A finite time extinction profile and optimal decay for a fast diffusive doubly nonlinear equation

Abstract

In this article, we consider a fast diffusive type doubly nonlinear parabolic equation and study the extinction behavior of a solution at a finite time. We show the complete extinction of a weak solution with a nonnegative initial datum, that is, a weak solution is positive before a finite time and vanishes after it, and derive the optimal decay estimates of extinction. Our key ingredient of the proof is a nonlinear intrinsic scaling and the expansion of positivity.

Introduction

We consider the finite time extinction phenomenon for the fast diffusive doubly nonlinear parabolic equations. The doubly nonlinear parabolic equations treated in the paper possess the p-Laplacian coupled with the porous medium operator. Precisely, let $\Omega \subset {\mathbb {R}}^n (n \ge 3)$ be a bounded domain with the smooth boundary $\partial \Omega$, and let $p \in (1, n)$ and $q \ge 1$ satisfy $p<q+1\le p^*$, where $p^*:=\frac{np}{n-p}$ is the Sobolev critical exponent. We shall deal with the following Cauchy-Dirichlet problem for doubly nonlinear parabolic equation:

$$\begin{aligned} {\left\{ \begin{array}{ll} \,\,\partial _t(|u|^{q-1}u)-\Delta _pu=0\quad &{}\text {in}\quad \Omega _\infty :=\Omega \times (0,\infty ) \\ \,\,u=0\quad &{}\text {on}\quad \!\!\partial \Omega \times (0,\infty ) \\ \,\,u(\cdot , 0)=u_0(\cdot ) \quad &{}\text {in}\quad \Omega . \end{array}\right. } \end{aligned}$$

(1.1)

Throughout this paper, $\Delta _p u:=\textrm{div}\left( |\nabla u|^{p-2}\nabla u\right)$ describes the p-Laplacian, where $\nabla u=\left( \partial _{x_i}u\right) _{1 \le i \le n}$ denotes the spatial gradient of u with respect to x, and we assume that the initial datum $u_0$ belongs to the Sobolev space $W^{1,p}_0 (\Omega )$, and is nonnegative, not identically zero, and bounded in $\Omega$. Our study for the non-homogeneous type doubly nonlinear equation (1.1) is motivated by that of p-Sobolev flow [18, 19, 25].

In order to formulate our main result, we briefly explain our complete extinction problem for (1.1), the precise notion is presented in Definition 3.4: A positive number $t^*$ is called the complete extinction time of (1.1) if a solution u of (1.1) is positive in $\Omega \times (0, t^*)$, and vanishes in $\Omega \times [t^*, \infty )$.

The complete extinction phenomenon actually holds true for a weak solution of (1.1) with a nonnegative initial data. Our proof employs the expansion of positivity and an intrinsic scaling, that are established in our previous works [18, 19, 25]. See [9, 10] on the positivity for the porous medium and p-Laplace equations. We also extend the expansion of positivity by a stretching transformation of time and a nonlinear scaling method to the subcritical case that $p<q+1<p^*$. See [19, Theorem A.6, Proposition 4.6] in the critical case that $q+1=p^*$.

The Hölder regularity is well-known to hold for nonnegative weak solutions to the porous medium type equations and the evolutionary p-Laplace equations; for instance see [8, 29]. The local regularity for doubly nonlinear parabolic equations also have been studied by Vespri [37, 38], Porzio and Vespri [26], and Ivanov [15, 16], where the case that $p \le q+1$ is treated, but the class of weak solutions is somehow different from ours. See further references [12]. The proofs of the regularity for general doubly nonlinear parabolic equations are based on De Giorgi’s alternative approach with the intrinsic scaling method, originally introduced by DiBenedetto (Fig. 1). For the so-called Trudinger’s equation in the homogeneous case that $q + 1 = p$, the Harnack inequality and local regularity for nonnegative weak solutions are proved by Kinnunen and Kuusi [17] et al. There also exists a viscosity approach for the doubly nonlinear equation by Bhattacharya and Marazzi ( [4]).

In the present paper, one of our main theorems is below.

FormalPara Theorem 1.1

(Finite complete extinction of (1.1)) Let $p \in (1, n)$ and $q \ge 1$ be such that $p < q + 1 \le p^*= \frac{n p}{n - p}$. Suppose that the initial datum $u_0$ belongs to $W_0^{1,p}(\Omega )$, and is nonnegative, not identically zero, and bounded in $\Omega$. Let u be a weak solution to (1.1) in the sense of Definition 3.1. Then there is a complete extinction time $t^*$ of u, that is, u is positive in $\Omega \times (0,t^*)$ and u vanishes in $\Omega \times [t^*, \infty )$. Moreover, the solution u and its gradient are locally Hölder continuous in $\Omega \times (0, t^*)$.

Fig. 1

Phenomenon of the complete extinction

Our second result is the extinction profile at the finite complete extinction time in the following:

FormalPara Theorem 1.2

(Asymptotic convergence) Let $p \in (1, n)$ and $q \ge 1$ be such that $p < q + 1 \le p^*= \frac{n p}{n - p}$. Suppose that the initial datum $u_0 \in W_0^{1,p}(\Omega )$ is bounded, nonnegative and not identically zero. Let u be a weak solution to (1.1) in the sense of Definition 3.1 and $t^*$ be the extinction time of u in the sense of Definition 3.4. Then the following statement holds:

• In the subcritical case $p< q + 1 < p^*$, for any increasing sequence $t_k \nearrow t^*$, there exist a subsequence $\{t_k\}$ with the same notation, and a nonnegative function $U \in W_0^{1,p}(\Omega ) {\setminus } \{0\}$ such that

  $$\begin{aligned} (t^*-t_k)^{-\frac{1}{q+1-p}}u(t_k) \rightarrow U\quad \text {strongly in} \quad W_0^{1,p}(\Omega ) \end{aligned}$$

  (1.2)

  as $k \rightarrow \infty$, where U is a weak solution, which is not identically zero but a nonnegative, to the Dirichlet problem

  $$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta _p U=\lambda _{p,q}|U|^{q-1}U &{}\quad \textrm{in} \quad \Omega \\ \quad \quad U=0 &{}\quad \textrm{on} \quad \partial \Omega \end{array}\right. } \end{aligned}$$

  (1.3)

  with a constant $\lambda _{p, q} = q/(q + 1 - p)$.

• In the critical case $q+1=p^*$, the weak convergence in $W^{1, p}_0 (\Omega )$ to a nonnegative solution of (1.3) holds true for some subsequence of any increasing time-sequence $t_k \nearrow t^*$.

FormalPara Remark 1.3

Let us comment on the critical case $q+1= p^*$ for Theorem 1.2.

• (a) In the critical case $q + 1 = p^*$, there is only shown to hold the weak convergence in $W^{1, p}_0 (\Omega )$ and its weak limit can be trivial; therefore the energy and volume gap at the weak limit U may appear in at most finitely many points in $\Omega$ by the energy and volume boundedness (see the end of proof of Theorem 1.2). Then, we may have the so-called energy and volume concentration at the limit (for $p = 2$ see [5, 34] and references therein). Moreover, the concentration of energy and volume is given as the limit of scaled solutions on space, where the scaling transformation makes invariant the elliptic parts of the evolution equation for the left-hand side of (1.2) (see  (5.2)), which is the same as stationary equation (1.3)$_1$. This scaling limit may be regarded as a microscopic limit of the left-hand side of (1.2). Therefore, the weak limit at infinity time is characterized as the sum of the macroscopic limit at finitely many concentration points and the reminder. The reminder is the weak limit of solutions on the left-hand side of (1.2). The proof of the phenomenon expected in the critical case $q+1 =p^*$ is based on the local boundedness of solutions on the left-hand side of (1.2) depending on the situation whether the local volume is uniformly bounded or not. This will be revealed in our forthcoming paper. The strong maximum principle for the positivity of scaled limit is in the literature, cf. [36]. If the convergence is strong as in (1.2), that is, no energy (volume) gap does not appear, the limit is a positive solution of (1.3). The existence of a positive solution of (1.3) may depend on the geometry of domain (refer to [30, 32] for ${\mathbb {R}}^n$ and a compact manifold, and also see [35, Chapter III, Sect. 3, pp. 183–193] for $p = 2$).

• (b) If the domain is star-shaped, a bounded weak solution of (1.3) satisfies the so-called Pohozaev identity and thus, it is identically zero. The proof of this fact is in [13, Theorem 1.1, Page 834; Corollary 1.3, Page 886]. The proof is based on the approximation as presented in Appendix C and the uniqueness of a bounded weak solution of (1.3) (refer to [13, Proposition 2.1, Page 886]). The solutions $v (\theta _k)$ only admit the boundedness with exponential growth on time $\theta _k$ as in Proposition 5.2, because of the power nonlinear term with the Sobolev conjugate exponent in the right-hand side of the equation.

As for the finite time extinction phenomenon of the plasma equations or the porous medium equations, those are given (1.1) with $p = 2$, a by now large literature is available; for instance, it is well known that this problem is originally addressed by Berryman and Holland [3]. After that, assuming the regularity for nonlinear term that mediates solution itself and geometric condition on the domain, Kwong ( [20,21,22]) established the finite time complete extinction of a continuous weak solution in terms of appropriate comparison function in any dimension, where the continuity of a weak solution of the porous medium equation is essentially used (see the regularity in [8, 29]). See also [2, 7] for details. In the Laplacian and 1D case, Sabinina proved a finite time extinction of the plasma equation [28]. Savaré and Vespri showed the asymptotic profile of doubly nonlinear parabolic equations in the case that $q+1< p^*$ (see [31, Remark 4.6]), where the equation is formally equivalent to (1.1) by a changing of unknown function $v=|u|^{q - 1} u$. This procedure enables to avoid the non-linearlity in the time-derivative and work in the $L^2$-framework. In contrast, we shall tackle the power nonlinearity in the time derivative in a direct fashion, which is a one of our motive. The large time behavior of solutions to Eq. (1.1) in the case $q<1$ is established by Stan and Vázquez [33]. However, the solution class is different from ours. For the p-Laplacian in 3D case and the critical case $q+1= p^*$, the complete extinction of a continuous weak solution to (1.1) in a convex domain is shown via constructing appropriate comparison functions [27]. The optimal decay estimate and stability of asymptotic profile in the Laplacian case $p = 2$ is studied by Akagi and Kajikiya [1], where the solution class is different from ours. Recently, in the sensational paper [5] by Bonforte and Figalli, the sharp extinction rates are established for (1.1) with $p=2$, whose approach is based on the so-called Nonlinear Entropy Method, combined with the spectrum analysis of the Dirichlet Laplacian in weighted $L^2$-spaces. We emphasize that Theorem 1.1 contains the results for the plasma or porous medium equations and the proof does not require any continuity of a weak solution, in contrast to those of the above results. Our approach based on the expansion of positivity may be of its own interest. The decay estimates rely on some energy estimates, obtained from an appropriate approximation of the equation (1.1). The approximation yields the rigorous derivation of energy estimates, that is demonstrated in Appendix B. The energy estimates lead to a monotonicity of the so-called Rayleigh quotient, that is the key to the decay estimates. See Lemma 4.1 and Appendix D below (Also refer to [1, 20, 21]).

Organization of the paper

The outline of the paper is in the following. In the next section, we give some notation and recall some fundamental tools used later. Section 3 is devoted to the global existence and regularity estimates for the doubly nonlinear parabolic equation (1.1). We further present the nonlinear intrinsic scaling, that transforms the prototype doubly nonlinear equations to the p-Sobolev type flow, and give the proof of our main result, Theorem 1.1. In Sect. 4, we derive the optimal decay estimate for Eq. (1.1). In Sect. 5, we prove our second result, Theorem 1.2. In Appendix A we give the uniqueness of nonnegative solutions to (1.1) and its transformed equation. Appendix B is devoted to construction of an approximating solution to (1.1) satisfying the energy estimate as in Proposition B.9. The quantitative estimates and convergence result for approximate equation (B.3) are given in Appendix C. In Appendices D and E, we collect the proof postponed in the previous section.

1 Preliminaries

1.1 Notation

In the following, we fix some notation which will be used throughout the paper. Let $\Omega \subset {\mathbb {R}}^n\,\,(n \ge 3)$ be a bounded domain with smooth boundary $\partial \Omega$. For a positive $T \le \infty$, let $\Omega _T:=\Omega \times (0,T)$ be the space-time domain.

From now on we denote by C, $C_1$, $C_2, \cdots$ different positive constants in a given context. Relevant dependencies on parameters will be emphasized using parentheses. For instance $C=C(n,p,\Omega ,\cdots )$ means that C depends on $n, p, \Omega \cdots$. Further, a general positive constant C will not necessarily be the same at different occurrences and can also change from line to line. For the sake of readability, the dependencies of the constants will be often omitted within the chains of estimates. In addition, the n-th line of the Eq. $(\,\cdot \,)$ is denoted by the symbol $(\,\cdot \,)_n$. With $S \subset {\mathbb {R}}^k$ being a finite measurable set with Lebesgue measure |S| and with f being an integrable function on S, we shall denote the integral average by

We also abbreviate the essential infimum and essential supremum as

$$\begin{aligned} {{\,\mathrm{ess\,inf}\,}}\equiv \inf \quad \text {and} \quad {{\,\mathrm{ess\,sup}\,}}\equiv \sup , \end{aligned}$$

respectively.

1.2 Function spaces

We recall some function spaces used throughout the paper. Let $1 \le p,q \le \infty$. For a Banach space ${\mathcal {X}}$ we use the space of Bochner $L^q (t_1, t_2)$-integrable functions $v: (t_1, t_2) \ni t \mapsto v (t) \in {\mathcal {X}}$, denoted by $L^q (t_1, t_2;{\mathcal {X}})$. Letting ${\mathcal {X}}$ as the Lebesgue space $L^p (\Omega )$, we have $L^{q}(t_1,t_2\,;\,L^{p}(\Omega ))$ with a finite norm

$$\begin{aligned} \Vert v\Vert _{L^{q}(t_1,t_2;\,L^{p}(\Omega ))}:= {\left\{ \begin{array}{ll} \displaystyle \left( \int _{t_1}^{t_2}\Vert v(t)\Vert _p^{q}\,dt \right) ^{1/q}\quad &{}(1 \le q<\infty ) \\ \displaystyle \sup _{t_1 \le t \le t_2}\Vert v(t)\Vert _p\quad &{}(q=\infty ), \end{array}\right. } \end{aligned}$$

where we abbreviate $\Vert v(t)\Vert _{L^p(\Omega )}$ to $\Vert v(t)\Vert _p$ for $1 \le p \le \infty$. If $p=q<+\infty$ then we have the identification as $L^p(\Omega \times (t_1,t_2))=L^{p}(t_1,t_2\,;\,L^{p}(\Omega ))$. To avoid confusion, we shall deal with the above Lebesgue spaces except $p=q=\infty$. Choosing ${\mathcal {X}}$ as the Sobolev space on $\Omega$, $W^{1, p}_0 (\Omega )$, we have the space $L^{q}(t_1,t_2;\,W_{0}^{1,p}(\Omega ))$ with a finite norm

$$\begin{aligned} \Vert v\Vert _{L^q(t_1,t_2\,;\,W_{0}^{1,p}(\Omega ))}&:=\left( \int _{t_1}^{t_2}\Vert v(t)\Vert _{W^{1,p}(\Omega )}^{q} \,dt\right) ^{1/q} \\&=\left( \int _{t_1}^{t_2}\left[ \int _{\Omega }|v|^{p}+|\nabla v|^{p}\,dx \right] ^{q/p}\,dt\right) ^{1/q}, \end{aligned}$$

provided $q < \infty$ and, if $q = \infty$

$$\begin{aligned} \Vert v\Vert _{L^\infty (t_1,t_2;\,W_{0}^{1,p}(\Omega ))}:=\sup _{t_1<t<t_2}\Vert v(t)\Vert _{W^{1,p}(\Omega )}. \end{aligned}$$

Again, we will omit to consider the case $p=q=\infty$.

In addition, for an interval $I \subset {\mathbb {R}}$, by $C(I; L^{q}(\Omega ))$ we denote the space of all continuous functions $I \ni t \mapsto u (t) \in L^q (\Omega )$.

1.3 Fundamental tools

We shall present the fundamental tools often used.

Let us define the Rayleigh quotient by

$$\begin{aligned} R(w):=\frac{\Vert \nabla w\Vert _p}{\,\Vert w\Vert _{q+1}}, \quad w \in W_0^{1,p}(\Omega ) \backslash \{0\}, \end{aligned}$$

(2.1)

which is involved the well-known Sobolev-Poincaré inequality

$$\begin{aligned} \Vert w\Vert _{q+1} \le C_{p,q} \Vert \nabla w\Vert _p, \quad w \in W_0^{1,p}(\Omega ), \end{aligned}$$

(2.2)

provided $q+1 \le p^*$. Notice that the best constant $C_{p,q}$ is achieved by

$$\begin{aligned} C_{p,q}=\sup _{w \in W_0^{1,p}(\Omega ) \backslash \{0\}} R(w)^{-1}. \end{aligned}$$

We record the following algebraic inequality throughout the paper.

