}$$

By Lemma 7.1, $(\mathcal {F}_{\epsilon }^{(0,0)})^{\frac{1}{2}}(z,\nabla u_{\epsilon }^{(m)}) D_ju_{\epsilon }^{(m)}$ converge pointwise to $(\mathcal {F}_{\epsilon }^{(0,0)})^{\frac{1}{2}}(z,\nabla u_{\epsilon })$ $D_ju_{\epsilon }$. It follows then from the Vitali convergence theorem that

$$\begin{aligned} (\mathcal {F}_\epsilon ^{(0,0)})^{\frac{1}{2}}(z,\nabla u_{\epsilon }^{(m)}) D_ju_{\epsilon }^{(m)}\rightarrow (\mathcal {F}_\epsilon ^{(0,0)})^{\frac{1}{2}}(z,\nabla u_{\epsilon }) D_ju_{\epsilon }\hbox { in }L^{2}(Q_T). \end{aligned}$$

For every $\psi \in C^{\infty }(Q_T)$ with ${\text {supp}}\psi \Subset Q_T$ and $i,j=1,\ldots ,N$

$$\begin{aligned} \begin{aligned}&\left( D_i \left( (\mathcal {F}_\epsilon ^{(0,0)})^{\frac{1}{2}}(z,\nabla u_{\epsilon }^{(m)}) D_ju_{\epsilon }^{(m)}\right) ,\psi \right) _{2,Q_T} \\&\qquad \quad = - \left( (\mathcal {F}_\epsilon ^{(0,0)})^{\frac{1}{2}}(z,\nabla u_{\epsilon }^{(m)}) D_ju_{\epsilon }^{(m)},D_i\psi \right) _{2,Q_T} \\&\qquad \quad \rightarrow - \left( (\mathcal {F}_{\epsilon }^{(0,0)})^{\frac{1}{2}}(z,\nabla u_{\epsilon }) D_ju_{\epsilon },D_i\psi \right) _{2,Q_T} \end{aligned} \end{aligned}$$

as $m\rightarrow \infty$. Thus, it is necessary that

$$\begin{aligned} \Theta _{ij}=D_i\left( (\mathcal {F}_\epsilon ^{(0,0)})^{\frac{1}{2}}(z,\nabla u_{\epsilon }) D_ju_{\epsilon }\right) \in L^{2}(Q_T)\quad \text {and}\quad \Vert \Theta _{ij}\Vert _{2,Q_T}^2\le M. \end{aligned}$$

(ii) Let us denote $\mathcal {D}_{p,q}=Q_T\cap \{z:\,\max \{p(z), q(z)\} < 2\}$ and take a set $\mathcal {B} \Subset \mathcal {D}_{p,q}$. By Young’s inequality, (6.2), (6.13) and due to the assumption $a(z) + b(z) \ge \alpha >0$, for every $i,j=1,2,\ldots ,N$

$$\begin{aligned} \begin{aligned}&\int _{\mathcal {B}}\vert D^2_{ij}u_{\epsilon }^{(m)}\vert ^{\underline{s}(z)}\,dz \\&\qquad \le C\left( 1 + \int _{\mathcal {B}}a(z)\vert D^2_{ij}u_{\epsilon }^{(m)}\vert ^{p(z)}\,dz + \int _{\mathcal {B}}b(z)\vert D^2_{ij}u_{\epsilon }^{(m)}\vert ^{q(z)}\,dz\right) \\ \end{aligned} \end{aligned}$$

(7.13)

with a constant $C'$ independent of $\epsilon$ and m. Using the Young inequality we estimate the first integral on the right-hand side with the help of the inequality

$$\begin{aligned} \begin{aligned} a\vert D_{ij}^{2}u_{\epsilon }^{(m)}\vert ^{p}&=\dfrac{a^{1-\frac{p}{2}}}{(\epsilon ^2+\vert \nabla u_\epsilon ^{(m)}\vert ^2)^{\frac{p-2}{2}\frac{p}{2}}}\left( a(\epsilon ^2+\vert \nabla u_\epsilon ^{(m)}\vert ^2)^{\frac{p-2}{2}}\vert D_{ij}^{2}u_\epsilon ^{(m)}\vert ^2\right) ^{\frac{p}{2}} \\&\le C_1 (\epsilon ^2+\vert \nabla u_\epsilon ^{(m)}\vert ^2)^{\frac{p}{2}} + C_2 a(\epsilon ^2+\vert \nabla u_\epsilon ^{(m)}\vert ^2)^{\frac{p-2}{2}}\vert D_{ij}^{2}u_\epsilon ^{(m)}\vert ^2. \end{aligned} \end{aligned}$$

The same inequality with p and a replaced by q and b is applied to the second integral. By virtue of (7.2) and (6.13) the right-hand side of (7.13) is bounded uniformly with respect to m and $\epsilon$ by a constant C. It follows that there exist $\theta _{ij} \in L^{\underline{s}(\cdot )}(\mathcal {D}_{p,q})$ such that $D^{2}_{ij}u_{\epsilon }^{(m)}\rightharpoonup \theta _{ij}$ in $L^{\underline{s}(\cdot )}(\mathcal {D}_{p,q})$ (up to a subsequence). Since $\nabla u_\epsilon ^{(m)}\rightharpoonup \nabla u_{\epsilon }$ in $L^{r(\cdot )}(Q_T)$ with $r(z)= \max \{2, \overline{s}(z)\}$, then for every $\psi \in C^{\infty }_0(\mathcal {D}_{p,q})$

$$\begin{aligned} \begin{aligned} (\theta _{ij},\psi )_{2,Q_T}&=\lim _{m\rightarrow \infty }(D^{2}_{ij}u^{(m)}_\epsilon ,\psi )_{2,Q_T} \\&=- \lim _{m\rightarrow \infty }(D_{i}u^{(m)}_\epsilon ,D_j\psi )_{2, Q_T}=-(D_iu_\epsilon ,D_j\psi )_{2,Q_T}. \end{aligned} \end{aligned}$$

It follows that $\theta _{ij}=D^{2}_{ij}u_{\epsilon }$, and $\Vert D^{2}_{ij}u_{\epsilon }\Vert _{\underline{s}(\cdot ),\mathcal {B}}\le C$ by the lower semicontinuity of the modular.$\square$

7.4 Regularized problem with $\partial \Omega \in C^2$

Let $\partial \Omega \in C^2$. By Proposition 5.3 and due to the density of $C^{\infty }([0,T];C_0^{\infty }(\Omega )$ in $L^2(0,T;W_0^{1,2}(\Omega ))$ there exist sequences $\{v_{0\delta }\}$, $\{f_\delta \}$ with the following properties:

$$\begin{aligned} \begin{aligned}&v_{0\delta }\in C_0^{\infty }(\Omega ),\quad \hbox {} v_{0\delta }\rightarrow u_0\, \hbox {in}\, W^{1,\mathcal {H}}_0(\Omega ),\quad {\text {supp}}v_{0\delta }\Subset \Omega , \\&f_\delta \in C^{\infty }([0,T];C_0^{\infty }(\Omega )),\quad \hbox {} f_\delta \rightarrow f\, \hbox {in}\, L^2(0,T;W^{1,2}_0(\Omega )), \\&{\text {supp}}f_\delta (\cdot ,t)\Subset \Omega \;\;\hbox { for all}\ t\in (0,T). \end{aligned} \end{aligned}$$

Let $d(x)={\text {dist}}(x,\partial \Omega )\equiv \inf \{\vert x-y\vert :\;y\in \partial \Omega \}$ be the distance from the point $x\in \Omega$ to the boundary. By [35, Lemma 14.16] there exists $\mu >0$ such that $d \in C^2(\Gamma _\mu )$, where $\Gamma _\mu =\{x\in \overline{\Omega }: d(x)<\mu \}$. For $x\in \Gamma _{\mu /2}{\setminus } \Gamma _{\mu /4}=\{x\in \Omega :\,\mu /4<d(z)<\mu /2\}$ and $0<\sigma <\mu /4$ we consider the mollified distance $d_\sigma (x)=d(y)\star \phi _\sigma (x-y)$ where $\phi _\sigma (\cdot )$ denotes the Friedrichs mollifier. Since $d \in C^2(\Gamma _\mu )$, it follows from the well known properties of the mollifier that for $x\in \Gamma _\beta {\setminus } \Gamma _{\gamma }$, $0<\sigma<\gamma <\beta$, and the multi-index $\alpha$, $0\le \vert \alpha \vert \le 2$,

$$\begin{aligned} \vert D^\alpha _xd_\sigma (x)-D_x^{\alpha }d(x)\vert \le \max \{\vert D^\alpha _xd(x)-D_y^{\alpha }d(y)\vert :\,\vert x-y\vert<\sigma <\gamma \}\rightarrow 0 \end{aligned}$$

as $\sigma \rightarrow 0$. Moreover, $d_\sigma \in C^{\infty }\left( \Gamma _\beta {\setminus } \Gamma _\gamma \right)$. Let us take the sequence of disjoint intervals $J_k=(a_k,b_k)$ with the endpoints $a_k=\mu 2^{-(2k+1)}$, $b_k=\mu 2^{-2k}$, $k\in \mathbb {N}$, and the centers $c_k=(a_k+b_k)/2$. Let us also take the sequence of co-centered intervals

$$\begin{aligned} I_k=\left( c_k-\dfrac{L_k}{4},c_k+\dfrac{L_k}{4}\right) \Subset J_k,\qquad L_k=b_k-a_k. \end{aligned}$$

Let $0<\sigma <c_k-L_k/4$. By Sard’s theorem the set of critical values of $d_\sigma (x)$ in $I_k$ has zero measure. It follows that for every sufficiently large $k\in \mathbb {N}$ we may find $\delta _k\in I_k$ and $\sigma _k>0$ so small that

$$\begin{aligned} \Sigma _{\delta _k}=\{x\in \Omega :\,d_{\sigma _k}(x)=\delta _k\}\subset \{x\in \Omega :\,a_k<d(x)<b_k\}. \end{aligned}$$

The surfaces $\Sigma _{\delta _k}$ have no common points, are $C^\infty$-smooth, and their parametrizations are uniformly bounded in $C^2$. The domains $\Omega _{\delta _k}$ bounded by $\Sigma _{\delta _k}$ form an expanding sequence covering $\Omega$ when $k\rightarrow \infty$.