Lemma 2.1

(Algebraic inequality) For every $p \in (1,\infty )$ there are positive constants $C_j=C_j(k,p)\,\,(j=1,2,3)$ such that for all $\xi ,\,\eta \in {\mathbb {R}}^k\,\,(k \ge 1)$

$$\begin{aligned} ||\xi |^{p-2}\xi -|\eta |^{p-2}\eta | \le C_1(|\xi |+|\eta |)^{p-2}|\xi -\eta | \end{aligned}$$

(2.3)

and

$$\begin{aligned} (|\xi |^{p-2}\xi -|\eta |^{p-2}\eta )\cdot (\xi -\eta ) \ge C_2 (|\xi |+|\eta |)^{p-2}|\xi -\eta |^2, \end{aligned}$$

(2.4)

where the symbol $\cdot$ denotes the inner product on ${\mathbb {R}}^k$. In particular, when $p\ge 2$

$$\begin{aligned} (|\xi |^{p-2}\xi -|\eta |^{p-2}\eta )\cdot (\xi -\eta ) \ge C_3 |\xi -\eta |^p. \end{aligned}$$

(2.5)

Proof

The proof of (2.3) and (2.4) can be derived from the proof of [14, Lemma 8.3, p.266]. Inequality (2.5) can be found in [9, Lemma 4.4 in Chapter I, p.13].

$\square$

2 Fundamental properties of a weak solution;  Proof of Theorem 1.1

In this section, we collect some results required for the proof of Theorem 1.1. We can refer to the proofs of results in [18, 19, 23, 25] (also see [24]). We first present some fundamental properties of a weak solution to (1.1) and then, we prove Theorem 1.1. Recall that $1< p < n$ and $q \ge 1$ satisfy $p < q + 1 \le p^*= \frac{n p}{n - p}$.

To begin, we give the definition of weak solutions of (1.1).

Definition 3.1

(Weak solutions)Let $u=u(x,t)$ be a measurable function defined on $\Omega _{\infty }:=\Omega \times (0,\infty )$. We call u as a weak supersolution (subsolution) of (1.1) in $\Omega _\infty$ provided that the following conditions (i)-(iii) are satisfied:

• (i) $u \in L^{\infty }(0,\infty \,;\,W^{1,p}(\Omega ))$,   $\partial _t \Big (|u|^{\frac{q-1}{2}}u \Big ) \in L^{2}(\Omega _\infty )$.

• (ii) For every $0\le \varphi \in C^{\infty }_0(\Omega _\infty )$

  $$\begin{aligned} \iint _{\Omega _\infty }\Big [-|u|^{q-1}u \,\partial _{t}\varphi +|\nabla u|^{p-2}\nabla u\cdot \nabla \varphi \Big ]\,dxdt \ge (\le )\, 0. \end{aligned}$$

• (iii) u attains the initial data continuously in the Sobolev space:

  $$\begin{aligned} \Vert u (t) - u_0\Vert _{W^{1,p} (\Omega )}\rightarrow 0 \quad \text {as}\quad t\searrow 0 \end{aligned}$$

  and satisfies the boundary condition in the trace sense:

  $$\begin{aligned} u(t) \in W^{1, p}_0 (\Omega ) \quad \text {for}\,\,\text { a.e.} \quad t \in (0, \infty ). \end{aligned}$$

We call a measurable function u defined on $\Omega _\infty$ as a weak solution to (1.1) if u is simultaneously a weak super and subsolution.

Theorem 3.2

(Global existence of (1.1) cf. [18, 25]) Let $p \in (1, n)$ and $q \ge 1$ satisfy $p < q + 1 \le p^*$. Suppose that the initial value $u_0$ is in the Sobolev space $W^{1, p}_0 (\Omega )$, nonnegative and bounded in $\Omega$. Then there is a global in time weak solution u of (1.1) in the sense of Definition 3.1, which is nonnegative and bounded in $\Omega _\infty$, that is,

$$\begin{aligned} 0 \le u\le \Vert u_{0}\Vert _{\infty } \quad \text {in}\,\,\Omega _\infty . \end{aligned}$$

(3.1)

Additionally, the energy identity

$$\begin{aligned} \Vert u(t_2)\Vert _{q+1}^{q+1}+\frac{q+1}{q}\int _{t_1}^{t_2}\Vert \nabla u(t)\Vert _p^p\,dt=\Vert u(t_1)\Vert _{q+1}^{q+1} \end{aligned}$$

(3.2)

holds whenever $t_1,t_2 \in [0,\infty )$ with $t_1<t_2$ and, the following integral inequalities hold true for every $t > 0$:

$$\begin{aligned} \Vert u(t)\Vert _{q+1}&\le \Vert u_0\Vert _{q+1}, \end{aligned}$$

(3.3)

$$\begin{aligned} \Vert \nabla u(t)\Vert _p&\le \Vert \nabla u_0\Vert _p, \end{aligned}$$

(3.4)

and

$$\begin{aligned} \int _0^\infty \left\| \partial _t \Big (u^{(q+1)/2}\Big )\right\| _2^2\,dt \le C \Vert \nabla u_0\Vert _p^p, \end{aligned}$$

(3.5)

with $C\equiv C(n,p)$ being a positive constant and $\Vert u(t)\Vert _{p}:=\Vert u(t)\Vert _{L^{p}(\Omega )}$ being shorthand notation.

Proof

The proof can be achieved similarly as in [24] and also [18, 25]; therefore, we shall only sketch it. Note that the arguments in [18, 25] are applicable to the both critical and subcritical cases $p<q+1 \le p^*$ with $1<p<n$ and $q \ge 1$. Equation (3.1) readily follows from [18, Propositions 3.4, 3.5]. From a similar argument to [18, Appendix B], we obtain (3.2) and thus,  (3.3) plainly follows. By the same argument as in [25, Lemma 3.2, (3.7); Lemma 4.1; Proof of Theorem 1.1], (3.4) is actually verified. Finally, (3.5) is obtained from [24, Lemma 3.4 and Sect. 5]. $\square$

Remark 3.3

Every weak solution u to (1.1) possesses the time continuity in $L^{q+1}(\Omega )$:

$$\begin{aligned} u \in C\left( [0,T]; L^{q+1}(\Omega )\right) , \quad T \in (0,\infty ). \end{aligned}$$

This is deduced by the construction of the approximate solutions or the exponential mollification method and the result can be extended to all $q>0$. The precise description is addressed in [24].

By the nonnegativity of the solution u to (3.1), we hereafter substitute the following equation (3.6) for (1.1):

$$\begin{aligned} {\left\{ \begin{array}{ll} \,\,\partial _tu^q-\Delta _pu=0\quad &{}\text {in}\quad \Omega _\infty :=\Omega \times (0,\infty ) \\ \,\,u=0\quad &{}\text {on}\quad \!\!\partial \Omega \times (0,\infty ) \\ \,\,u(\cdot , 0)=u_0(\cdot ) \quad &{}\text {in}\quad \Omega . \end{array}\right. } \end{aligned}$$

(3.6)

We now recall the definition of the extinction time of a weak solution u to (3.6) (see [19]).

Definition 3.4

(Extinction time) Let u be a nonnegative weak solution to (3.6) in $\Omega _{\infty }$ in the sense of Definition 3.1. We call a positive number $t^{*}$ the extinction time of u if it satisfies

• (i) u(x, t) is nonnegative and not identically zero on $\Omega \times (0,t^{*})$

• (ii) $u(x,t)=0$ for any $x \in {\overline{\Omega }}$ and all $t \ge t^{*}$.

The finite time extinction for (3.6) actually holds true, as stated below.

Proposition 3.5

(Finite time extinction for (3.6)) Let $1< p < n$ and $q \ge 1$ satisfy $p <q + 1 \le p^*$. Let u be a nonnegative weak solution to (3.6) in $\Omega _{\infty }$ in the sense of Definition 3.1. Then there is a extinction time $t^*> 0$ of u in the sense of Definition 3.4, which is bounded from above as follows:

$$\begin{aligned} t^*\le \lambda _{p, q}C_{p,q}^p \left( \,\int _\Omega u_0^{q+1}\,dx \right) ^{\frac{q + 1 - p}{q+1}}, \end{aligned}$$

where $\lambda _{p, q}:=q/(q+1-p)$ and $C_{p,q}$ is the best constant as in the Sobolev-Poincaré inequality (2.2).

Proof

The proof simply follows from (3.2) in Theorem 3.2 and the Sobolev-Poincaré inequality (2.2). The full proof can be seen in [23, Proposition 3.4].

$\square$

We shall present the following nonlinear intrinsic scaling (see [19, Proposition 4.1] and also [23, Proposition 4.1]), which is the key ingredient to derive the strict positivity of the solution u before the extinction time.

Proposition 3.6

(Nonlinear intrinsic scaling) Let u be a nonnegative weak solution to the equation (3.6) in $\Omega _{\infty }$ and let $t^*<+\infty$ be a finite extinction time of u. There exist unique $\Lambda \in C^1[0,\infty )$ solving

$$\begin{aligned} {\left\{ \begin{array}{ll} \Lambda ^\prime (\theta ) = (t^*)^{-1}\left( \,\displaystyle \int _{\Omega } u^{q+1}\left( x,t^*\left( 1-e^{-\Lambda (\theta )} \right) \right) \, dx \right) ^{\frac{q+1-p}{q+1}}\\ \Lambda (0) = 0 \end{array}\right. } \end{aligned}$$

(3.7)

and, subsequently, $g \in C^1[0,\infty )$ solving

$$\begin{aligned} {\left\{ \begin{array}{ll} g^\prime (\tau ) = e^{\Lambda (g(\tau ))} \\ g(0) = 0 \end{array}\right. } \end{aligned}$$

(3.8)

such that the following is valid: Let

$$\begin{aligned} h(\tau )=t^*\left( 1-e^{-\Lambda (g(\tau ))}\right) \end{aligned}$$

(3.9)

and set

$$\begin{aligned} w(x,\tau ):= \frac{u(x,h(\tau ))}{\gamma (\tau )}, \quad \gamma (\tau ):= \left( \int _{\Omega } u^{q+1}(x,h(\tau )) \, dx \right) ^{\frac{1}{q+1}}. \end{aligned}$$

(3.10)

Then w is a nonnegative weak solution of the doubly nonlinear parabolic equation (3.11) on $\Omega _\infty$ with the volume constraint on $[0, \infty )$

$$\begin{aligned} {\left\{ \begin{array}{ll} \,\,\partial _\tau w^q-\Delta _pw=\lambda (\tau ) w^q\quad &{}\textrm{in}\quad \Omega _{\infty } \\ \,\,\displaystyle \Vert w(\tau ) \Vert _{L^{q+1}(\Omega )} =1\quad &{}\mathrm {for \,\,all}\,\, \tau >0 \\ \,\,w=0\quad &{}\textrm{on}\quad \!\!\partial \Omega \times (0,\infty ) \\ \,\,w(\cdot , 0)=w_0(\cdot ) \quad &{}\textrm{in}\quad \Omega , \end{array}\right. } \end{aligned}$$

(3.11)

where $\displaystyle \lambda (\tau ):= -q \frac{\gamma '(\tau )}{\gamma (\tau )}=\int _{\Omega }|\nabla w(x,\tau )|^{p}\,dx$ and the initial value $w_0$ is defined as $u_0 / \Vert u_0\Vert _{q+1}$ with $u_0$ being the initial data as in (3.6). The definition of a weak solution to (3.11) is defined similarly as in Definition 3.1 for the equation (1.1) or (3.6).

Proof

The proof of Proposition 3.6 is exactly similar to that in [19, Proposition 4.1, pp.254–258] and we omit the details here (also see [23, Proposition 4.3]).

$\square$

Theorem 3.7

(Strict positivity for solutions w of (3.11)) Let w be a nonnegative weak solution to (3.11) with initial data $w_0=u_0 / \Vert u_0\Vert _{q+1}$, defined by  (3.10), where the original initial data $u_0$ is as in (3.6). Then there holds true that

$$\begin{aligned} w=w(x,\tau )>0 \quad \textrm{in}\quad \Omega _\infty \equiv \Omega \times (0,\infty ). \end{aligned}$$

(3.12)

Proof

Notice that, by the definitions  (3.8)–(3.10), $h(\tau ) \nearrow t^*$ if and only if $\tau \nearrow +\infty$. By use of the volume conservation as in  (3.11)$_2$ and the expansion of positivity, we reach the conclusion. See [18, Proposition 5.4] and [19, Propositions 5.4, C.5] (also see [23, Proposition 5.8]). $\square$

We are now in position to prove Theorem 1.1.

Proof of Theorem 1.1

We will prove Theorem 1.1 separately as follows.

Step 1 (Complete extinction)

From (3.12) in Theorem 3.7 it follows that

$$\begin{aligned} u=u(x,t)>0 \quad \textrm{in}\quad \Omega \times (0,t_0] \end{aligned}$$

for every $t_0<t^*$ because $\gamma (t) > 0$ for any nonnegative $t < t^*$ in the definition (3.10). Hence the complete extinction of solution u is actually verified.

Step 2 (Regularity of solution w to (3.11))

At first we derive the boundedness of a weak solution w to (3.11), where we use the fact that $\lambda (\tau ) = \Vert \nabla w (\tau )\Vert _{p}^p$ in (3.11). See the proof in [18, Propositions 3.5 and 5.3].

Proposition 3.8

(Boundedness of (3.11)) Let w be a nonnegative weak solution of (3.11) in $\Omega _{T}$ for any positive $T<\infty$. Then w is bounded from above in $\Omega _{T}$ and

$$\begin{aligned} \Vert w(\tau )\Vert _{\infty } \le \exp \left( \frac{1}{q}\int _0^T\Vert \nabla w(\tau )\Vert _{p}^{p}\,d\tau \right) \Vert w_{0}\Vert _{\infty } \quad \text {for every} \quad 0\le \tau < T. \end{aligned}$$

Following the exactly similar argument to [18, Sect. 5.2], we can deduce the Hölder and the spatial gradient Hölder regularity of solution w to (3.11) with initial datum $w_0=u_0 / \Vert u_0\Vert _{q+1}$ (also see [19, Sect. 5] and [23, Theorem 6.4]).

Let $T\in (0, \infty )$ and $\Omega ^\prime$ be any subdomain compactly contained in $\Omega$. As is noticed in Theorem 3.5, we have the expansion of positivity for the solution w, that together with Proposition 3.8 yields the bounds from above and below of a solution w to (3.11)

$$\begin{aligned} 0<m \le w \le M:=\exp \left( {\frac{1}{q}\int _0^T\Vert \nabla w(\tau )\Vert _p^p\,d\tau }\right) \Vert w_0\Vert _\infty \quad \text {in}\quad \Omega ^\prime \times (0, T]\nonumber \\ \end{aligned}$$

(3.13)

for some $m>0$. Equation (3.11)$_1$ can be written as follows: Let $W:=w^q \iff w=W^\frac{1}{q}$ and set $G(W):=\frac{1}{q} W^{1/q-1}$. Equation (3.11)$_1$ becomes

$$\begin{aligned} \partial _\tau W-\textrm{div} \big (|\nabla W|^{p-2}G(W)^{p-1}\nabla W \big )=\lambda (\tau )W \quad \text {in}\quad \Omega ^\prime _T:=\Omega ^\prime \times (0,T)\nonumber \\ \end{aligned}$$

(3.14)

and hence, W is a positive and bounded weak solution of the evolutionary p-Laplacian equation (3.14) in $\Omega ^\prime _T$. By (3.13) G is actually uniformly elliptic and bounded in $\Omega ^\prime _T$. The right-hand side of (3.14) is bounded, which is assured by (3.13), $\lambda (\tau ) = \Vert \nabla w (\tau )\Vert _p^p$ and $w \in L^\infty (0, \infty ; W^{1, p}_0 (\Omega ))$ (also refer to [19, Lemma C.3]).

The following Hölder continuity is verified by the local energy inequality for a local weak solution W to (3.14) ( [18, Lemma C.1]) and standard iterative real analysis methods. See also [9, Chapter III] more details.

Theorem 3.9

(Hölder continuity) Let W be a positive and bounded weak solution to (3.14) and let $\Omega ^\prime \Subset \Omega$ be a subdomain. Then W is locally Hölder continuous in $\Omega ^\prime \times (0,T]$ with a Hölder exponent $\beta \in (0, 1)$ on the parabolic metric $|x|+|\tau |^{1/p}$.

By the positivity and boundedness (3.13) and the Hölder continuity in Theorem 3.9, the coefficient function $G^{p-1}$ is lower, upper bounded and Hölder continuous and thus, a Hölder continuity of the spatial gradient holds true.

Theorem 3.10

(Gradient Hölder continuity) Let W be a positive and bounded weak solution to (3.14) and let $\Omega ^\prime \Subset \Omega$ be a subdomain. Then, there exist a positive constant C depending only on n, p, m, M, $\lambda (0)$, $\beta$, $\Vert \nabla W\Vert _{L^p(\Omega ^\prime _T)}$, $[G]_{\beta , \Omega ^\prime _T}$ and $[W]_{\beta , \Omega ^\prime _T}$ and a positive exponent $\alpha <1$ depending only on n, p and $\beta$ such that $\nabla W$ is locally Hölder continuous in $\Omega ^\prime \times (0,T]$ with an exponent $\alpha$ on the usual parabolic distance. Furthermore, its Hölder constant is bounded above by C. The symbol $[f]_\beta$ denotes the Hölder semi-norm for a Hölder continuous function f with a Hölder exponent $\beta$.