Let $\{u_{\epsilon ,\delta _k}\}$ be the sequence of strong solutions of problem (5.1) in the cylinders $Q^{(\delta _k)}_T=\Omega _{\delta _k}\times (0,T)$ with the data $v_{0\delta _k}$, $f_{\delta _k}$, ${\text {supp}} v_{0\delta _k}\Subset \Omega _{\delta _k}$, ${\text {supp}} f_{\delta _k}(\cdot ,t)\Subset \Omega _{\delta _k}$ for a.e. $t\in (0,T)$. By ${\widetilde{u}}_{\epsilon ,\delta _k}$ we denote the zero continuations of $u_{\epsilon ,\delta _k}$ from $Q_T^{(\delta _k)}$ to the cylinder $Q_T$. By Theorems 7.1, 7.2 the continued functions satisfy the uniform estimates

$$\begin{aligned} \begin{aligned} \mathrm {(i)}&\qquad \Vert \partial _t{\widetilde{u}}_{\epsilon ,\delta _k}\Vert _{2,Q_T} \\&\qquad \quad +{\text {ess}} \sup _{(0,T)}\int _{\Omega }\left( 1+\mathcal {F}_\epsilon ^{(0,0)}(z,\nabla {\widetilde{u}}_{\epsilon ,\delta _k})\right) \vert \nabla \widetilde{u}_{\epsilon ,\delta _k}\vert ^2\,dx \le C_{1}, \\ \mathrm {(ii)}&\qquad {\text {ess}} \sup _{(0,T)}\Vert {\widetilde{u}}_{\epsilon ,\delta _k}\Vert ^2_{2,\Omega } + \int _{Q_T}\mathcal {F}_\epsilon ^{(0,0)}(z,\nabla \widetilde{u}_{\epsilon ,\delta _k})\vert \nabla \widetilde{u}_{\epsilon ,\delta _k}\vert ^2\,dz\le C_{2}, \\ \mathrm {(iii)}&\qquad \left\| D_{x_i}\left( \left( \mathcal {F}_\epsilon ^{(0,0)}(z,\nabla {\widetilde{u}}_{\epsilon ,\delta _k})\right) ^{\frac{1}{2}}D_{x_j}\widetilde{u}_{\epsilon ,\delta _k}\right) \right\| _{2,Q_T}\le C_{3},\quad \\ \mathrm {(iv)}&\qquad \int _{Q_{T}}\vert \nabla \widetilde{u}_{\epsilon ,\delta _k}\vert ^{\underline{s}(z)+r}\,dz\le C_4,\qquad r\in (0,r^\sharp ), \end{aligned}\nonumber \\ \end{aligned}$$

(7.14)

with constants $C_i$ depending on the data, but independent of $\epsilon$ and $\delta _k$; $C_1$, $C_3$, $C_4$ depend also on the $C^2$-norm of the parametrization of $\partial \Omega$. Using (7.14) we may choose a subsequence with the following convergence properties:

$$\begin{aligned} \begin{aligned}&{\widetilde{u}}_{\epsilon ,\delta _k}\rightarrow u_{\epsilon } \star \hbox {-weakly in }L^{\infty }(0,T;L^2(\Omega )), \quad \partial _t{\widetilde{u}}_{\epsilon ,\delta _k}\rightharpoonup \partial _tu_{\epsilon }\hbox { in }L^2(Q_T), \\&\nabla {\widetilde{u}}_{\epsilon ,\delta _k}\rightharpoonup \nabla u_\epsilon \hbox { in }L^{\underline{s}(\cdot )+r}, \quad a(z)\vert \nabla {\widetilde{u}}_{\epsilon ,\delta _k}\vert ^{p-2}\nabla {\widetilde{u}}_{\epsilon ,\delta _k} \rightharpoonup \mathcal {A}_{1,\epsilon }\hbox { in }L^{p'(\cdot )}(Q_T), \\&b(z)\vert \nabla {\widetilde{u}}_{\epsilon ,\delta _k})\vert ^{q-2}\nabla {\widetilde{u}}_{\epsilon ,\delta _k} \rightharpoonup \mathcal {A}_{2,\epsilon }\hbox { in }L^{q'(\cdot )}(Q_T). \end{aligned} \end{aligned}$$

To identify the limits $\mathcal {A}_{i,\epsilon }$ and to prove that $u_{\epsilon }$ is a solution of problem (5.1) in $Q_T$ we imitate the proof of Theorem 7.1 and use Proposition 5.3. The proof of uniqueness does not require any changes.