By using an elementary algebraic estimate and a interior positivity, boundedness, the Hölder regularity of W and its gradient $\nabla W$ in Theorems 3.9 and 3.10, we can bring out the local Hölder regularity of the weak solution w to (3.11) and its gradient $\nabla w$.

Theorem 3.11

(Hölder regularity for (3.11) c.f. [18, Theorem 5.7]) Let w be a weak solution to (3.11) and let $\Omega ^\prime \Subset \Omega$ be a subdomain. Then, there exist a positive exponent $\gamma <1$ depending only on $n,p,\beta , \alpha$ and a positive constant C depending only on $n, p, m, M, \lambda (0), \beta , \alpha ,\Vert \nabla w\Vert _{L^p(\Omega ^\prime _T)}$, $[W]_{\beta , \Omega ^\prime _T}$ and $[w]_{\beta , \Omega ^\prime _T}$ such that, both w and $\nabla w$ are locally Hölder continuous in $\Omega ^\prime \times (0,T]$ with an exponent $\gamma$ on a parabolic metric $|x|+|\tau |^{1/p}$ and $|x| + |\tau |^{1/2}$, respectively. The Hölder constants are bounded from above by C.

Step 3 (Regularity of original solution u to (3.6))

Finally, the local Hölder and gradient local Hölder regularity of the original solution u to (3.6) follow from Theorem 3.11 and the definition of w in (3.10), where $\gamma (t)$ in (3.10) is continuous for any nonnegative $t < \infty$ by (3.2) in Theorem 3.2 and the regularity of transformations in (3.10) with (3.7)–(3.9).

Theorem 3.12

(Hölder continuity) Let u be a nonnegative and bounded weak solution to (3.6) and $t^*$ be the extinction time of u. Then the following implications hold:

• (i) u is locally Hölder continuous in $\Omega \times (0,t^*)$ with an exponent $\alpha \in (0, 1)$ on a parabolic metric $|x|+|t|^{1/p}$.

• (ii) The spatial gradient $\nabla u$ is locally Hölder continuous in $\Omega \times (0,t^*)$ with an exponent $\beta \in (0,1)$ on the usual parabolic metric $|x|+|t|^{1/2}$.

Therefore, the proof of Theorem 1.1 is concluded. $\square$

3 Decay estimates

In this section, we shall derive the decay estimates for a nonnegative weak solution of (1.1).

Lemma 4.1

Let u be a nonnegative weak solution to (3.6). Let $t^*$ be an extinction time of u. Then for any $t\in [0,t^*)$

$$\begin{aligned} R(u(t)) \le R(u_0). \end{aligned}$$

This lemma is shown by an approximating argument as in [23, Sect. 7], and the proof will be given in Appendix D.

The main result of this section is condensed in the optimal decay estimates.

Proposition 4.2

(Estimates from above and below) Let $1< p < n$ and $q \ge 1$ satisfy $p < q + 1 \le p^*$. Let u be a nonnegative weak solution to (3.6) and $t^*$ be the extinction time of u. Then, for every nonnegative $t\le t^*$, the following estimations hold true:

$$\begin{aligned}{} & {} \Vert u(t)\Vert _{q+1} \le \left( \frac{R(u_0)^p}{\lambda _{p,q}} \right) ^{\frac{1}{q+1-p}} (t^*-t)^{\frac{1}{q+1-p}}, \end{aligned}$$

(4.1)

$$\begin{aligned}{} & {} \Vert u(t)\Vert _{q+1} \ge \left( \frac{C_{p,q}^{-p}}{\lambda _{p,q}} \right) ^{\frac{1}{q+1-p}} (t^*-t)^{\frac{1}{q+1-p}}, \end{aligned}$$

(4.2)

$$\begin{aligned}{} & {} \left( \frac{C_{p,q}^{-(q+1)}}{\lambda _{p,q}} \right) ^{\frac{1}{q+1-p}} (t^*-t)^{\frac{1}{q+1-p}} \le \Vert \nabla u(t)\Vert _p \nonumber \\{} & {} \quad \le \left( \frac{R(u_0)^{q+1}}{\lambda _{p,q}} \right) ^{\frac{1}{q+1-p}} (t^*-t)^{\frac{1}{q+1-p}}. \end{aligned}$$

(4.3)

Proof

We shall shorten $\Phi (t):=\Vert u(t)\Vert _{q+1}^{q+1}$. Multiplying (3.6) by u and integration-by-parts render us

$$\begin{aligned} \frac{d}{dt}\int _\Omega u^{q+1}(t)\,dx+\frac{q+1}{q}\int _\Omega |\nabla u(t)|^p\, dx =0, \end{aligned}$$

that can be written in terms of $\Phi (t)$ as

$$\begin{aligned} \frac{q}{q+1}\Phi ^\prime (t)+R(u(t))^p \Vert u(t)\Vert _{q+1}^{p}=0 \end{aligned}$$

(4.4)

with denoting $^\prime :=\frac{d}{dt}$. By Lemma 4.1R(u(t)) is bounded from above by $R(u_0)$ and it readily follows from (4.4) that

$$\begin{aligned} \frac{q}{q+1}\Phi ^\prime (t)+R(u_0)^p \Phi (t)^{\frac{p}{q+1}} \ge 0 \end{aligned}$$

that is solved

$$\begin{aligned} \Phi (t)^{\frac{q+1-p}{q+1}}\le \frac{1}{\lambda _{p,q}}R(u_0)^p(t^*-t) \end{aligned}$$

namely,

$$\begin{aligned} \Vert u\Vert _{q+1}\le \left( \frac{R(u_0)^p}{\lambda _{q,p}}\right) ^{\frac{1}{q+1-p}}(t^*-t)^{\frac{1}{q+1-p}} \end{aligned}$$

for any $t\in [0,t^*]$. Thus, the first assertion (4.1) is actually verified.

Again, by (4.4) and the Sobolev-Poincaré inequality (2.2) we gain

$$\begin{aligned} \frac{q}{q+1}\Phi ^\prime (t)+C_{p,q}^{-p} \Phi (t)^{\frac{p}{q+1}} \le 0. \end{aligned}$$

Integrating this over $(0, t^*)$ again yields

$$\begin{aligned} \Phi (t)^\frac{1}{q+1}=\Vert u(t)\Vert _{q+1}\ge \left( \frac{C_{q,p}^{-p}}{\lambda _{q,p}}\right) ^{\frac{1}{q+1-p}}(t^*-t)^{\frac{1}{q+1-p}}, \end{aligned}$$

which is the desired result (4.2).

Finally, we now turn to prove (4.3). Using the definition of Rayleigh quotient and (4.1), we get

$$\begin{aligned} \frac{\Vert \nabla u(t)\Vert _p}{R(u(t))}\le \left( \frac{R(u_0)^p}{\lambda _{p,q}}\right) ^{\frac{1}{q+1-p}}(t^*-t)^{\frac{1}{q+1-p}}. \end{aligned}$$

Since by Proposition 4.1R(u(t)) is bounded from above by $R(u_0)$, the above inequality leads to

$$\begin{aligned} \Vert \nabla u(t)\Vert _p\le \left( \frac{R(u_0)^{q+1}}{\lambda _{q,p}}\right) ^{\frac{1}{q+1-p}}(t^*-t)^{\frac{1}{q+1-p}}. \end{aligned}$$

(4.5)

By use of (4.2) and the Sobolev-Poincaré inequality (2.2) again, we also bound

$$\begin{aligned} \Vert \nabla u(t)\Vert _p\ge \left( \frac{C_{q,p}^{-(q+1)}}{\lambda _{q,p}}\right) ^{\frac{1}{q+1-p}}(t^*-t)^{\frac{1}{q+1-p}}, \end{aligned}$$

(4.6)

which together with (4.5) in turn implies the last inequality (4.3) and therefore the proof is complete. $\square$

Proposition 4.2 yields the optimal extinction rate and the estimations of the extinction time $t^*\equiv t^*(u_0)$ from above and below.

Corollary 4.3

(Extinction rate and extinction time) Let u be a nonnegative weak solution of (3.6). Then the optimal extinction rate of u is given by

$$\begin{aligned} (t^*-t)^{\frac{1}{q+1-p}} \quad \text {as}\quad t \rightarrow t^*. \end{aligned}$$

(4.7)

Furthermore, the extinction time $t^*\equiv t^*(u_0)$, depending on the initial datum $u_0$, is estimated as

$$\begin{aligned} \lambda _{p,q}\;\frac{\Vert u_0\Vert _{q+1}^{q+1}}{\Vert \nabla u_0\Vert _p^p}\le t^*\le \lambda _{p,q}\;C_{p,q}^p\;\Vert u_0\Vert _{q+1}^{q+1-p}. \end{aligned}$$

Proof

By Proposition 4.2, the optimal extinction rate is given by

$$\begin{aligned} (t^*-t)^{\frac{1}{q+1-p}}\qquad \text {as}\qquad t\rightarrow t^*. \end{aligned}$$

By (4.1) and (4.2) in Proposition 4.2, it holds that

$$\begin{aligned} \frac{C_{p,q}^{-p}}{\lambda _{p,q}}(t^*-t)\le \Vert u(t)\Vert _{q+1}^{q+1-p}\le \frac{R(u_0)^p}{\lambda _{p,q}}(t^*-t). \end{aligned}$$

Passing to the limit as $t \searrow 0$ in the display above, the assertion is readily follows. $\square$

4 Asymptotic profile; Proof of Theorem 1.2

In this section we prove the asymptotic convergence as stated in Theorem 1.2.

4.1 Transformation stretching the time-interval

We introduce a transformation stretching time-interval, that extends time-interval up to the extinction time into infinite one. The extinction profile is clearly determined by the stationary problem associated with the transformed evolution equation (Fig. 2). The method is nowadays well known for singular parabolic equations (refer to [20, 21, 31]). Let $t^*$ be an extinction time of a nonnegative weak solution u to (3.6). We define, for any $t<t^*$,

$$\begin{aligned} v(x,s):=(t^*-t)^{-\frac{1}{q+1-p}}u(x,t) \end{aligned}$$

(5.1)

with

$$\begin{aligned} \displaystyle s:=\log \left( \frac{t^*}{t^*-t} \right) \quad \iff \quad t=(1-e^{-s})t^*. \end{aligned}$$

Fig. 2

Graphs of $s=\log \left( \frac{t^*}{t^*-t} \right)$ and $t=t^*(1-e^{-s})$

By simple manipulation, $v=v(x,s)$ solves the following equation

$$\begin{aligned} {\left\{ \begin{array}{ll} \,\,\partial _sv^q-\Delta _pv=\lambda _{p,q}v^q\quad &{}\text {in}\quad \Omega _\infty :=\Omega \times (0,\infty ) \\ \,\,v=0\quad &{}\text {on}\quad \!\!\partial \Omega \times (0,\infty ) \\ \,\,v(\cdot , 0)=v_0(\cdot ):=(t^*)^{-\frac{1}{q+1-p}}u_0(\cdot ) \quad &{}\text {in}\quad \Omega , \end{array}\right. } \end{aligned}$$

(5.2)

where $\lambda _{p,q}:=q/(q+1-p)$ is a fixed constant. More precisely, v is a nonnegative weak solution to (5.2) in the following sense.

Definition 5.1

(Weak solutions of (5.2))A measurable function $v=v(x,s)$, defined on $\Omega _{\infty }:=\Omega \times (0,\infty )$, is called a nonnegative weak solution of (5.2) in $\Omega _\infty$ provided that the following conditions are satisfied:

• (D1) $v \in L^{\infty }(0,\infty ;\,W^{1,p}(\Omega ))$,    $\partial _s v^q \in L^1 (\Omega _S)$ for any positive $S < \infty$.

• (D2) For every $\varphi \in C^{\infty }_{0}(\Omega _\infty )$

  $$\begin{aligned} \iint _{\Omega _{\infty }}\Big [-v^q \,\partial _{s}\varphi +|\nabla v|^{p-2}\nabla v\cdot \nabla \varphi \Big ]\,dxds =\lambda _{p,q}\iint _{\Omega _\infty } v^q \varphi \,dxds. \end{aligned}$$

• (D3) v attains the initial datum continuously in the Sobolev space:

  $$\begin{aligned} \Vert v (s) - v_0\Vert _{W^{1,p} (\Omega )}\rightarrow 0 \quad \text {as}\quad s\searrow 0 \end{aligned}$$

  and satisfies the boundary condition in the trace sense:

  $$\begin{aligned} v(s) \in W^{1, p}_0 (\Omega ) \quad \text {for}\,\,\text {a.e.} \quad s \in (0, \infty ) \end{aligned}$$

We shall show the crucial properties satisfied by a nonnegative weak solution to (5.2).

Proposition 5.2

Let v be given by formula (5.1) with u being the nonnegative weak solution to (1.1) in the sense of Definition 3.10 and $t^*$ being the extinction time of u. Then v is a nonnegative weak solution of (5.2) with being strictly positive on $\Omega \times (0,s_0]$ for every positive $s_0 < \infty$. Moreover, the following quantitative estimates hold true for any $s \ge 0$:

$$\begin{aligned} \left( \frac{C_{p,q}^{-p}}{\lambda _{p,q}} \right) ^{\frac{1}{q+1-p}} \le \Vert v(s)\Vert _{q+1} \le \left( \frac{R(u_0)^p}{\lambda _{p,q}} \right) ^{\frac{1}{q+1-p}}, \end{aligned}$$

(5.3)

$$\begin{aligned} \left( \frac{C_{p,q}^{-(q+1)}}{\lambda _{p,q}} \right) ^{\frac{1}{q+1-p}} \le \Vert \nabla v(s)\Vert _p \le \left( \frac{R(u_0)^{q+1}}{\lambda _{p,q}} \right) ^{\frac{1}{q+1-p}} \end{aligned}$$

(5.4)

and v is bounded from above by

$$\begin{aligned} 0 \le v(s) \le e^{\lambda _{p,q}s/q}\left( (t^*)^{-\frac{1}{q+1-p}}\Vert u_0\Vert _\infty \right) \quad \text {in}\,\,\Omega . \end{aligned}$$

(5.5)

Proof

We shall verify that the conditions (D1)–(D3) in Definition 5.1 are fulfilled for the transformed solution v. The inequalities  (5.3) and (5.4) follow from the definition (5.1) and (4.1)–(4.3) in Proposition 4.2. Thus, the condition (D1)$_1$ is confirmed by (5.3) and (5.4). From the boundedness (3.1) and regularity (3.5) of the solution u, obtained in Theorem 3.2, the condition (D1)$_2$ follows. Indeed, since by $q \ge 1$ the function ${\mathbb {R}}_+ \ni w \mapsto f(w)=w^{\frac{2q}{q+1}}$ is locally Lipschitz continuous, it is

$$\begin{aligned} \partial _sv^{q}=\partial _s\left( v^{\frac{q+1}{2}}\right) ^{\frac{2q}{q+1}}=\tfrac{2q}{q+1}\left( v^{\frac{q+1}{2}}\right) ^{\frac{q-1}{q+1}}\partial _sv^{\frac{q+1}{2}}=\tfrac{2q}{q+1}v^{\frac{q-1}{2}}\partial _sv^{\frac{q+1}{2}}. \end{aligned}$$

(5.6)

Formally, the direct calculation shows that

$$\begin{aligned}&\partial _sv^{\frac{q+1}{2}}=\partial _t\left( (t^*-t)^{-\frac{q+1}{2(q+1-p)}}u(x,t)^{\frac{q+1}{2}}\right) \cdot \partial _st \\&\quad =\left[ \frac{q+1}{2(q+1-p)}(t^*-t)^{-\frac{q+1}{2(q+1-p)}-1}u(x,t)^{\frac{q+1}{2}} +(t^*-t)^{-\frac{q+1}{2(q+1-p)}}\partial _t u(x,t)^{\frac{q+1}{2}}\right] \partial _st \\&\quad =\frac{q+1}{2(q+1-p)}v(x,s)^{\frac{q+1}{2}}+(t^*-t)^{-\frac{q+1}{2(q+1-p)}}\partial _t u(x,t)^{\frac{q+1}{2}}\partial _st . \end{aligned}$$

Combining the preceding formulae, we have that $\partial _s t = t^*- t$ and

$$\begin{aligned} \partial _sv^{q}&=\frac{q}{q+1-p}v^{q}+\tfrac{2q}{q+1}(t^*-t)^{-\frac{q-1}{2(q+1-p)}-\frac{q+1}{2(q+1-p)}}u^{\frac{q-1}{2}}\partial _tu^{\frac{q+1}{2}}\partial _st \\&=\frac{q}{q+1-p}v^{q}+\tfrac{2q}{q+1}(t^*-t)^{-\frac{p-1}{q+1-p}}u^{\frac{q-1}{2}}\partial _tu^{\frac{q+1}{2}}. \end{aligned}$$

It is certain by (5.3) that $v \in L^\infty (0, \infty \,; L^{q+1} (\Omega ))$. Hence, for given positive $S<\infty$, setting $T:=t^*\left( 1-e^{-S}\right)$ and using  (3.1), (3.5) and Hölder’s inequality, we gain

$$\begin{aligned}&\iint _{\Omega _S} \partial _sv^{q}\,dxds=\frac{q}{q+1-p}\iint _{\Omega _S}v^{q}\,dxds\\&\qquad +\frac{2q}{q+1}\iint _{\Omega _T}(t^*-t)^{-\frac{q}{q+1-p}}u^{\frac{q-1}{2}}\partial _tu^{\frac{q+1}{2}}\,dxdt \\&\quad \le \frac{q}{q+1-p}|\Omega _S|^{\frac{1}{q+1}}\left( \iint _{\Omega _S}v^{q+1}\,dxds\right) ^{\frac{q}{q+1}} \\&\quad \quad +\frac{2q}{q+1}(t^*-T)^{-\frac{q}{q+1-p}}\Vert u_0\Vert _\infty ^{\frac{q-1}{2}}|\Omega _T|^{\frac{1}{2}}\iint _{\Omega _T}(\partial _tu^{\frac{q+1}{2}})^2\,dxdt<\infty , \end{aligned}$$

proving (D1)$_2$.

In change of variable $s=\log \left( \dfrac{t^*}{t^*-t}\right)$, keeping

$$\begin{aligned} (x,s) \in \Omega _ \infty \quad \iff \quad (x,t) \in \Omega _{0, t^*}:=\Omega \times (0,t^*) \end{aligned}$$

in mind, we observe that, for every testing function $\psi =\psi (x,s) \in C^\infty _0(\Omega _\infty )$,

$$\begin{aligned} \iint _{\Omega _\infty }\partial _s v^q\cdot \psi \,dxds&=-\iint _{\Omega _\infty }v^q\psi _s\,dxds \\&=-\iint _{\Omega _{0,t^*}}(t^*-t)^{-\frac{q}{q+1-p}}u(x,t)^q\psi _t\left( x,\log \left( \tfrac{t^*}{t^*-t}\right) \right) \,dxdt. \end{aligned}$$

By choosing $\varphi (x,t):=(t^*-t)^{-\frac{q}{q+1-p}}\psi \left( x,\log \left( \tfrac{t^*}{t^*-t}\right) \right) \in C^\infty _0(\Omega \times (0,t^*))$, it is

$$\begin{aligned}&\iint _{\Omega _\infty }\partial _s v^q\cdot \psi \,dxds \\&\quad =-\iint _{\Omega _{0,t^*}}(t^*-t)^{-\frac{q}{q+1-p}}u(x,t)^q\Big [(t^*-t)^{\frac{q}{q+1-p}}\varphi _t-\lambda _{p,q}(t^*-t)^{\frac{q}{q+1-p}-1}\varphi \Big ]\,dxdt \\&\quad =-\iint _{\Omega _{0,t^*}}u(x,t)^q\varphi _t\,dxdt+\iint _{\Omega _{0,t^*}}\lambda _{p,q}u(x,t)^q\varphi (x,t)\,dx\underbrace{\left( (t^*-t)^{-1}dt\right) }_{=ds} \\&\quad =\iint _{\Omega _{0,t^*}}|\nabla u|^{p-2}\nabla u\cdot \nabla \varphi \,dxdt + \iint _{\Omega _\infty }\lambda _{p,q}v^q(x,s)\psi (x,s)\,dxds \\&\quad =\iint _{\Omega _{0,t^*}}(t^*-t)^{-\frac{q}{q+1-p}}|\nabla u|^{p-2}\nabla u(x,t)\cdot \nabla \psi \left( x,\log \left( \tfrac{t^*}{t^*-t}\right) \right) \,dxdt \\&\qquad + \iint _{\Omega _\infty }\lambda _{p,q}v^q(x,s)\psi (x,s)\,dxds \\&\quad =\iint _{\Omega _\infty }\Big [|\nabla v|^{p-2}\nabla v\cdot \nabla \psi +\lambda _{p,q}v^q\psi \Big ]\,dxds, \end{aligned}$$

which claims the condition (D2).

The initial boundary condition (D3) plainly follows by the condition (iii) in Definition 3.1 and therefore the transformed v is a weak solution to (3.6). Further, from the definition (5.1) of v and the positivity of u in $\Omega \times (0, t^*)$ stated in Theorem 1.1, it follows that

$$\begin{aligned} v=v(x,s)>0 \quad \textrm{in}\quad \Omega \times (0,s_0] \end{aligned}$$

(5.7)

for every $s_0=\log \left( \dfrac{t^*}{t^*-t_0} \right) <+\infty$ and any positive $t_0<t^*$ with $t^*$ being the extinction time of u.

In the final step, we show the sketch of the proof of (5.5). For $0\le s_1<s_2<\infty$ and $\delta >0$ small enough, we define the following Lipschitz cut-off function:

$$\begin{aligned} \chi _{\delta }(s):= {\left\{ \begin{array}{ll} 0, \quad &{}s \in [0, s_1),\\ \frac{1}{\delta }(s-s_1), \quad &{}s \in [s_1, s_1+\delta ),\\ 1, \quad &{}s \in [s_1+\delta , s_2-\delta ],\\ -\frac{1}{\delta }(s-s_2), \quad &{}s \in (s_2-\delta , s_2], \\ 0, \quad &{}s \in (s_2, \infty ). \end{array}\right. } \end{aligned}$$

We further define, for any $\delta >0$

$$\begin{aligned} \varphi _{\delta }(v):=\min \left\{ 1,\frac{\left( e^{-\lambda _{p,q}s/q}v-M\right) _+}{\delta }\right\} \quad \text {with}\quad M:=(t^*)^{-\frac{1}{q+1-p}}\Vert u_0\Vert _\infty . \end{aligned}$$

We choose a testing function as $e^{-\lambda _{p,q}s}\chi _{\delta }(s)\varphi _{\delta }(v)$ in the weak formulation (D2) and take the limit as $\delta \searrow 0$ similarly as [18, Proposition 3.5] to have that

$$\begin{aligned} \left[ \int _\Omega \left( e^{-\lambda _{p,q}s}v^q-M^q\right) _+\,dx\right] _{s_1}^{s_2} \le 0, \end{aligned}$$

where we use that the time-derivative of powered v is integrable in space-time, that is stated in (D1)$_2$ and checked before. Thus, letting $s_1 \searrow 0$ gives that

$$\begin{aligned} \int _{\Omega } \left( e^{-\lambda _{p,q}s_2}v(s_2)^q-M^q\right) _+\,dx&\le \int _{\Omega } \left( e^{-\lambda _{p,q}s_2}v(s_1)^q-M^q\right) _+\,dx \\&\le \int _{\Omega } \left( v(s_1)^q-M^q\right) _+\,dx\rightarrow 0, \end{aligned}$$

that is

$$\begin{aligned} e^{-\lambda _{p,q}s}v(s)^q-M^q \le 0\qquad \forall (x,s) \in \Omega _\infty , \end{aligned}$$

as desired.

Therefore the proof is complete. $\square$

We are now ready to prove Theorem 1.2.

Proof of Theorem 1.2

The proof of theorem 1.2 now goes in several steps.

Step 1:   For any increasing sequence $0=\tau _0 \le \tau _k \nearrow t^*$, we set $\displaystyle s_k:=\log \left( \frac{t^*}{t^*-\tau _k} \right)$. First, we shall prove that

$$\begin{aligned} \lim _{k \rightarrow \infty }\int _\Omega \left| \partial _s v^q (\theta _k)\right| \,dx=0 \end{aligned}$$

(5.8)

for some number $\theta _k \in (s_k,s_k+1) \setminus {\mathcal {N}}$, where ${\mathcal {N}}$ is negligible with respect to the one-dimensional Lebesgue measure. Indeed, recalling (5.1) and Lemma C.4, there holds

$$\begin{aligned} \displaystyle \int _0^\infty \!\!\! \int _\Omega \Big [\partial _s v^{(q+1)/2}\Big ]^2\,dxds<+\infty . \end{aligned}$$

Using this together with

and the mean value theorem, there exist a number $\theta _k \in (s_k, s_k+1) {\setminus } {\mathcal {N}}$ such that

$$\begin{aligned} \lim _{k \rightarrow \infty } \int _\Omega \Big [\partial _sv^{(q + 1)/2} (\theta _k)\Big ]^2\,dx=0, \end{aligned}$$

where ${\mathcal {N}}$ is a negligible subset in $(0,\infty )$ with respect to Lebesgue measure on ${\mathbb {R}}$. By means of Hölder’s inequality, (5.6) and (5.3), it is

$$\begin{aligned} \int _\Omega |\partial _s v^q (\theta _k)| \,dx&\le \frac{2q}{q+1}\left( \int _\Omega v^{q-1}(\theta _k)\,dx\right) ^{1/2}\left( \int _\Omega \Big [\partial _sv^{(q+1)/2}(\theta _k)\Big ]^2\,dx\right) ^{1/2} \\&\le \frac{2q}{q+1}|\Omega |^{1 / (q + 1)} \left( \sup _{s \in (0,\infty )} \Vert v (s)\Vert _{q+1}^{(q+1)/2}\right) \left\| \partial _s v^{(q + 1)/2} (\theta _k)\right\| _{2} \\&\!\!{\mathop {\le }\limits ^{(5.3)}} \,\,\,\frac{2q}{q+1}|\Omega |^{1/(q+1)} \left( \frac{R(u_0)^p}{\lambda _{p,q}} \right) ^{\frac{q-1}{2(q+1-p)}} \left\| \partial _s v^{(q+1)/2}(\theta _k)\right\| _{2} \rightarrow 0 \end{aligned}$$

in the limit $k \rightarrow \infty$, which validates (5.8).

With this $\theta _k$, by letting $t_k:=t^*(1-e^{-\theta _k})$ it is

$$\begin{aligned} \theta _k \nearrow \infty \quad \iff \quad t_k \nearrow t^*\end{aligned}$$

as $k\rightarrow \infty$, and therefore, via (D2) in Definition 5.1 it holds that

$$\begin{aligned} \int _\Omega \partial _sv^q(\theta _k)\varphi \,dx+\int _\Omega |\nabla v|^{p-2}\nabla v(\theta _k) \cdot \nabla \varphi \,dx=\lambda _{p,q}\int _\Omega v^q(\theta _k)\varphi \,dx \end{aligned}$$

(5.9)

for every $\varphi =\varphi (x) \in C^\infty _0(\Omega )$. Since by (5.3) and (5.4) $\{v(\theta _k)\}$ is bounded in $W_0^{1,p}(\Omega )$, there exist a (non-relabeled) subsequence $\{\theta _k\}$ and a limit function $U \in W_0^{1,p}(\Omega )$ so that

$$\begin{aligned} v(\cdot , \theta _k) \rightarrow U\quad \text {weakly}\,\,\text {in}\,\,W_0^{1,p}(\Omega ) \end{aligned}$$

(5.10)

in the limit $k \rightarrow \infty$, where we used Mazur’s theorem implying that the closed subspace $W^{1, p}_0 (\Omega )$ of $W^{1, p} (\Omega )$ is weakly closed in $W^{1, p} (\Omega )$. Moreover, this together with the compact embedding $W_0^{1,p}(\Omega ) \hookrightarrow L^{r}(\Omega )$ for all $r \in [1,p^*)$ yields the strong convergence

$$\begin{aligned} v(\cdot , \theta _k) \rightarrow U\quad \text {strongly}\,\,\text {in}\,\,L^{r}(\Omega ), \quad \forall r \in [1,p^*), \end{aligned}$$

(5.11)

therefore, up to extract a (non-relabeled) subsequence, we deduce that

$$\begin{aligned} v(\cdot , \theta _k) \rightarrow U\quad \text {a.e.}\,\,\text {in}\,\,\Omega . \end{aligned}$$

(5.12)

Step 2:   In this step, we are going to show the above weak limit U is actually nonnegative weak solution to (1.3). Fort this, we first deduce the strong convergence of the gradient in order to derive the convergence of the weak form of p-Laplacian and lower order terms appearing in (5.9).

Lemma 5.3

(Strong convergence of the gradient) For all $r \in [1,p)$, as $k \rightarrow \infty$,

$$\begin{aligned} \nabla v(\theta _k) \rightarrow \nabla U\quad strongly\,\,\,in \,\,\,L^r(\Omega ), \end{aligned}$$

where U is the limit function as in the procedure leading to (5.10).

The proof of this lemma is postponed, and will be given in Appendix E.

The Hölder inequality, the algebraic inequality (2.3), the estimate (5.4) and the strong convergence of the gradient, Lemma 5.3 give the estimations: For all $p\ge 2$

$$\begin{aligned}&\int _\Omega \big ||\nabla v|^{p-2}\nabla v(\theta _k)-|\nabla U|^{p-2} \nabla U\big ||\nabla \varphi |\,dx \\&\le \quad C\Vert \nabla \varphi \Vert _\infty \int _\Omega \left[ |\nabla v(\theta _k)|^{p-2}+|\nabla U|^{p-2} \right] |\nabla v(\theta _k)-\nabla U|\,dx \\&\le \quad C\Vert \nabla \varphi \Vert _\infty \left( \int _\Omega \left[ |\nabla v(\theta _k)|^{p-1}+|\nabla U|^{p-1} \right] \,dx \right) ^\frac{p-2}{p-1} \left( \int _\Omega |\nabla v(\theta _k)-\nabla U|^{p-1}\,dx \right) ^\frac{1}{p-1} \\&\le \quad C\Vert \nabla \varphi \Vert _\infty |\Omega |^\frac{p-2}{p(p-1)} \left( \Vert \nabla v(\theta _k)\Vert _p^{p-2}+\Vert \nabla U\Vert _p^{p-2} \right) \Vert \nabla v(\theta _k)-\nabla U\Vert _{p-1} \\&\!\!{\mathop {\le }\limits ^{(5.4)}} \,\,\,C\Vert \nabla \varphi \Vert _\infty |\Omega |^\frac{p-2}{p(p-1)} \left( \left[ \frac{R(u_0)^{q+1}}{\lambda _{p,q}}\right] ^{\frac{p-2}{q+1-p}}+\Vert \nabla U\Vert _p^{p-2} \right) \Vert \nabla v(\theta _k)-\nabla U\Vert _{p-1} \rightarrow 0 \end{aligned}$$

as $k \rightarrow \infty$ and therefore, it is

$$\begin{aligned} \int _\Omega |\nabla v|^{p-2}\nabla v(\theta _k)\cdot \nabla \varphi \,dx \rightarrow \int _\Omega |\nabla U|^{p-2}\nabla U\cdot \nabla \varphi \,dx. \end{aligned}$$

(5.13)

When $1< p < 2$, from $|\nabla v(\theta _k)|+|\nabla U| \ge |\nabla v(\theta _k)-U|$ and (2.3), it follows that

$$\begin{aligned}{} & {} \int _\Omega \left| |\nabla v|^{p - 2} \nabla v (\theta _k)- |\nabla U|^{p - 2} \nabla U\right| |\nabla \varphi | dx\\{} & {} \quad \le C \Vert \nabla \varphi \Vert _\infty \int _\Omega \left| \nabla v(\theta _k) - \nabla U\right| ^{p - 1} d x\rightarrow 0 \end{aligned}$$

as $k \rightarrow \infty$, which claims (5.13) in the case $1< p < 2$.

Similarly as above, by means of (2.3) and (5.3), we have for $q \ge 1$,

$$\begin{aligned}&\lambda _{p,q}\int _\Omega \big |v^q(\theta _k)-U^q\big ||\varphi |\,dx \\&\quad \le \quad C\Vert \varphi \Vert _\infty \int _\Omega \left[ |v(\theta _k)|^{q-1}+|U|^{q-1} \right] |v(\theta _k)-U|\,dx \\&\quad \le \quad C\Vert \varphi \Vert _\infty \left( \int _\Omega \left[ |v(\theta _k)|^q+|U|^q \right] \,dx \right) ^\frac{q-1}{q} \left( \int _\Omega |v(\theta _k)-U|^q\,dx \right) ^\frac{1}{q} \\&\quad \!\!{\mathop {\le }\limits ^{(5.3)}} \quad C\Vert \varphi \Vert _\infty \left( \left[ \frac{R(u_0)}{\lambda _{p,q}}\right] ^{\frac{q-1}{q+1-p}}+\Vert U\Vert _q^{q-1} \right) \Vert v(\theta _k)- U\Vert _q \rightarrow 0 \end{aligned}$$

and therefore,

$$\begin{aligned} \lambda _{p,q}\int _\Omega v^q(\theta _k)\varphi \,dx \rightarrow \lambda _{p,q}\int _\Omega U^q\varphi \,dx \end{aligned}$$

(5.14)

as $k \rightarrow \infty$. On the whole, merging (5.8), (5.9), (5.13) and (5.14), we find that the limit function U satisfies, for every $\varphi \in C_0^\infty (\Omega )$,

$$\begin{aligned} \int _\Omega |\nabla U|^{p-2}\nabla U\cdot \nabla \varphi \,dx=\lambda _{p,q}\int _\Omega U^q\varphi \,dx, \end{aligned}$$

(5.15)

and the nonnegativity of U directly follows from that of $v(\theta _k)$ and convergence (5.12).