The higher integrability of the gradient follows immediately from (7.14) (iv). To prove the second-order regularity we need the pointwise convergence of $\nabla {\widetilde{u}}_{\epsilon ,\delta _k}$ in $Q_T$. We mimic the proof of Lemma 7.1. By Mazur’s Lemma there is a sequence of convex combinations of $\{{\widetilde{u}}_{\epsilon ,\delta _1}, \widetilde{u}_{\epsilon ,\delta _2}, \dots ,{\widetilde{u}}_{\epsilon ,\delta _k}\}$ that converges to $u_\epsilon$ strongly in $\mathbb {W}_{\overline{s}(\cdot )}(Q_T)$. Let us denote this sequence by $\{w_{\delta _k}\}$, ${\text {supp}} w_{\delta _k}\subseteq Q_{T}^{(\delta _k)}$. Let $W_{\delta _k}\in \mathbb {W}_{\overline{s}(\cdot )}(Q_T)$ with ${\text {supp}}W_{\delta _k}\subseteq Q_T^{(\delta _k)}$ be defined as follows:

$$\begin{aligned} \begin{aligned}&\Vert W_{\delta _k}-u_\epsilon \Vert _{\mathbb {W}_{\overline{s}(\cdot )}(Q_T)} \\&\quad =\inf \left\{ \Vert w-u_\epsilon \Vert _{\mathbb {W}_{\overline{s}(\cdot )}(Q_T)}\,\vert \, w\in \mathbb {W}_{\overline{s}(\cdot )}(Q_T), \ {\text {supp}}w \subseteq Q_T^{(\delta _k)} \right\} \rightarrow 0\quad \text {as }\delta _k\rightarrow 0. \end{aligned} \end{aligned}$$

In the identities

$$\begin{aligned} \begin{aligned}&\int _{Q_T} \left( \phi \partial _tu_\epsilon +\mathcal {F}_{\epsilon }^{(0,0)}(z,\nabla u_\epsilon )\nabla u_{\epsilon }\cdot \nabla \phi \right) \,dz=\int _{Q_T}f\phi \,dz\qquad \phi \in \mathbb {W}_{\overline{s}(\cdot )}(Q_T), \\&\int _{Q_T} \left( \phi \partial _t\widetilde{u}_{\epsilon ,\mu }+\mathcal {F}_{\epsilon }^{(0,0)}(z,\nabla \widetilde{u}_{\epsilon ,\mu })\nabla {\widetilde{u}}_{\epsilon ,\mu }\cdot \nabla \phi \right) \,dz=\int _{Q_T}{\widetilde{f}}_{\mu }\phi \,dz \qquad \phi \in \mathbb {W}_{\overline{s}(\cdot )}(Q_T^{(\mu )}) \end{aligned} \end{aligned}$$

we may take for the test function $\phi =W_{\delta _k}-\widetilde{u}_{\epsilon ,\mu }$ with $\delta _k \ge \mu$, which means that ${\text {supp}}W_{\delta _k}\subseteq {\text {supp}}{\widetilde{u}}_{\epsilon ,\mu }$. Combining the results we obtain

$$\begin{aligned} \begin{aligned}&\mathcal {G}_{\epsilon }(\nabla u_\epsilon ,\nabla \widetilde{u}_{\epsilon ,\mu }) \\&\qquad = \int _{Q_T} \left( \mathcal {F}_{\epsilon }^{(0,0)}(z,\nabla u_{\epsilon })\nabla u_\epsilon - \mathcal {F}_{\epsilon }^{(0,0)}(z,\nabla {\widetilde{u}}_{\epsilon , \mu })\nabla {\widetilde{u}}_{\epsilon ,\mu }\right) \cdot \nabla (u_\epsilon -{\widetilde{u}}_{\epsilon ,\mu })\,dz \\&\qquad = \int _{Q_T}(f-{\widetilde{f}}_{\mu })(W_{\delta _k}-\widetilde{u}_{\epsilon ,\mu }) - \int _{Q_T}(W_{\delta _k}-\widetilde{u}_{\epsilon ,\mu })\partial _t(u_\epsilon -\widetilde{u}_{\epsilon ,\mu })\,dz \\&\qquad \quad -\int _{Q_T} \left( \mathcal {F}_{\epsilon }^{(0,0)}(z,\nabla u_{\epsilon })\nabla u_{\epsilon } - \mathcal {F}_{\epsilon }^{(0,0)}(z,\nabla \widetilde{u}_{\epsilon ,\mu })\nabla \widetilde{u}_{\epsilon ,\mu }\right) \cdot \nabla (W_{\delta _k}-u_\epsilon )\,dz. \end{aligned} \end{aligned}$$

All terms on the right-hand side tend to zero as $\mu \rightarrow 0$. By virtue of Proposition 3.2, Lemma 3.1 and Lemma 3.2 we have

$$\begin{aligned} \int _{Q_T}\vert \nabla (u_\epsilon -{\widetilde{u}}_{\epsilon , \mu })\vert ^{\underline{s}(z)}\,dz\rightarrow 0\quad \text {and a.e. in }Q_T \hbox {as }\mu \rightarrow 0. \end{aligned}$$

The second-order regularity follows now exactly as in the proof of Theorem 7.2. The above arguments are summarized in the following assertion.