In the subcritical case $q + 1 < p^*$, we shall prove the strong convergence of $v (\theta _k)$ to U in $W^{1, p}_0 (\Omega )$. For this we first prove that

$$\begin{aligned} \int _\Omega \left| \partial _s v^q(\theta _k) (v(\theta _k) - U)\right| \,dx \rightarrow 0 \end{aligned}$$

(5.16)

as $k \rightarrow \infty$. In fact, since by $q \ge 1$ the chain rule of weak differential implies

$$\begin{aligned} \partial _s v^q = \frac{2 q}{q + 1} v^{(q - 1)/2} \partial _s v^{(q + 1)/2}. \end{aligned}$$

Then, by means of Hölder’s inequality and (5.3), it is

$$\begin{aligned}&\int _\Omega \left| \partial _s v^q(\theta _k) (v(\theta _k) - U)\right| \,dx \\&\quad \le \quad \frac{2 q}{q + 1} \int _\Omega |\partial _s v^{(q + 1)/2}(\theta _k)|\left( v^{(q + 1)/2}(\theta _k) + v^{(q - 1)/2}(\theta _k) |U|\right) \,dx \\&\quad \le \quad \frac{2 q}{q + 1} \left\| \partial _s v^{(q + 1)/2}(\theta _k) \right\| _2 \Big (\Vert v(\theta _k)\Vert _{q+1}^{(q+1)/2} +\Vert v (\theta _k)\Vert _{q + 1}^{q-1} \Vert U \Vert _{q + 1}^q\Big ) \\&\quad \!\!{\mathop {\le }\limits ^{(5.3)}} \,\,\, \frac{2 q}{q + 1}\left\| \partial _s v^{(q + 1)/2}(\theta _k) \right\| _2\\&\qquad \left( \left[ \frac{R(u_0)^p}{\lambda _{p,q}}\right] ^{\frac{q+1}{q+1-p}}+\left[ \frac{R(u_0)^p}{\lambda _{p,q}}\right] ^{\frac{q-1}{2(q+1-p)}}\Vert U \Vert _{q + 1}^q \right) , \end{aligned}$$

which converges to zero, since $\Vert \partial _sv^{(q + 1)/2} (\theta _k) \Vert _2 \rightarrow 0$ in the limit $k \rightarrow \infty$.

At this stage, we now subtract (5.15) from (5.9) and use the test function $\varphi = v (\theta _k) - U$ in the resulting equation. Similarly as the manipulation leading to (5.13) and (5.14), by the algebraic inequality (2.5) if $p \ge 2$ then

$$\begin{aligned}&C\int _\Omega |\nabla v (\theta _k) - \nabla U|^p \,dx \\&\quad \le \int _\Omega \left( |\nabla v (\theta _k)|^{p - 2}\nabla v (\theta _k)-|\nabla U|^{p - 2} \nabla U\right) \cdot (\nabla v (\theta _k) - \nabla U)\,dx \\&\quad =-\int _\Omega \partial _s v^q (\theta _k) (v (\theta _k) - U)dx+\lambda _{p, q}\int _\Omega (v^q (\theta _k) - U^q) (v (\theta _k) - U)\,dx\\&\quad \le \int _\Omega \left| \partial _s v^q(\theta _k) (v(\theta _k) - U)\right| \,dx+C \big \Vert |v (\theta _k)| + |U|\big \Vert _{q + 1}^{q - 1}\Vert v (\theta _k) - U\Vert _{q + 1}^2, \\&\quad \rightarrow 0 \end{aligned}$$

as $k \rightarrow \infty$, where in the final line we used (5.16) and the strong convergence of $v (\theta _k)$ to U in  (5.11). In the remaining case $1<p<2$, by the Hölder inequality and (2.4) we see that

$$\begin{aligned}&\int _\Omega |\nabla v(\theta _k)-\nabla U|^p\,dx \\&\quad \le \left( \int _\Omega |\nabla v(\theta _k)-\nabla U|^2\left[ |\nabla v(\theta _k)|+|\nabla U| \right] ^{p-2}\,dx \right) ^{\frac{p}{2}} \\&\qquad \cdot \left( \int _\Omega \left[ |\nabla v(\theta _k)|+|\nabla U| \right] ^p\,dx \right) ^{\frac{2-p}{2}} \\&\quad \le C\left( \int _\Omega \left( |\nabla v(\theta _k)|^{p-2}\nabla v(\theta _k)-|\nabla U|^{p-2}\nabla U \right) \cdot \left( \nabla v(\theta _k)-\nabla U\right) \,dx \right) ^{\frac{p}{2}} \\&\qquad \cdot \left( \int _\Omega \left[ |\nabla v(\theta _k)|+|\nabla U|\right] ^p\,dx \right) ^{\frac{2-p}{2}} \\&\quad \le C\Bigg (\int _\Omega \left| \partial _s v^q(\theta _k) (v(\theta _k) - U)\right| \,dx+C \big \Vert |v (\theta _k)| + |U|\big \Vert _{q + 1}^{q - 1}\Vert v (\theta _k) - U\Vert _{q+1}^2\Bigg )^{\frac{p}{2}} \\&\qquad \cdot \left( \left[ \frac{R(u_0)^{q+1}}{\lambda _{p,q}}\right] ^{\frac{p(2-p)}{2(q+1-p)}}+\Vert \nabla U\Vert _p^{\frac{p(2-p)}{2}} \right) \\&\quad \rightarrow 0 \end{aligned}$$

as $k \rightarrow \infty$, where in the last line we used (5.4) and the strong convergence of $v (\theta _k)$ to U in  (5.11) again.

In the subcritical case $q + 1 < p^*$, the solution U to (1.3) obtained as the limit of $v (\theta _k)$ also inherits the same boundedness as in  (5.3) and (5.4) from the solutions $v (\theta _k)$ by virtue of the strong convergence of $v (\theta _k)$ to U in $W^{1, p}_0 (\Omega )$. By (5.3) and (5.4) for U, the solution U is not identically zero and nonnegative in $\Omega$.

On the other hand, in the critical case $q + 1 = p^*$, the solution U does not satisfy the lower boundedness in  (5.3) and (5.4) because of the restriction of convergences  (5.10) and (5.11). Thus, as well-known, the energy and volume gap may appear along the limitation of $v (\theta _k)$ to U as $k \rightarrow \infty$.

This completes the proof of Theorem 1.2. $\square$

Appendices

Appendix A: Uniqueness of a weak solution to (3.6) and (5.2)

Now we shall state the uniqueness of a weak solution to (3.6) and (5.2).

Theorem A.1

(Uniqueness for  (3.6) and (5.2)) A weak solution of (3.6) is unique. So is that of (5.2).

The proof is based on the comparison theorem, retrieved from [18, Theorem 3.6]. Here it is decisive that the weak time-derivative of powered solution in the equation is integrable in space-time.

Theorem A.2

(Comparison theorem) Let $0<S \le \infty$. Suppose that $v^+$ and $v^-$ are weak supersolution and subsolution to (3.6) or (5.2), respectively. If $v^+ \ge v^-$ on the parabolic boundary $\partial _{\text {par}}\Omega _S:=\partial \Omega \times (0,S) \cup {\overline{\Omega }} \times \{0\}$, then

$$\begin{aligned} v^+ \ge v^-\quad \text {in}\,\,\,\Omega _S. \end{aligned}$$

Appendix B: Approximation of the doubly nonlinear equation (3.6)

This appendix is devoted to regularity estimates for an approximation solution of the doubly nonlinear equation (3.6), which is the key to deriving energy estimates for the original solution u to (3.6), as stated in Proposition B.9 in Sect. 5.1 below. Following [23, Sect. 7.2], we proceed argument as follows.

Let $\varepsilon , \,\delta$ be arbitrary positive numbers such that $\varepsilon \searrow 0, \,\delta \searrow 0$ later. We shall show the existence and some regularity estimates for nonnegative solutions $u=u_{\varepsilon , \delta }$ to the equation

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t (u+\varepsilon )^q-\textrm{div}\,\left( (\delta +|\nabla u|^2 )^{\frac{p-2}{2}} \nabla u\right) =0 \quad &{}\text {in} \,\,\Omega _\infty :=\Omega \times (0,\infty ) \\ u=0 \quad &{}\text {on}\,\,\partial \Omega \times (0,\infty ) \\ u(0)=u_0 \quad \,\,&{}\text {in}\,\,\Omega , \end{array}\right. } \end{aligned}$$

(B.1)

where the initial value $u_0$ is as in (3.6), which in turn becomes, by setting $U_\varepsilon :=u+\varepsilon >0$,

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t U_\varepsilon ^q-\textrm{div}\,\left( (\delta +|\nabla U_\varepsilon |^2 )^{\frac{p-2}{2}}\nabla U_\varepsilon \right) =0 \quad &{}\text {in} \,\,\Omega _\infty \\ U_\varepsilon =\varepsilon \quad &{}\text {on}\,\,\partial \Omega \times (0,\infty ) \\ U_\varepsilon (0)=u_0+\varepsilon \quad \,\,&{}\text {in}\,\,\Omega . \end{array}\right. } \end{aligned}$$

(B.2)

We now fix $T \in (0,\infty )$. For every $t<T$, we define

$$\begin{aligned} v(x,s)\equiv v_{\varepsilon ,\delta }(x,s):=(T-t)^{-\frac{1}{q+1-p}}U_\varepsilon (x,t)-\varepsilon ,\qquad s=\log \left( \frac{T}{T-t}\right) \end{aligned}$$

and

$$\begin{aligned} V_\varepsilon :=v+\varepsilon . \end{aligned}$$

The straightforward computation shows that $V_\varepsilon$ solves

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _s V_\varepsilon ^q-\textrm{div}\,\left( \left( \delta (Te^{-s})^{-\frac{2}{q+1-p}}+|\nabla V_\varepsilon |^2 \right) ^{\frac{p-2}{2}}\nabla V_\varepsilon \right) =\lambda _{p,q}V_\varepsilon ^q \quad &{}\text {in} \,\,\Omega _\infty \\ V_\varepsilon =\varepsilon (T-t)^{-\frac{1}{q+1-p}} \quad &{}\text {on}\,\,\partial \Omega \times (0,\infty ) \\ V_\varepsilon (0)=T^{-\frac{1}{q+1-p}}\left( u_0+\varepsilon \right) \quad \,\,&{}\text {in}\,\,\Omega \end{array}\right. }\nonumber \\ \end{aligned}$$

(B.3)

with the constant $\lambda _{p,q}=\frac{q}{q+1-p}$.

1.1 B.1. Existence and quantitative estimates of solutions to (B.1)

We now show the existence and energy estimates of solutions to  (B.1).

Proposition B.1

(Existence of a weak solution to (B.1)) For any nonnegative $u_0 \in W_0^{1,p} \cap L^\infty (\Omega )$, there exists a unique weak solution u to (B.1) such that u is nonnegative and bounded as

$$\begin{aligned} 0 \le u \le \Vert u_0\Vert _\infty \quad \text {in}\,\,\Omega _\infty \end{aligned}$$

(B.4)

and u satisfies the integral estimates, for all $t \ge 0$,

$$\begin{aligned}&\Vert u(t)+\varepsilon \Vert _{q+1} \le C \left( \Vert u_0\Vert _{q+1}+1 \right) , \nonumber \\&\Vert \left( \delta +|\nabla u(t)|^2 \right) ^{\frac{1}{2}} \Vert _p \le \Vert \nabla u_0\Vert _{p}+1, \nonumber \\&\Vert \partial _t (u+\varepsilon )^q\Vert _2^2 \le C \left( \Vert u_0\Vert _\infty +1 \right) ^{q-1}\left( \Vert \nabla u_0\Vert _p^p+1 \right) . \end{aligned}$$

(B.5)

Proof

The existence of a weak solution and the integral estimates are obtained from the approximation as in [25, Lemma 3.2, (3.7); Lemma 4.1; Proof of Theorem 1.1]. The nonnegativity and boundedness follow from the comparison principle (see [25, Theorem 3.6]) for (B.2)

$$\begin{aligned} \varepsilon \le U_\varepsilon \le \varepsilon +\Vert u_0\Vert _\infty , \end{aligned}$$

finishing the proof. $\square$

1.2 B.2. Regularity of solutions to Eq. (B.1)

For the reader’s convenience, following [23, Sect. 7.2], we collect the regularity results for Eq. (B.1), which enable us to deduce the energy inequality, as stated in Lemma B.9.

Let u be a weak solution of (B.1). Set

$$\begin{aligned} \mu :=(u+\varepsilon )^q \iff u=\mu ^\frac{1}{q}-\varepsilon . \end{aligned}$$

This procedure transforms (B.1) into the following problem

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t \mu -\textrm{div}\,\left( \left( \dfrac{\mu }{q} \right) ^{(\frac{1}{q}-1)(p-1)}(q^2\mu ^{2(1-\frac{1}{q})}\delta +|\nabla \mu |^2 )^{\frac{p-2}{2}}\nabla \mu \right) =0 \quad &{}\text {in} \,\,\Omega _\infty \\ \mu =\varepsilon ^q \quad &{}\text {on}\,\,\partial \Omega \times (0,\infty ) \\ \mu (0)=(u_0+\varepsilon )^q \quad \,\,&{}\text {in}\,\,\Omega . \end{array}\right. }\nonumber \\ \end{aligned}$$

(B.6)

Due to (B.4) in Proposition B.1 there holds that

$$\begin{aligned} \varepsilon ^q \le \mu \le (\Vert u_0\Vert _\infty +\varepsilon )^q \end{aligned}$$

(B.7)

Set $\mu -\varepsilon ^q=\nu$, then (B.6) becomes

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t \nu -\textrm{div}\,\left( \left( \dfrac{\nu +\varepsilon ^q}{q} \right) ^{(\frac{1}{q}-1)(p-1)}\left( q^2(\nu +\varepsilon ^q)^{2(1-\frac{1}{q})}\delta +|\nabla \nu |^2 \right) ^{\frac{p-2}{2}}\nabla \nu \right) =0 \quad &{}\text {in} \,\,\Omega _\infty \\ \nu =0 \quad &{}\text {on}\,\,\left( \partial \Omega \right) _\infty \\ \nu (0)=(u_0+\varepsilon )^q-\varepsilon ^q \quad \,\,&{}\text {in}\,\,\Omega ,\end{array}\right. }\nonumber \\ \end{aligned}$$

(B.8)

where we used shorthand notation $\Omega _\infty :=\Omega \times (0,\infty )$ and $\left( \partial \Omega \right) _\infty :=\partial \Omega \times (0,\infty )$.

The condition (B.7) is written for the solution $\nu$ to (B.8) as

$$\begin{aligned} \varepsilon ^q\le \nu +\varepsilon ^q \le \left( \Vert u_0\Vert _\infty +\varepsilon \right) ^q \quad \text {in}\quad \Omega _\infty . \end{aligned}$$

(B.9)

The boundary regularity can be transformed into that in the interior case in the usual way (refer to [9, Chapter X, Sect. 1–2, pp. 292–296] and also [11, Appendix C.1, Page 710]). Take an arbitrary point $(x_0,t_0) \in \partial \Omega \times (0,T)$ and introduce a change of coordinates which maps a small portion of the boundary $\partial \Omega$ around $(x_0, t_0)$ into a portion on hyperplane. We consider the equation (B.8) in the local parabolic cylinder

$$\begin{aligned} Q_R(x_0,t_0):=B_R(x_0) \times (t_0-R,t_0), \end{aligned}$$

where the positive number R satisfies $R < t_0$. If necessary, the radius R is chosen small and the portion of the boundary $\partial \Omega \cap B_R(x_0)$ can be represented as the graph

$$\begin{aligned} x_n=\gamma ({\bar{x}})\equiv \gamma (x_1,\ldots ,x_{n-1}) \end{aligned}$$

of a smooth function $\gamma :{\mathbb {R}}^{n-1} \rightarrow {\mathbb {R}}$. Let us define the following change of variables

$$\begin{aligned} {\left\{ \begin{array}{ll} y_i=x_i=:\Phi ^i(x)\quad (i=1,\ldots ,n-1) \\ y_n=x_n-\gamma ({\bar{x}})=:\Phi ^n(x) \end{array}\right. } \end{aligned}$$

and assume that $\nabla _{{{\bar{x}}}} \gamma ({\bar{x}}_0) = 0$; therefore $\nabla _{{{\bar{y}}}} \gamma ({\bar{y}}_0) = 0$, by letting the graph of $\gamma$ at $x_0$ tangent to the $(n - 1)$-dimensional ${\bar{x}}$-plane. We shall shorten

$$\begin{aligned} y=\Phi (x). \end{aligned}$$

Let

$$\begin{aligned} {\left\{ \begin{array}{ll} x_i=y_i=:\Psi ^i(y)\quad (i=1,\ldots ,n-1) \\ x_n=y_n+\gamma ({\bar{y}})=:\Psi ^n(y) \end{array}\right. } \end{aligned}$$

and denote

$$\begin{aligned} x=\Psi (y). \end{aligned}$$

Then $\Phi =\Psi ^{-1}$ and the mapping $x \mapsto \Phi (x)=y$ is flattening boundary near $x_0$, that is, the portion $\partial \Omega \cap B_R(x_0)$ is mapped to $\{y_n=0\} \cap {\mathcal {B}}_R(y_0)$ and it is $\Omega \cap {\mathcal {B}}_R(y_0) \subset \{y_n>0\}$, where $y_0:=\Phi (x_0)$ and ${\mathcal {B}}_R(y_0):=\Phi (B_R(x_0))$. Furthermore, the Jacobi matrix of the map $x=\Psi (y)$ is

$$\begin{aligned} D\Psi (y)=\left( \begin{array}{c|c}I_{n-1} &{} 0 \\ \hline \nabla _{{\bar{y}}}\gamma ({\bar{y}}) &{} 1\end{array}\right) \end{aligned}$$

with the $(n-1) \times (n-1)$ identity matrix $I_{n-1}$ and its Jacobian is given by

$$\begin{aligned} \det D\Psi (y)=1. \end{aligned}$$

Under this change of variables (B.8)$_1$ and (B.8)$_2$ become

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t \nu -\nabla _{y_\ell }\left( f(\nu ,\nabla \nu )\,a_{\ell ,k}\,\nabla _{y_k}\nu \right) =0 \quad &{}\text {in} \,\,{\mathcal {Q}}_R(z_0) \cap \{y_n > 0\}\\ \nu =0 \quad &{}\text {on}\,\,{\mathcal {Q}}_R(z_0) \cap \{y_n=0\}, \end{array}\right. } \end{aligned}$$