Theorem 7.3

The assertions of Theorems 7.1 and 7.2 remain true for the domain $\Omega$ with $\partial \Omega \in C^2$.

8 The degenerate problem

8.1 Existence and uniqueness of strong solution: proof of Theorem 2.1

Let $\{u_\epsilon \}$ be the family of strong solutions of the regularized problems (5.1) satisfying estimates (7.1). These uniform in $\epsilon$ estimates enable one to extract a sequence $\{u_{\epsilon _k}\}$ and find functions $u\in {\mathbb {W}_{\overline{s}(\cdot )}(Q_T)}$, $\mathcal {A}_1,\,\mathcal {A}_2 \in (L^{(\overline{s}(\cdot ))^{'}}(Q_T))^N$ with the following properties:

$$\begin{aligned} \begin{aligned}&u_{\epsilon _k} \rightarrow u \quad \star \hbox {-weakly in } L^\infty (0,T;L^2(\Omega )), \ \qquad u_{\epsilon _k t} \rightharpoonup u_{t}\hbox { in }L^2(Q_T), \\&\nabla u_{\epsilon _k} \rightharpoonup \nabla u\hbox { in } (L^{r(\cdot )}(Q_T))^N \quad \text {with} \ r(z)= \max \{2, \overline{s}(z)\}, \\&\beta _{\epsilon _k}^{\frac{p(z)-2}{2}}(\nabla u_{\epsilon _k}) \nabla u_{\epsilon _k}\rightharpoonup \mathcal {A}_{1}\hbox { in }(L^{(\overline{s}(\cdot ))^{'}}(Q_T))^N, \\&\beta _{\epsilon _k}^{\frac{q(z)-2}{2}}(\nabla u_{\epsilon _k}) \nabla u_{\epsilon _k}\rightharpoonup \mathcal {A}_{2}\hbox { in }(L^{(\overline{s}(\cdot ))^{'}}(Q_T))^N, \end{aligned}\nonumber \\ \end{aligned}$$

(8.1)

where in third and fourth lines we make use of the uniform estimates (7.1). Moreover, $u\in C^0([0,T];L^2(\Omega ))$. Each of $u_{\epsilon _k}$ satisfies the identity

$$\begin{aligned} \int _{Q_T} u_{\epsilon _k t} \xi \,dz + \int _{Q_T} \mathcal {F}^{(0,0)}_{\epsilon _k}(z,\nabla u_{\epsilon _k})\nabla u_{\epsilon _k} \cdot \nabla \xi \,dz = \int _{Q_T} f \xi \,dz \end{aligned}$$

(8.2)

for every $\xi \in \mathbb {W}_{\overline{s}(\cdot )}(Q_T)$, which yields

$$\begin{aligned} \int _{Q_T} u_{ t} \xi \,dz + \int _{Q_T} (a(z) \mathcal {A}_1 + b(z) \mathcal {A}_2) \cdot \nabla \xi \,dz = \int _{Q_T} f \xi \,dz. \end{aligned}$$

(8.3)

Identification of $\mathcal {A}_1$ and $\mathcal {A}_2$ is based on the monotonicity of the flux $\mathcal {F}_{\epsilon }^{(0,0)}(z,\xi )\xi$ with $\epsilon \in [0,1)$, the argument is a literal repetition of the proof given in [8, Theorem 2.1]. We obtain the following equality:

$$\begin{aligned} \int _{Q_T}\left( a(z) \mathcal {A}_{1} + b(z) \mathcal {A}_{2}-\mathcal {F}_{0}^{(0,0)}(z,\nabla u)\right) \nabla u \cdot \nabla \phi \,dz=0\quad \forall \phi \in \mathbb {W}_{\overline{s}(\cdot )}(Q_T). \end{aligned}$$

Since $u\in C([0,T];L^{2}(\Omega ))$, the initial condition is fulfilled by continuity. Estimates (2.23) follow from the uniform in $\epsilon$ estimates of Theorem 7.1 and the lower semicontinuity of the modular. Uniqueness of a strong solution is an immediate consequence of the monotonicity of the flux.