(B.10)

where for brevity, ${\mathcal {Q}}_R(z_0):={\mathcal {B}}_R(y_0) \times (s_0-R,s_0)$ and by the transformed solution

$$\begin{aligned} \nu ({{{\bar{y}}}}, y_n+\gamma ({{{\bar{y}}}}),\,t) \quad \text {with}\quad {{{\bar{y}}}} = (y_1, \ldots , y_{n-1}) \end{aligned}$$

we denote the transformation with the same notation as the original $\nu$ in (B.8). Moreover, we set

$$\begin{aligned}&f(\nu ,\nabla \nu ):=\left( \frac{\nu +\varepsilon ^q}{q}\right) ^{-(1-\frac{1}{q})(p-1)}\left\{ q^2\delta (\nu +\varepsilon ^q)^{2(1-\frac{1}{q})}+(\nabla _{y_n}\nu )^2+\left| \nabla _{{\bar{y}}}\nu -\nabla _{y_n}\nu \nabla _{{\bar{y}}}\gamma \right| ^2 \right\} ^\frac{p-2}{2} \\&\quad =\left( \frac{\nu +\varepsilon ^q}{q}\right) ^{-(1-\frac{1}{q})(p-1)}\left\{ q^2\delta (\nu +\varepsilon ^q)^{2(1-\frac{1}{q})}+\sum \limits _{k^\prime ,\ell ^\prime =1}^na_{k^\prime ,\ell ^\prime }(y)\nabla _{y_{k^\prime }}\nu \nabla _{y_{\ell ^\prime }}\nu \right\} ^\frac{p-2}{2} \end{aligned}$$

and, the coefficient matrix $(a_{\ell ,k}(y))$ is given by

$$\begin{aligned} (a_{\ell ,k}(y)):=\left( \begin{array}{c|c}I_{n-1} &{} -(\nabla _{{\bar{y}}} \gamma )^\top \\ \hline -\nabla _{{\bar{y}}} \gamma &{} 1+|\nabla _{{\bar{y}}} \gamma |^2 \end{array}\right) . \end{aligned}$$

(B.11)

Hereafter, the summation convention on repeated indices is used and the dependence on approximation parameters $\varepsilon$ and $\delta$ of the function f will be kept in mind.

Here we notice the uniformly ellipticity and boundedness condition for the coefficient $(a_{k, \ell } (y))$: For any $\eta = ({\bar{\eta }}, \eta _n) \in {\mathbb {R}}^n$

$$\begin{aligned} \frac{1}{2} |\eta |^2 \le \sum \limits _{k,\ell =1}^na_{k, \ell } \eta _k \eta _{\ell } = |{{{\bar{\eta }}}}|^2 + \eta _n^2 (1 + |\nabla _{\bar{y}} \gamma |^2)- 2\eta _n {{{\bar{\eta }}}}\cdot \nabla _{{{\bar{y}}}} \gamma \le 2 |\eta |^2,\nonumber \\ \end{aligned}$$

(B.12)

where we used Cauchy’s inequality and the size R of the local parabolic cylinder can be chosen so that $|\nabla _{{{\bar{y}}}} \gamma ({{{\bar{y}}}})|$ is sufficient small for any ${\bar{y}} \in {\mathcal {B}}_R(y_0) \cap \{y_n=0\}$, since $\nabla _{{{\bar{y}}}} \gamma ({{{\bar{y}}}}_0) = 0$ and $\gamma ({\bar{y}})$ is smooth.

Note that the solution $\nu$ to (B.8) is nonnegative and we may treat (B.10)$_1$ as

$$\begin{aligned} \partial _t \nu -\nabla _{y_\ell }\left( {\bar{f}}(\nu ,\nabla \nu )\,a_{\ell ,k}\,\nabla _{y_k}\nu \right) =0 \quad \text {in} \,\,{\mathcal {Q}}_R(z_0) \cap \{y_n > 0\} \end{aligned}$$

with

$$\begin{aligned}&{\bar{f}}(\nu ,\nabla \nu ):=\left( \frac{|\nu |+\varepsilon ^q}{q}\right) ^{-(1-\frac{1}{q})(p-1)}\nonumber \\&\qquad \left( q^2\delta (|\nu |+\varepsilon ^q)^{2(1-\frac{1}{q})}+(\nabla _{y_n}\nu )^2+\left| \nabla _{{\bar{y}}}\nu -\nabla _{y_n}\nu \nabla _{{\bar{y}}}\gamma \right| ^2 \right) ^\frac{p-2}{2} \nonumber \\&=\left( \frac{|\nu |+\varepsilon ^q}{q}\right) ^{-(1-\frac{1}{q})(p-1)}\left\{ q^2\delta (|\nu |+\varepsilon ^q)^{2(1-\frac{1}{q})}+\sum \limits _{k^\prime ,\ell ^\prime =1}^na_{k^\prime ,\ell ^\prime }(y)\nabla _{y_{k^\prime }}\nu \nabla _{y_{\ell ^\prime }}\nu \right\} ^\frac{p-2}{2}. \end{aligned}$$

(B.13)

With the notation ${\bar{y}}=(y_1,\ldots ,y_{n-1})$, ${\tilde{\nu }}$ denotes the odd extension of $\nu$ in the cylinder ${\mathcal {Q}}_R(z_0)$, that is,

$$\begin{aligned} {\tilde{\nu }}(y,t):= {\left\{ \begin{array}{ll} \nu ({\bar{y}},y_n,t) \quad &{}\text {in}\quad {\mathcal {Q}}_R(z_0) \cap \{y_n \ge 0\} \\ -\nu ({\bar{y}},-y_n,t) \quad &{}\text {in}\quad {\mathcal {Q}}_R(z_0) \cap \{y_n \le 0\} \end{array}\right. } \end{aligned}$$

(B.14)

and therefore, by the reflexion principle, ${\tilde{\nu }}$ solves

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t {\tilde{\nu }}-\nabla _{y_\ell }\left( {\bar{f}}({\tilde{\nu }},\nabla {\tilde{\nu }})\,a_{\ell ,k}\,\nabla _{y_k}{\tilde{\nu }} \right) =0 \quad &{}\text {in} \,\,{\mathcal {Q}}_R(z_0)\\ {\tilde{\nu }}=0 \quad &{}\text {on}\,\,{\mathcal {Q}}_R(z_0) \cap \{y_n=0\}. \end{array}\right. } \end{aligned}$$

(B.15)

Under this setting, by (B.9)

$$\begin{aligned} ~ \varepsilon ^q \le |{\tilde{\nu }}|+\varepsilon ^q \le \left( \Vert u_0\Vert _\infty +\varepsilon \right) ^q, \end{aligned}$$

(B.16)

which in turn implies that, for any $\eta =(\eta _1,\ldots , \eta _n) \in {\mathbb {R}}^n$, if $p \ge 2$, then

$$\begin{aligned}&\frac{1}{2}\left( \frac{\Vert u_0\Vert _\infty + \varepsilon }{q}\right) ^{-\left( q-1\right) (p - 1)} \left( q^2 \delta \varepsilon ^{2(q-1)} + \frac{1}{2} |\eta |^2 \right) ^{\frac{p-2}{2}}|\eta |^2 \nonumber \\&\le {{{\bar{f}}}} ({\tilde{\nu }}, \eta ) a_{k, \ell } \eta _k\eta _{\ell } \nonumber \\&= \left( \frac{|{\tilde{\nu }}| + \varepsilon ^q}{q} \right) ^{\!\!-\left( 1 - \frac{1}{q}\right) (p - 1)}\!\! \left\{ q^2 \delta (|{\tilde{\nu }}| + \varepsilon ^q)^{2 (1 - \frac{1}{q})} + \sum \limits _{k^\prime , \ell ^\prime =1}^na_{k^\prime , \ell ^\prime } (y)\eta _{k^\prime } \eta _{\ell ^\prime }\right\} ^{\frac{p-2}{2}}\!\!\!\!a_{k,\ell }\eta _k \eta _{\ell } \nonumber \\&=\left( \frac{|{\tilde{\nu }}| + \varepsilon ^q}{q} \right) ^{\!\!-\left( 1 - \frac{1}{q}\right) (p - 1)}\!\! \left\{ q^2 \delta (|{\tilde{\nu }}| + \varepsilon ^q)^{2 (1 - \frac{1}{q})} + (\eta _n)^2+|{\bar{\eta }}-\nabla _{{\bar{y}}}\gamma \,\eta _n|^2\right\} ^{\frac{p-2}{2}}\!\!\!\!a_{k,\ell }\eta _k \eta _{\ell } \nonumber \\&\le 2\left( \frac{\varepsilon ^q}{q}\right) ^{-(1 - \frac{1}{q}) (p - 1)}\left\{ q^2 \delta (\Vert u_0\Vert _\infty + \varepsilon )^{2 (q - 1)}+2 |\eta |^2\right\} ^{\frac{p-2}{2}}|\eta |^2, \end{aligned}$$

(B.17)

where in the second and penultimate lines we used condition (B.12). If $1< p < 2$, then it holds that the inequalities with the upper and lower bounds replaced each other in (B.17).

We will collect the regularity of the solution ${{{\tilde{\nu }}}}$.

Theorem B.2

(Hölder continuity I, [23, Theorem 7.3]) Let ${\tilde{\nu }}$ be a nonnegative and bounded weak solution to (B.15). Then ${\tilde{\nu }}$ is locally Hölder continuous in ${\mathcal {Q}}_R(z_0)$ with an exponent $\alpha \in (0, 1)$ on a parabolic metric $|x|+|t|^{1/p}$.

Proof

Let us consider the nonnegative weak solution $\nu$ in (B.10). A local Hölder continuity of nonnegative weak solution $\nu$ to (B.10) up to the lateral boundary ${\mathcal {Q}}_R(z_0) \cap \{y_n = 0\}$ is obtained from De Giorgi’s local energy estimates for truncated solutions on the intrinsically scaled local parabolic cylinder. This intrinsic scaling method was originally introduced by Di Benedetto, that is the fundamental to the regularity for degenerate and singular parabolic equations such as the evolutionary p-Laplace type equations, treated here (See [9, Chapter III] for details). Thus, the odd-reflected function ${\tilde{\nu }}$ of $\nu$ on the boundary as in (B.14) is also Hölder continuous in ${\mathcal {Q}}_R(z_0)$, since the reflection on the boundary is $C^1$-extension. $\square$

Theorem B.3

(Hölder continuity II, [23, Theorem 7.4]) Let ${\tilde{\nu }}$ be a nonnegative and bounded weak solution to (B.15). Then the spatial gradient $\nabla _y {\tilde{\nu }}$ is locally Hölder continuous in ${\mathcal {Q}}_R(z_0)$ with an exponent $\gamma \in (0,1)$ on the usual parabolic distance.

Proof

The Hölder continuity of ${\tilde{\nu }}$, Theorem B.2, which together with the absolutely continuous of $|\cdot |$, implies that $|{\tilde{\nu }}|$ is also Hölder continuous. Then by (B.16) the function $|{\tilde{\nu }}|+\varepsilon ^q$ appeared in (B.13) is actually lower and upper bounded and Hölder continuous, $(a_{k,\ell }(y))$ is smooth coefficient and the coefficient function ${{{\bar{f}}}} ({{{\tilde{\nu }}}}, \nabla {{{\tilde{\nu }}}})$ satisfies (B.17). As a result, the Hölder continuity of the spatial gradient is obtained from the argument based on Campanato’s perturbation method. $\square$

In the following, the weak differentiability as $\nabla ^2_y {\tilde{\nu }}$ holds true.

Proposition B.4

[23, Proposition 7.5] Let ${\tilde{\nu }}$ be a nonnegative weak solution to (B.15). Then, there exists a weak derivative $\nabla ^2_y {\tilde{\nu }}$ such that

$$\begin{aligned} \left( q^2\delta (|{\tilde{\nu }}|+\varepsilon ^q)^{2(1-\frac{1}{q})}+(\nabla _{y_n}{\tilde{\nu }})^2+\left| \nabla _{{\bar{y}}}{\tilde{\nu }}-\nabla _{{\bar{y}}}\gamma \nabla _{y_n}{\tilde{\nu }}\right| ^2 \right) ^\frac{p-2}{4} \nabla ^2_y{\tilde{\nu }}\in L^2_{\textrm{loc}}({\mathcal {Q}}_R(z_0)) \end{aligned}$$

and thus, by (B.16) $\nabla ^2_y{\tilde{\nu }}$ is also squared-integrable locally in ${\mathcal {Q}}_R(z_0)$.

Proof

The assertion is proved by the difference quotient method, whose detail is found in the proof of [23, Proposition 7.5] (also refer to [9, Chapter VIII, Sect. 3, pp. 223–230]). $\square$

Furthermore, we deduce the Schauder estimate for the equation (B.15).

By the weak second differentiability Proposition B.4, we can differentiate (B.15) with respect to $y_j,\,\,j=1,\ldots , n$ to get

$$\begin{aligned} \partial _t\left( \nabla _{y_j}{\tilde{\nu }}\right) -\nabla _{y_\ell }\bigg ({\mathcal {A}}_{\ell ,k}\nabla _{y_k}\nabla _{y_j}{\tilde{\nu }}+{\mathcal {F}}_{\ell ,k}\nabla _{y_k}{\tilde{\nu }}\nabla _{y_j}{\tilde{\nu }}+{\mathcal {G}}_{\ell ,k}^{(j)}\nabla _{y_k} {\tilde{\nu }}\bigg )=0 \quad \text {in} \,\,{\mathcal {Q}}_R(z_0)\nonumber \\ \end{aligned}$$

(B.18)

in the distribution sense, where

$$\begin{aligned} {\mathcal {A}}_{\ell ,k}&:={\bar{f}}({\tilde{\nu }},\nabla {\tilde{\nu }}) \left( a_{\ell ,k}+(p-2)\,\dfrac{\left( \textstyle \sum \limits _{k^\prime }a_{\ell ,k^\prime }\nabla _{y_{k^\prime }}{\tilde{\nu }}\right) \left( \textstyle \sum \limits _{\ell ^\prime }a_{k, \ell ^\prime }\nabla _{y_{\ell ^\prime }}{\tilde{\nu }}\right) }{\,\,q^2\delta (|{\tilde{\nu }}|+\varepsilon ^q)^{2(1-\frac{1}{q})}+\sum \limits _{k^\prime ,\ell ^\prime =1}^n a_{k^\prime ,\ell ^\prime }\nabla _{y_{k^\prime }}{\tilde{\nu }}\nabla _{y_{\ell ^\prime }}{\tilde{\nu }}\,\,} \right) \end{aligned}$$

and set the coefficient matrices of the lower order terms as

$$\begin{aligned} {\mathcal {F}}_{\ell ,k}&:=-\frac{q-1}{q^2}(\textrm{sgn}\,{\tilde{\nu }})\left( \dfrac{|{\tilde{\nu }}|+\varepsilon ^q}{q}\right) ^{-(1-\frac{1}{q})(p-1)-1}\\&\quad \quad \quad \cdot \left\{ q^2\delta (|\nu |+\varepsilon ^q)^{2(1-\frac{1}{q})}+\sum \limits _{k^\prime ,\ell ^\prime =1}^na_{k^\prime ,\ell ^\prime }(y)\nabla _{y_{k^\prime }}\nu \nabla _{y_{\ell ^\prime }}\nu \right\} ^\frac{p-4}{2} \\&\quad \quad \quad \quad \cdot \left[ q^2\delta (|{\tilde{\nu }}|+\varepsilon ^q)^{2(1-\frac{1}{q})}+(p-1)\sum \limits _{k^\prime ,\ell ^\prime =1}^n a_{k^\prime ,\ell ^\prime }\nabla _{y_{k^\prime }}{\tilde{\nu }}\nabla _{y_{\ell ^\prime }}{\tilde{\nu }}\right] \,a_{\ell ,k}, \\ {\mathcal {G}}_{\ell ,k}^{(j)}&:={\bar{f}}({\tilde{\nu }},\nabla {\tilde{\nu }})\!\!\left( \nabla _{y_j}a_{\ell ,k}+\frac{p-2}{2}\,\dfrac{\left( \textstyle \sum \limits _{k^\prime , \ell ^\prime =1}^n\nabla _{y_j}a_{\ell ^\prime ,k^\prime }\nabla _{y_{k^\prime }}{\tilde{\nu }}\nabla _{y_{\ell ^\prime }}{\tilde{\nu }}\right) \,a_{\ell ,k}}{\,\,q^2\delta (|{\tilde{\nu }}|+\varepsilon ^q)^{2(1-\frac{1}{q})}+\sum \limits _{k^\prime ,\ell ^\prime =1}^n a_{k^\prime ,\ell ^\prime }\nabla _{y_{k^\prime }}{\tilde{\nu }}\nabla _{y_{\ell ^\prime }}{\tilde{\nu }}} \right) . \end{aligned}$$