8.2 Continuity with respect to the data. Proof of Theorem 2.2

Let u, v be the strong solutions of problem (1.1) with the data $\{u_0,f\}$ and $\{v_0,g\}$. The energy identity (2.21) follows if we take u for the test function in (2.17). Let us take some $t,t+h\in [0,T]$. The function $w=u-v$ is an admissible test-function in identities (2.17) for u and v. Combining these identities in the cylinder $\Omega \times (t,t+h)$ we obtain

$$\begin{aligned} \begin{aligned} \int _{t}^{t+h}\int _{\Omega }w_tw\,dz&+ \int _{t}^{t+h}\int _{\Omega } \left( \mathcal {F}_0^{(0,0)}(z,\nabla u)\nabla u- \mathcal {F}_0^{(0,0)}(z,\nabla v)\nabla v\right) \cdot \nabla w\,dz \\&= \int _{t}^{t+h}\int _{\Omega }(f-g)w\,dz. \end{aligned} \end{aligned}$$

Let us divide this equality by h and send $h\rightarrow 0$. By the Lebesgue differentiation theorem, for a.e. $t\in (0,T)$ each term has a limit, whence for a.e. $t\in (0,T)$

$$\begin{aligned} \begin{aligned} \dfrac{1}{2}\dfrac{d}{dt}\left( \Vert w\Vert ^{2}_{2,\Omega }(t)\right)&+\int _{\Omega } \left( \mathcal {F}_0^{(0,0)}(z,\nabla u)\nabla u- \mathcal {F}_0^{(0,0)}(z,\nabla v)\nabla v\right) \cdot \nabla w\,dx \\&= \int _{\Omega }(f-g)w\,dx. \end{aligned} \end{aligned}$$

(8.4)

By Proposition 3.1 the second term on the left-hand side is nonnegative. Dropping this term and applying the Cauchy inequality to the right-hand side we arrive at the differential inequality

$$\begin{aligned} \dfrac{d}{dt}\left( \Vert w(t)\Vert ^{2}_{2,\Omega }\right) \le \Vert w(t)\Vert ^2_{2,\Omega } + \Vert (f-g)(t)\Vert ^2_{2,\Omega }, \end{aligned}$$

which can be integrated: for all $t\in [0,T]$

$$\begin{aligned} \begin{aligned} \Vert w(t)\Vert _{2,\Omega }^2&\le \textrm{e}^{t}\Vert w(0)\Vert ^2_{2,\Omega }+ \textrm{e}^t \int _0^t\textrm{e}^{-\tau }\Vert f-g\Vert ^2_{2,\Omega }(\tau )\,d\tau \\&\le \textrm{e}^T\left( \Vert w(0)\Vert ^2_{2,\Omega }+\Vert f-g\Vert _{2,Q_T}^2\right) . \end{aligned} \end{aligned}$$

(8.5)

Let us revert to (8.4), integrate it in t, and then drop the first nonnegative term in the resulting relation. In the notation (3.6)

$$\begin{aligned} 2\mathcal {G}_0(\nabla u-\nabla v)\le \Vert u_0-v_0\Vert _{2,\Omega }^2+\Vert f-g\Vert ^2_{2,Q_T}+\Vert w\Vert _{2,Q_T}^2. \end{aligned}$$

According to (8.5), the last term on the right-hand side tends to zero if the first two terms tend to zero. The assertion follows now from Proposition 3.2 and Lemma 3.2.

8.3 Strong convergence of the gradients

Lemma 8.1

Let the conditions of Theorem 2.1 be fulfilled and $\{u_\epsilon \}$ be the sequence of solutions of the regularized problems (5.1). Then $\nabla u_\epsilon \rightarrow \nabla u$ in $L^{\underline{s}(\cdot )}(Q_T)$ and a.e. in $Q_T$ (up to a subsequence).

Proof

The limit function $u\in \mathbb {W}_{\overline{s}(\cdot )}(Q_T)$ satisfies the identity

$$\begin{aligned} \int _{Q_T} u_t\xi \,dz+\int _{Q_T} \mathcal {F}_0^{(0,0)}(z,\nabla u)\nabla u\cdot \nabla \xi \,dz=\int _{Q_T} f \xi \,dz\qquad \forall \xi \in \mathbb {W}_{\overline{s}(\cdot )}(Q_T). \end{aligned}$$

(8.6)

As distinguished from the case of the finite-dimensional Galerkin’s approximations considered in Lemma 7.1, the inclusions $u_\epsilon ,u\in \mathbb {W}_{\overline{s}(\cdot )}(Q_T)$ allow us to take $u_\epsilon -u$ for the test-function in identities (5.2) and (8.6). Let us subtract these identities and rearrange the result in the following way:

$$\begin{aligned} \begin{aligned} \mathcal {G}_{\epsilon }&(\nabla u_\epsilon ,\nabla u) \equiv \int _{Q_T}\left( \mathcal {F}^{(0,0)}_{\epsilon }(z,\nabla u_\epsilon )\nabla u_\epsilon -\mathcal {F}^{(0,0)}_{\epsilon }(z,\nabla u)\nabla u\right) \cdot \nabla (u_\epsilon -u)\,dz \\&= -\int _{Q_T}(u_{\epsilon }-u)_t(u_{\epsilon }-u)\,dz \\&\qquad - \int _{Q_T}\left( \mathcal {F}^{(0,0)}_{\epsilon }(z,\nabla u)\nabla u- \mathcal {F}^{(0,0)}_{0}(z,\nabla u)\nabla u \right) \cdot \nabla (u_\epsilon -u)\,dz \\&\equiv J_1(\epsilon )+J_2(\epsilon ). \end{aligned}\nonumber \\ \end{aligned}$$