We have the uniform ellipticity and boundedness for the coefficient $({\mathcal {A}}_{\ell ,k})$: For any $\eta =(\eta _k) \in {\mathbb {R}}^n$

$$\begin{aligned} \min \{1, p-1\} {\bar{f}}({\tilde{\nu }},\nabla {\tilde{\nu }})\,a_{\ell ,k}\,\eta _{\ell }\eta _k \le {\mathcal {A}}_{\ell ,k}\eta _{\ell }\eta _k \le \max \{1, p-1\} {\bar{f}}({\tilde{\nu }},\nabla {\tilde{\nu }})\,a_{\ell ,k}\,\eta _{\ell }\eta _k. \end{aligned}$$

By (B.12) for $(a_{\ell ,k})$ and (B.17) for ${\bar{f}}({\tilde{\nu }},\nabla {\tilde{\nu }})$ it holds that, whenever $p \ge 2$

$$\begin{aligned}&\frac{1}{2}\left( \frac{\Vert u_0\Vert _\infty + \varepsilon }{q}\right) ^{-\left( q-1\right) (p - 1)} \left( q^2 \delta \varepsilon ^{2(q-1)} \right) ^{\frac{p-2}{2}}|\eta |^2 \nonumber \\&\le {\mathcal {A}}_{\ell ,k}\eta _{\ell }\eta _k \nonumber \\&\le 2\left( \frac{\varepsilon ^q}{q}\right) ^{-(1 - \frac{1}{q}) (p - 1)}\bigg \{q^2 \delta (\Vert u_0\Vert _\infty + \varepsilon )^{2 (q - 1)}+2 \Vert \nabla _y{\tilde{\nu }}\Vert _\infty ^2\bigg \}^{\frac{p-2}{2}}|\eta |^2, \end{aligned}$$

(B.19)

and, whenever $1<p<2$,

$$\begin{aligned}&\frac{p-1}{2}\left( \frac{\Vert u_0\Vert _\infty + \varepsilon }{q}\right) ^{-\left( q-1\right) (p - 1)}\bigg \{q^2 \delta (\Vert u_0\Vert _\infty + \varepsilon )^{2 (q - 1)}+2 \Vert \nabla _y{\tilde{\nu }}\Vert _\infty ^2\bigg \}^{\frac{p-2}{2}} |\eta |^2 \nonumber \\&\le {\mathcal {A}}_{\ell ,k}\eta _{\ell }\eta _k \nonumber \\&\le 2\left( \frac{\varepsilon ^q}{q}\right) ^{-(1 - \frac{1}{q}) (p - 1)}\left( q^2 \delta \varepsilon ^{2(q-1)} \right) ^{\frac{p-2}{2}}|\eta |^2, \end{aligned}$$

(B.20)

where we use the boundedness of $\nabla _y{\tilde{\nu }}$ by Theorem B.3, again. By Theorem B.3 the coefficients ${\mathcal {A}}_{\ell ,k}$ and lower-order terms are also locally Hölder continuous in ${\mathcal {Q}}_R (z_0)$. Thus, the equation (B.18) can be regarded as the parabolic equation on ${\mathcal {Q}}_R (z_0)$, having the uniformly elliptic, bounded as in (B.19) and (B.20) and Hölder continuous coefficients ${\mathcal {A}}_{\ell , k}$, and the Hölder continuous lower-order terms ${\mathcal {F}}_{\ell , k}$ and ${\mathcal {G}}_{\ell , k}^{(j)}$.

Then we have the following Schauder’s estimate.

Proposition B.5

([23, Proposition 7.6]) Let ${\tilde{\nu }}$ be a nonnegative weak solution to (B.15). Then $\partial _s {\tilde{\nu }}$ and $\nabla ^2_y {\tilde{\nu }}$ are locally Hölder continuous in ${\mathcal {Q}}_R(z_0)$.

Proof

We can apply the Campanato estimates for (B.18) to get the assertion of Proposition  B.5. The detail is found in the proof of [23, Proposition 7.6].

$\square$

The same regularity as in Theorems B.2 and B.3 and Proposition B.5 holds true for a weak solution $\mu =\nu +\varepsilon ^q$ to (B.6) and thus, $w=\mu ^{\frac{1}{q}}-\varepsilon$ does so since, by (B.4) in Proposition B.1, $\varepsilon ^q \le \mu =(u+\varepsilon )^q \le (\varepsilon +\Vert u_0\Vert _\infty )^q$.

Proposition B.6

([23, Proposition 7.7]) Let $u=u_{\varepsilon , \delta }$ be a nonnegative weak solution to (B.1). Then there hold that all of $u,\,\partial _t u,\,\nabla u$ and $\nabla ^2 u$ are locally Hölder continuous in $({\bar{\Omega }})_\infty ={\bar{\Omega }} \times (0,\infty )$.

1.3 B.3. Energy estimates for solutions to (B.1)

Here we derive the energy estimates for weak solutions to (B.1). This energy estimates will play a central role in the weak compactness theory for constructing a weak solution to the limiting equations  (3.6) of (B.1) as $\varepsilon \searrow 0$ and $\delta \searrow 0$.

Firstly, we establish the energy identities for weak solutions to (B.1), whose proof can been seen in [23, Lemma 7.8].

Lemma B.7

(Energy identities for solutions to (B.1)) Let $u=u_{\varepsilon ,\delta }$ be a weak solution to  (B.1). Then, for any nonnegative $t_1<t_2<\infty$, the following energy identities are valid.

$$\begin{aligned}&\left[ \frac{q}{q+1}\int _\Omega (u(t)+\varepsilon )^{q+1}\,dx -\varepsilon \int _\Omega (u(t)+\varepsilon )^q\,dx \right] _{t_1}^{t_2}\nonumber \\&\qquad +\iint _{\Omega _{t_1,t_2}}\left( \delta +|\nabla u|^2 \right) ^{\frac{p-2}{2}}|\nabla u|^2\,dxdt=0, \end{aligned}$$

(B.21)

$$\begin{aligned}&q\iint _{\Omega _{t_1,t_2}}(u+\varepsilon )^{q-1}(\partial _t u)^2\,dxdt+\left[ \int _\Omega \frac{1}{p}\left( \delta +|\nabla u(t)|^2 \right) ^{\frac{p}{2}}\,dx \right] _{t_1}^{t_2}=0, \end{aligned}$$

(B.22)

where $\Omega _{t_1,t_2}:=\Omega \times (t_1,t_2)$. The above identity (B.22) is rewritten as

$$\begin{aligned} \frac{4q}{(q+1)^2}\iint _{\Omega _{t_1,t_2}}\left( \partial _t (u+\varepsilon )^{\frac{q+1}{2}} \right) ^2\,dxdt+\left[ \int _\Omega \frac{1}{p}\left( \delta +|\nabla u(t)|^2 \right) ^{\frac{p}{2}}\,dx \right] _{t_1}^{t_2}=0.\nonumber \\ \end{aligned}$$

(B.23)

Proof

Following [23], we present the short proof. Multiplying u by (B.1)$_1$ and integrating over $\Omega \times (t_1,t_2)$ give that

$$\begin{aligned} \iint _{\Omega _{t_1,t_2}}\partial _t(u+\varepsilon )^qu\,dxdt+\iint _{\Omega _{t_1,t_2}}\left( \delta +|\nabla u|^2 \right) ^{\frac{p-2}{2}}|\nabla u|^2\,dxdt=0. \end{aligned}$$

Rewriting $\partial _t(u+\varepsilon )^qu=\frac{q}{q+1}\partial _t(u+\varepsilon )^{q+1}-\varepsilon \partial _t(u+\varepsilon )^q$ and inserting this into the above display, we obtain the first identity (B.21).

Note that Eq.s (B.1) for u and  (B.2) for $U_\varepsilon =u+\varepsilon$ are equivalent. The second identity (B.22) is shown by multiplying $\partial _tU_\varepsilon$ by (B.2)$_1$ and integrating over $\Omega \times (t_1,t_2)$. Indeed, we first remark that $\partial _tU_\varepsilon \in L^2(\Omega _\infty )$, which is guaranteed via the identities

$$\begin{aligned} \partial _tU_\varepsilon =\partial _t\left( U_\varepsilon ^q \right) ^{1/q}=\frac{1}{q}U_\varepsilon ^{1-q}\partial _t U_\varepsilon ^q, \end{aligned}$$

and $\partial _tU_\varepsilon ^q=\partial _t(u+\varepsilon )^q \in L^2(\Omega _\infty )$ by (B.5)$_3$ in Proposition B.1, and the inequality

$$\begin{aligned} 0 \le U_\varepsilon ^{1-q}=(u+\varepsilon )^{1-q}\le \varepsilon ^{1-q} \end{aligned}$$

by (B.4) in Proposition B.1. Noticing that $\partial _tU_\varepsilon =0$ on $\partial \Omega \times (0,\infty )$, we get

$$\begin{aligned} 0&=\iint _{\Omega _{t_1,t_2}}\partial _tU_\varepsilon ^q\,\partial _t U_\varepsilon \,dxdt+\iint _{\Omega _{t_1,t_2}}\left( \delta +|\nabla U_\varepsilon |^2 \right) ^{\frac{p-2}{2}} \nabla U_\varepsilon \cdot \partial _t \nabla U_\varepsilon \,dxdt \\&=q\iint _{\Omega _{t_1,t_2}}U_\varepsilon ^{q-1}\left( \partial _t U_\varepsilon \right) ^2\,dxdt+\iint _{\Omega _{t_1,t_2}}\dfrac{d}{dt}\,\dfrac{1}{p}\left( \delta +|\nabla U_\varepsilon |^2 \right) ^{\frac{p}{2}}\,dxdt, \end{aligned}$$

which will in turn imply the second identity (B.22).

Finally, Eq. (B.23) directly follows from (B.22) and

$$\begin{aligned}q(u+\varepsilon )^{q-1}(\partial _t u)^2=\frac{4q}{(q+1)^2}\left( \partial _t (u+\varepsilon )^{\frac{q+1}{2}} \right) ^2. \end{aligned}$$

$\square$

1.4 B.4. Passing to the limit in (B.1)

This subsection presents the construction for a solution to the limiting equation (3.6) of (B.1). We stress that the limit of approximating solutions $u=u_{\varepsilon , \delta }$ to (B.1) satisfying the inequalities as stated in Proposition B.9, coincides with the weak solution to (3.6) by the uniqueness of solution to (3.6).

Lemma B.8

(Convergence result I) Let $\{u_{\varepsilon , \delta }\}$ be the family of weak solutions to (B.1). Then, there exist a subsequence $\{u_{\varepsilon , \delta }\}$ without changing notation, and a limit function $u_\infty \in L^\infty (0,\infty ; W_0^{1,p}(\Omega ))$ such that for every positive $T<\infty$, as $\varepsilon , \delta \searrow 0$,

$$\begin{aligned} u_{\varepsilon ,\delta } \rightarrow u_\infty \quad&*\text {-weakly}\,\,in\,\,L^\infty (0,\infty \,; W_0^{1,p}(\Omega ))\,\,; \end{aligned}$$

(B.24)

$$\begin{aligned} u_{\varepsilon ,\delta } \rightarrow u_\infty \quad&strongly\,\,in\,\,L^r(\Omega _T)\,\,\text {for}\,\,any \,\,r \ge 1\,\,; \end{aligned}$$

(B.25)

$$\begin{aligned} u_{\varepsilon ,\delta } \rightarrow u_\infty \quad&in\,\,C([0,T]\,;\,L^{q+1}(\Omega ))\,\,; \end{aligned}$$

(B.26)

$$\begin{aligned} \nabla u_{\varepsilon ,\delta } \rightarrow \nabla u_\infty \quad&strongly\,\,in \,\,L^r(\Omega _T)\,\,\text {for}\,\,any\,\,1 \le r <p\,\,; \end{aligned}$$

(B.27)

$$\begin{aligned} \partial _t\left( u_{\varepsilon , \delta }+\varepsilon \right) ^{\frac{q+1}{2}} \rightarrow \partial _t u_\infty ^{\frac{q+1}{2}} \quad&weakly \,\,in\,\,L^2(\Omega _\infty )\,\,; \end{aligned}$$

(B.28)

$$\begin{aligned} \partial _t\left( u_{\varepsilon , \delta }+\varepsilon \right) ^q \rightarrow \partial _t u_\infty ^q \quad&weakly \,\,in\,\,L^2(\Omega _\infty ) \end{aligned}$$

(B.29)

and thus, $u_\infty$ is a weak solution of (3.6).

Proof

For reader’s convenience, following [25, Sect. 4, pp. 164–167] and [23, Sect. 7.4], we again give the shorthand proof. Let $u=u_{\varepsilon , \delta }$ be a weak solution to (B.1). We first verify a boundedness of $u_{\varepsilon , \delta }$ in $L^\infty (0,\infty ;L^{q+1}(\Omega ))$. Letting $u=u_{\varepsilon , \delta }$, by Young’s inequality, we have

$$\begin{aligned} \varepsilon \int _\Omega (u(t)+\varepsilon )^{q}\,dx \le \dfrac{1}{2}\,\dfrac{q}{q+1}\int _\Omega (u(t)+\varepsilon )^{q+1}\,dx+C(q^{-1})\varepsilon ^{q+1}|\Omega |. \end{aligned}$$

Inserting this into (B.21), we obtain

$$\begin{aligned} \sup _{t>0}\int _\Omega (u(t)+\varepsilon )^{q+1}\,dx&+\iint _{\Omega _T}\left( \delta +|\nabla u|^2 \right) ^{\frac{p-1}{2}}\,dxdt \nonumber \\&\le \int _\Omega (u_0+\varepsilon )^{q+1}\,dx+C\varepsilon ^{q+1}|\Omega |, \end{aligned}$$

(B.30)

yielding the boundedness of $u_{\varepsilon , \delta }$ in $L^\infty (0,\infty ; L^{q+1}(\Omega ))$. By (B.22) and (B.23), we have,

$$\begin{aligned} \iint _{\Omega _T}\left( \partial _t (u+\varepsilon )^{\frac{q+1}{2}} \right) ^2\,dxdt+\sup _{t>0}\Vert \left( \delta +|\nabla u(t)|^2 \right) ^{\frac{1}{2}}\Vert _p^p \le C\Vert \left( \delta +|\nabla u_0|^2 \right) ^{\frac{1}{2}}\Vert _p^p. \end{aligned}$$

(B.31)

From (B.30) and (B.31) there is a subsequence $\{u_{\varepsilon , \delta }\}$ without changing notation, and a limit function $u_\infty \in L^p (0,\infty ; W_0^{1,p}(\Omega ))$ such that

$$\begin{aligned} u_{\varepsilon ,\delta } \rightarrow u_\infty \quad *\text {--}\,weakly\,\,in\,\,L^\infty (0, \infty ; W_0^{1,p}(\Omega )). \end{aligned}$$

(B.32)

and

$$\begin{aligned} u_{\varepsilon ,\delta },\,\nabla u_{\varepsilon ,\delta }\rightarrow u_\infty ,\,\nabla u_\infty \quad weakly\,\,in\,\,L^p(0, \infty ; L^p(\Omega )), \end{aligned}$$

(B.33)

since, by Mazur’s theorem, $W_0^{1,p}(\Omega )$ is weakly closed in $W^{1,p}(\Omega )$. The compact Sobolev embedding $W^{1,p}_0(\Omega ) \hookrightarrow L^\gamma (\Omega )$ for all $1 \le \gamma <p^*=np / (n-p)$ and the maximum principle (B.4) imply the convergence  (B.25).

Applying (B.31) for the 2nd term in the left hand side of (B.21) in Lemma B.7, we have that, for $0 \le t_1<t_2<\infty$,

$$\begin{aligned} \Bigg | \Bigg [\dfrac{q}{q+1}\Vert u(t)+\varepsilon \Vert _{q+1}^{q+1}-\varepsilon \Vert u(t)+\varepsilon \Vert _q^q \Bigg ]_{t_1}^{t_2} \Bigg | \le C\left( \Vert \nabla u_0\Vert _p^p+1 \right) (t_2-t_1).\nonumber \\ \end{aligned}$$

(B.34)

From (B.34) and (B.30), $\Vert u(t)+\varepsilon \Vert _{q+1}$ is continuous on $t \ge 0$, uniformly on small positive $\delta$ and $\varepsilon$, and $u(t)+\varepsilon$ is also weakly continuous for all $t \ge 0$ in $L^{q+1}(\Omega )$, uniformly for positive $\delta$ and $\varepsilon$. Therefore we have that $u(t)+\varepsilon$ is continuous on $t \ge 0$ in $L^{q+1}(\Omega )$, uniformly on small positive $\delta$ and $\varepsilon$. Accordingly, the convergence (B.26) is follows.

Applying the argument as in [6, Theorem 2.1, pp.31–33] and [25, Lemma 4.2, pp165–167], we can deduce the strong convergence of gradient (B.27).