(8.7)

By the choice of $\{u_\epsilon \}$

$$\begin{aligned} J_1(\epsilon )=-\int _{Q_T}(u_{\epsilon t}-u_t)(u_\epsilon -u)\,dz\rightarrow 0 \quad \hbox { when}\ \epsilon \rightarrow 0 \end{aligned}$$

as the product of weakly and strongly convergent sequences. By the generalized Hölder inequality (2.3)

$$\begin{aligned} \begin{aligned} \vert J_2(\epsilon )\vert&\le 2 a^+\Vert \nabla (u_\epsilon -u)\Vert _{p(\cdot ),Q_T} \Vert (\epsilon ^2+\vert \nabla u\vert ^2)^{\frac{p(\cdot )-2}{2}}\nabla u-\vert \nabla u\vert ^{p(\cdot )-2}\nabla u\Vert _{p'(\cdot ),Q_T} \\&\qquad + 2 b^+\Vert \nabla (u_\epsilon -u)\Vert _{q(\cdot ),Q_T} \Vert (\epsilon ^2+\vert \nabla u\vert ^2)^{\frac{q(\cdot )-2}{2}}\nabla u-\vert \nabla u\vert ^{q(\cdot )-2}\nabla u\Vert _{q'(\cdot ),Q_T}. \end{aligned} \end{aligned}$$

By (8.1), the first factors of both terms on the right-hand side are bounded by an independent of $\epsilon$ constant C. To show that the second factors tend to zero as $\epsilon \rightarrow 0$, it is sufficient to check that this is true for the modulars

$$\begin{aligned} \mathcal {M}_{1, \epsilon }=\int _{Q_T} \sigma _{1,\epsilon }^{p'(z)}(z)\,dz, \qquad \sigma _{1, \epsilon }(z)\equiv \vert (\epsilon ^2+\vert \nabla u\vert ^2)^{\frac{p(z)-2}{2}}\nabla u- \vert \nabla u\vert ^{p(z)-2}\nabla u\vert , \\ \mathcal {M}_{2, \epsilon }=\int _{Q_T} \sigma _{2,\epsilon }^{q'(z)}(z)\,dz, \qquad \sigma _{2,\epsilon }(z)\equiv \vert (\epsilon ^2+\vert \nabla u\vert ^2)^{\frac{q(z)-2}{2}}\nabla u- \vert \nabla u\vert ^{q(z)-2}\nabla u\vert . \end{aligned}$$

We will consider in detail the integral $\mathcal {M}_{1,\epsilon }$. It is asserted that $\sigma _{1, \epsilon }(z) \rightarrow 0$ as $\epsilon \rightarrow 0$ for a.e. $z\in Q_T$. Indeed: for every $z\in Q_T$ either $\vert \nabla u(z)\vert =0$, whence $\sigma _{1,\epsilon }(z)=0$ for all $\epsilon$, or $\vert \nabla u(z)\vert =\delta >0$ and

$$\begin{aligned} \sigma _{1,\epsilon }(z)\le \left| (\epsilon ^2 +\delta ^2)^{\frac{p(z)-2}{2}}- \delta ^{p(z)-2}\right| \delta =\left| (\epsilon ^2 +\delta ^2)^{\frac{p(z)-1}{2}}- \delta ^{p(z)-1}\right| \rightarrow 0\quad \hbox { as}\ \epsilon \rightarrow 0. \end{aligned}$$

The functions $\sigma _{1,\epsilon }^{p'(z)}(z)$ have the independent of $\epsilon$ integrable majorant:

$$\begin{aligned} \begin{aligned} \sigma ^{p'(z)}_{1,\epsilon }(z)&\le \left( \vert \nabla u\vert ^{p(z)-1}+\vert \nabla u\vert (\epsilon ^2+\vert \nabla u\vert ^2)^{\frac{p(z)-2}{2}}\right) ^{\frac{p(z)}{p(z)-1}} \\&\le \left( \vert \nabla u\vert ^{p(z)-1}+(\epsilon ^2+\vert \nabla u\vert ^2)^{\frac{p(z)-1}{2}}\right) ^{\frac{p(z)}{p(z)-1}} \le C\left( 1+\vert \nabla u\vert ^{p(z)}\right) \end{aligned} \end{aligned}$$

with a constant $C=C(p^\pm )$. By the dominated convergence theorem $\mathcal {M}_{1,\epsilon }\rightarrow 0$ as $\epsilon \rightarrow 0$. The same arguments show that $\mathcal {M}_{2,\epsilon }\rightarrow 0$ as $\epsilon \rightarrow 0$. Returning to (8.7) we conclude that $\mathcal {G}_\epsilon (\nabla u_\epsilon ,\nabla u)\rightarrow 0$ as $\epsilon \rightarrow 0$. By Lemma 3.2$\vert \nabla (u_\epsilon -u)\vert \rightarrow 0$ in $L^{\underline{s}(\cdot )}(Q_T)$, whence the pointwise convergence $\nabla u_\epsilon \rightarrow \nabla u$ a.e. in $Q_T$. $\square$

8.4 Second-order regularity. Proof of Theorem 2.3

Fix $i,j\in \{1,2,\ldots ,N\}$. By Theorem 7.2 and Lemma 8.1, there exists $\eta _{ij}\in L^{2}(Q_T)$ such that

$$\begin{aligned} \begin{aligned}&D_{j}\left( (\mathcal {F}_{\epsilon _k}^{(0,0)})^{\frac{1}{2}}(z,\nabla u_{\epsilon _k})D_i u_{\epsilon _k}\right) \rightharpoonup \eta _{ij} \hbox {in }L^{2}(Q_T), \\&\qquad (\mathcal {F}^{(0,0)}_{\epsilon _k})^{\frac{1}{2}}(z,\nabla u_{\epsilon _k})\nabla u_{\epsilon _k}\rightarrow (\mathcal {F}^{(0,0)}_{0})^{\frac{1}{2}}(z,\nabla u)\nabla u \\&\qquad \quad \equiv \left( a(z) \vert \nabla u\vert ^{p(z)-2} + b(z) \vert \nabla u\vert ^{q(z)-2} \right) ^{\frac{1}{2}}\nabla u\quad \hbox { a.e. in}\ Q_T.\qquad \qquad \end{aligned} \end{aligned}$$

By virtue of (7.2) $\Vert (\mathcal {F}_{\epsilon _k}^{(0,0)})^{\frac{1}{2}}(z,\nabla u_{\epsilon _k})\nabla u_{\epsilon _k}\Vert _{2+\delta ,Q_T}$ are uniformly bounded for some $\delta >0$ whence by the Vitali convergence theorem

$$\begin{aligned} (\mathcal {F}_{\epsilon _k}^{(0,0)})^{\frac{1}{2}}(z,\nabla u_{\epsilon _k})\nabla u_{\epsilon _k}\rightarrow (\mathcal {F}_{0}^{(0,0)})^{\frac{1}{2}}(z,\nabla u)\nabla u\quad \hbox { in}\ L^{2}(Q_T). \end{aligned}$$

It follows that $\eta _{ij}=D_{j}\left( (\mathcal {F}_{0}^{(0,0)})^{\frac{1}{2}}(z,\nabla u)D_i u\right)$: for every $\phi \in C_0^{\infty }(\overline{Q}_T)$

$$\begin{aligned} \begin{aligned} -(\eta _{ij},\phi )_{2,Q_T}&=-\lim _{k\rightarrow \infty }\left( D_{j}\left( (\mathcal {F}_{\epsilon _k}^{(0,0)})^{\frac{1}{2}}(z,\nabla u_{\epsilon _k})D_i u_{\epsilon _k}\right) ,\phi \right) _{2,Q_T} \\&= \lim _{k\rightarrow \infty }\left( (\mathcal {F}_{\epsilon _k}^{(0,0)})^{\frac{1}{2}}(z,\nabla u_{\epsilon _k})D_i u_{\epsilon _k},D_j\phi \right) _{2,Q_T} \\&= \left( (\mathcal {F}_{0}^{(0,0)})^{\frac{1}{2}}(z,\nabla u)D_i u,D_j\phi \right) _{2,Q_T}. \end{aligned} \end{aligned}$$

Let $\inf _{Q_T}\overline{s}(z)< 2$ and, thus, $\mathcal {D}_{p,q}=Q_T\cap \{z:\,\overline{s}(z)< 2\}\not =\emptyset$. Arguing as in the proof of Theorem 7.2 we find that for every $\mathcal {B} \Subset \mathcal {D}_{p,q}$

$$\begin{aligned} \begin{aligned} \int _{\mathcal {B}}\vert D^2_{ij}u_{\epsilon }\vert ^{\underline{s}(z)}\,dz&\le C \end{aligned} \end{aligned}$$

with a constant C independent of $\epsilon$ and $\mathcal {B}$. It follows that $D^2_{ij}u_{\epsilon _k}\rightharpoonup \zeta _{ij}\in L^{\underline{s}(\cdot )}(\mathcal {B})$ (up to a subsequence). Because of the weak convergence $\nabla u_{\epsilon _k}\rightharpoonup \nabla u$ in $L^{r(\cdot )}(Q_T)$ with $r(z)= \max \{2, \overline{s}(z)\}$, it is necessary that $\zeta _{ij}=D^{2}_{ij}u$. The estimate $\Vert D^{2}_{ij}u\Vert _{\underline{s}(\cdot ),\mathcal {B}}$ $\le C$ follows from the uniform estimate on $D^2_{ij}u_\epsilon$ as in the proof of Theorem 7.2.