Due to the energy identity (B.23), there exist a function $\omega \in L^2(\Omega _\infty )$ such that, as $\varepsilon ,\,\delta \searrow 0$,

$$\begin{aligned} \partial _t\left( u_{\varepsilon , \delta }+\varepsilon \right) ^{\frac{q+1}{2}} \rightarrow \omega \quad weakly \,\,in\,\,L^2(\Omega _\infty ) \end{aligned}$$

(B.35)

and

$$\begin{aligned} \iint _{\Omega _\infty } \partial _t\left( u_{\varepsilon , \delta }+\varepsilon \right) ^{\frac{q+1}{2}}\cdot \psi \,dxdt=-\iint _{\Omega _\infty } \left( u_{\varepsilon , \delta }+\varepsilon \right) ^{\frac{q+1}{2}}\cdot \partial _t\psi \,dxdt \end{aligned}$$

for every $\psi \in C^\infty _0(\Omega _\infty )$. By the strong convergence (B.25) and (B.35), as $\varepsilon , \delta \searrow 0$,

$$\begin{aligned} \iint _{\Omega _\infty } \omega \cdot \psi \,dxdt=-\iint _{\Omega _\infty } u_\infty ^{\frac{q+1}{2}}\cdot \partial _s\psi \,dxdt \end{aligned}$$

for every $\psi \in C^\infty _0(\Omega _\infty )$. Therefore we conclude that $\omega =\partial _t u_\infty ^{\frac{q+1}{2}}$ in $L^2(\Omega _\infty )$, which together with (B.35) implies (B.28). Analogously, the weak convergence (B.29) can be shown by (B.5)$_3$.

Now, we will verify that the limit function $w_\infty$ obtained above is a weak solution to (3.6). The limit function $w_\infty$ satisfies that

$$\begin{aligned} u_\infty \in L^\infty (0,\infty ; W_0^{1,p}(\Omega )) ; \quad \partial _t u_\infty ^{\frac{q+1}{2}} \in L^2(\Omega _\infty ), \end{aligned}$$

that is exactly (i) in Definition 5.1. We have the weak form of (B.1)$_1$

$$\begin{aligned} \iint _{\Omega _T}\partial _t (u_{\varepsilon , \delta }+\varepsilon )^{q}\varphi \,dxdt+\iint _{\Omega _T}\left( \delta +|\nabla u_{\varepsilon , \delta }|^2 \right) ^{\frac{p-2}{2}} \nabla u_{\varepsilon , \delta }\cdot \nabla \varphi \,dxdt=0\nonumber \\ \end{aligned}$$

(B.36)

for every $\varphi \in C^\infty _0(\Omega _T)$. Employing the strong convergence of gradient (B.27) and the weak convergence (B.29) in Lemma B.8 and taking the limit as $\varepsilon ,\,\delta \searrow 0$ in (B.36), we deduce that, for every $\varphi \in C^\infty _0(\Omega _T)$,

$$\begin{aligned} \iint _{\Omega _T}\partial _t u_\infty ^q\varphi \,dxdt+\iint _{\Omega _T}|\nabla u_\infty |^{p-2}\nabla u_\infty \cdot \nabla \varphi \,dxdt=0, \end{aligned}$$

that is (ii) in Definition 5.1.

Finally, the initial condition (iii) as in Definition 5.1 can be plainly obtained from  (B.38) in Proposition B.9 below. $\square$

From Lemmata B.7 and B.8 the limit function $u_\infty$ satisfies the following integral estimates.

Proposition B.9

(Energy estimates for $u_\infty$) Let $u_\infty$ be defined as in Lemma B.8. Then, it holds true that, for any nonnegative $t<\infty$,

$$\begin{aligned}{} & {} \int _\Omega u_\infty (t)^{q+1}\,dx+\frac{q+1}{q}\iint _{\Omega _{0,t}} |\nabla u_\infty |^p\,dxdt =\int _\Omega u_0^{q+1}\,dx \,\,; \end{aligned}$$

(B.37)

$$\begin{aligned}{} & {} \dfrac{4q}{(q+1)^2}\iint _{\Omega _{0,t}}\left( \partial _t u_\infty ^{\frac{q+1}{2}} \right) ^2\,dxdt+\dfrac{1}{p}\int _\Omega |\nabla u_\infty (t)|^p\,dx \le \dfrac{1}{p}\int _\Omega |\nabla u_0|^p\,dx\nonumber \\ \end{aligned}$$

(B.38)

with the shorthand notation $\Omega _{0,t}:=\Omega \times (0,t)$.

Proof

As shown in Lemma B.8, $u_\infty$ is the weak solution to (3.6) and thus, multiplying (3.6)$_1$ by $u_\infty$ and integration by parts lead to the identity (B.37). Inequality (B.38) follows from the weak convergence in (B.28) and the identity (B.23) in Lemma B.7. $\square$

Finally, we state the uniqueness for the equation (3.6).

Lemma B.10

Let u be a weak solution of (3.6)and $u_\infty$ be obtained in Lemma B.8. Then it holds that $u \equiv u _\infty$ in $\Omega _\infty$. Thus, the solution u satisfy the integral inequalities as (B.37) and  (B.38).

Proof

The uniqueness for a weak solution to (3.6) follows from Theorem A.2 with considering $\lambda _{p,q}=0$. Thus, the energy estimates hold true for the weak solution to (3.6) by Proposition B.9. $\square$

Appendix C: Quantitative estimates for the transformed doubly nonlinear equation (B.3)

In this appendix, using the higher regularity for u as in Lemma B.7, we shall derive the energy inequality for a solution of (B.3).

1.1 C.1. Energy estimate for (B.3)

Proposition B.6 enables us to deduce the following energy estimate for solutions to (B.3).

Lemma C.1

Let $V_\varepsilon =v_{\varepsilon ,\delta }+\varepsilon$ be a solution to  (B.3). Then, for every positive $S<\infty$, there holds that

$$\begin{aligned}&\sup _{0<s<S}\frac{q}{q+1}\int _\Omega V_\varepsilon (s)^{q+1}\,dx +\iint _{\Omega _{S}}\left( \delta \left( Te^{-s}\right) ^{-\frac{2}{q+1-p}}+|\nabla V_\varepsilon |^2 \right) ^{\frac{p-2}{2}}|\nabla V_\varepsilon |^2\,dxds \nonumber \\&\quad \quad \le \frac{2q}{q+1}T^{-\frac{q+1}{q+1-p}}\left( \int _\Omega (u_0+\varepsilon )^{q+1}\,dx\right) e^{\frac{q+1}{q}\lambda _{p,q}S} \end{aligned}$$

(C.1)

and

$$\begin{aligned}&\frac{4q}{(q+1)^2}\iint _{\Omega _S}\left( \partial _s V_\varepsilon ^{\frac{q+1}{2}} \right) ^2\,dxds +\frac{1}{p}\sup _{0<s<S}\int _\Omega \left( \delta \left( Te^{-s}\right) ^{-\frac{2}{q+1-p}}+|\nabla V_\varepsilon |^2 \right) ^{\frac{p}{2}}\,dx \nonumber \\&\quad \quad \le 2e^{\frac{p}{q+1-p}S}\left[ T^{-\frac{p}{q+1-p}}\int _\Omega \frac{1}{p}\left( \delta +|\nabla u_0|^2\right) ^{\frac{p}{2}}\,dx \right. \nonumber \\&\quad \quad \quad \left. +\frac{\lambda _{p,q}}{q+1}T^{-\frac{q+1}{q+1-p}}\left( \int _\Omega (u_0+\varepsilon )^{q+1}\,dx\right) e^{\frac{q+1}{q}\lambda _{p,q}S}\right] . \end{aligned}$$

(C.2)

Proof

Take ${\bar{S}} \in (0,S)$ arbitrarily. Multiplying (B.3) by $V_\varepsilon$ and integrating over $\Omega _{{\bar{S}}}:=\Omega \times (0,{\bar{S}})$, we have

$$\begin{aligned} \left[ \frac{q}{q+1}\int _{\Omega }V_\varepsilon ^{q+1}\,dx\right] _{s=0}^{{\bar{S}}}&-\iint _{\Omega _{{\bar{S}}}}\textrm{div}\,\left( \left( \delta (Te^{-s})^{-\frac{2}{q+1-p}}+|\nabla V_\varepsilon |^2 \right) ^{\frac{p-2}{2}}\nabla V_\varepsilon \right) V_\varepsilon \,dxds \\&=\lambda _{p,q}\iint _{\Omega _{{\bar{S}}}}V_\varepsilon ^{q+1}\,dxds. \end{aligned}$$

Since by Lemma B.6, $\nabla V_\varepsilon =(T-t)^{-\frac{1}{q+1-p}}\nabla u=(Te^{-s})^{-\frac{1}{q+1-p}} \nabla u$ is continuous up to the boundary $\partial \Omega$ and $V_\varepsilon \ge (T e^{-s})^{-\frac{1}{q + 1 - p}}\varepsilon$ in $\Omega _\infty$ and $V_\varepsilon = (Te^{-s})^{-\frac{1}{q+1-p}}\varepsilon$ on $\partial \Omega \times (0,\infty )$, the inner product $\nu \cdot \nabla V_\varepsilon$ is nonpositive on the boundary $\partial \Omega$ for the outward unit normal vector $\nu$ on $\partial \Omega$ so that, the second term on the left-hand side of the above display is estimated as

$$\begin{aligned}&-\iint _{\Omega _{{\bar{S}}}}\textrm{div}\,\left( \left( \delta (Te^{-s})^{-\frac{2}{q+1-p}}+|\nabla V_\varepsilon |^2 \right) ^{\frac{p-2}{2}}\nabla V_\varepsilon \right) V_\varepsilon \,dxds \\&=-\int _0^S\int _{\partial \Omega }\left( \delta (Te^{-s})^{-\frac{2}{q+1-p}}+|\nabla V_\varepsilon |^2 \right) ^{\frac{p-2}{2}}\left( \nu \cdot \nabla V_\varepsilon \right) V_\varepsilon \,dxdS_x \\&\quad \quad +\iint _{\Omega _{{\bar{S}}}}\left( \delta (Te^{-s})^{-\frac{2}{q+1-p}}+|\nabla V_\varepsilon |^2 \right) ^{\frac{p-2}{2}}|\nabla V_\varepsilon |^2\,dxds \\&\ge \iint _{\Omega _{{\bar{S}}}}\left( \delta (Te^{-s})^{-\frac{2}{q+1-p}}+|\nabla V_\varepsilon |^2 \right) ^{\frac{p-2}{2}}|\nabla V_\varepsilon |^2\,dxds. \end{aligned}$$

This estimate in turn leads to

$$\begin{aligned} \frac{q}{q+1}\int _{\Omega }V_\varepsilon ({\bar{S}})^{q+1}\,dx&+\iint _{\Omega _{{\bar{S}}}}\left( \delta (Te^{-s})^{-\frac{2}{q+1-p}}+|\nabla V_\varepsilon |^2 \right) ^{\frac{p-2}{2}}|\nabla V_\varepsilon |^2\,dxds \nonumber \\&\le \frac{q}{q+1}\int _{\Omega }V_\varepsilon (0)^{q+1}\,dx+\lambda _{p,q}\iint _{\Omega _{{\bar{S}}}}V_\varepsilon ^{q+1}\,dxds. \end{aligned}$$

(C.3)

From Gronwall’s inequality, it follows that

$$\begin{aligned} \lambda _{p,q}\iint _{\Omega _{{\bar{S}}}}V_\varepsilon ^{q+1}\,dxds \le \frac{q}{q+1}\left( \int _\Omega V_\varepsilon (0)^{q+1}\,dx\right) \left( e^{\frac{q+1}{q}\lambda _{p,q}S}-1\right) \end{aligned}$$

and inserting this into (C.3) gives

$$\begin{aligned} \frac{q}{q+1}\int _{\Omega }V_\varepsilon ({\bar{S}})^{q+1}\,dx&+\iint _{\Omega _{{\bar{S}}}}\left( \delta (Te^{-s})^{-\frac{2}{q+1-p}}+|\nabla V_\varepsilon |^2 \right) ^{\frac{p-2}{2}}|\nabla V_\varepsilon |^2\,dxds \\&\le \frac{q}{q+1}\left( \int _\Omega V_\varepsilon (0)^{q+1}\,dx\right) e^{\frac{q+1}{q}\lambda _{p,q}S} \\&=\frac{q}{q+1}T^{-\frac{q+1}{q+1-p}}\left( \int _\Omega (u_0+\varepsilon )^{q+1}\,dx\right) e^{\frac{q+1}{q}\lambda _{p,q}S}. \end{aligned}$$

Finally, taking the supremum over ${\bar{S}} \in (0,S)$ in both side of the above inequality yields inequality (C.1).

We next show inequality (C.2). Multiplying (B.3) by $\partial _sV_\varepsilon$ and integrating over $\Omega _{{\bar{S}}}$, we get

$$\begin{aligned}&(\textrm{I})+(\textrm{II}) \nonumber \\&\quad :=\iint _{\Omega _{{\bar{S}}}}\partial _sV_\varepsilon ^q\partial _sV_\varepsilon \,dxds\nonumber \\&\qquad -\iint _{\Omega _{{\bar{S}}}}\textrm{div}\,\left( \left( \delta (Te^{-s})^{-\frac{2}{q+1-p}}+|\nabla V_\varepsilon |^2 \right) ^{\frac{p-2}{2}}\nabla V_\varepsilon \right) \partial _s V_\varepsilon \,dxds \nonumber \\&\quad =\lambda _{p,q}\iint _{\Omega _{{\bar{S}}}}V_\varepsilon ^q\partial _sV_\varepsilon \,dxds \end{aligned}$$

(C.4)

$$\begin{aligned}&\quad =:(\textrm{III}). \end{aligned}$$

(C.5)

$(\textrm{I})$ is calculated as

$$\begin{aligned} (\textrm{I})=\iint _{\Omega _{{\bar{S}}}}qV_\varepsilon ^{q-1}\left( \partial _s V_\varepsilon \right) ^2\,dxds=\frac{4q}{(q+1)^2}\iint _{\Omega _{{\bar{S}}}}\left( \partial _s V_\varepsilon ^{\frac{q+1}{2}}\right) ^2\,dxds. \end{aligned}$$

and, by using  (C.1), we infer that

$$\begin{aligned}&(\textrm{III})=\left[ \frac{\lambda _{p,q}}{q+1}\int _\Omega V_\varepsilon (s)^{q+1}\,dx\right] _{s=0}^{{\bar{S}}} \le \frac{\lambda _{p,q}}{q+1}\int _\Omega V_\varepsilon ({\bar{S}})^{q+1}\,dx \nonumber \\&\quad \le \frac{\lambda _{p,q}}{q+1}T^{-\frac{q+1}{q+1-p}}\left( \int _\Omega (u_0+\varepsilon )^{q+1}\,dx\right) e^{\frac{q+1}{q}\lambda _{p,q}S}. \end{aligned}$$

(C.6)

As discussed above, since $\nu \cdot \nabla V_\varepsilon \le 0$ for any outward unit normal vector $\nu$ on $\partial \Omega$ and $\partial _s V_\varepsilon =\frac{1}{q+1-p}(Te^{-s})^{-\frac{1}{q+1-p}}\varepsilon \ge 0$ on $\partial \Omega \times (0,\infty )$, the integral term $(\textrm{II})$ of divergence form is estimated as

$$\begin{aligned} (\textrm{II})&=-\int _0^S\int _{\partial \Omega } \left( \delta (Te^{-s})^{-\frac{2}{q+1-p}}+|\nabla V_\varepsilon |^2 \right) ^{\frac{p-2}{2}}\left( \nu \cdot \nabla V_\varepsilon \right) \partial _s V_\varepsilon \,dxdS_x \\&\quad +\iint _{\Omega _{{\bar{S}}}}\left( \delta (Te^{-s})^{-\frac{2}{q+1-p}}+|\nabla V_\varepsilon |^2 \right) ^{\frac{p-2}{2}}\nabla V_\varepsilon \cdot \partial _s \nabla V_\varepsilon \,dxds \\&\ge \iint _{\Omega _{{\bar{S}}}}\left( \delta (Te^{-s})^{-\frac{2}{q+1-p}}+|\nabla V_\varepsilon |^2 \right) ^{\frac{p-2}{2}}\partial _s\frac{1}{2}|\nabla V_\varepsilon |^2\,dxds. \end{aligned}$$

This together with the fact that

$$\begin{aligned}&\left( \delta (Te^{-s})^{-\frac{2}{q+1-p}}+|\nabla V_\varepsilon |^2 \right) ^{\frac{p-2}{2}}\partial _s\frac{1}{2}|\nabla V_\varepsilon |^2 \\&\quad =\partial _s \frac{1}{p}\left( \delta (Te^{-s})^{-\frac{2}{q+1-p}}+|\nabla V_\varepsilon |^2 \right) ^{\frac{p}{2}} \\&\qquad -\frac{1}{q+1-p}\delta \left( Te^{-s}\right) ^{-\frac{2}{q+1-p}}\left( \delta (Te^{-s})^{-\frac{2}{q+1-p}}+|\nabla V_\varepsilon |^2 \right) ^{\frac{p-2}{2}} \end{aligned}$$

yields

$$\begin{aligned}&(\textrm{II}) \ge \left[ \int _\Omega \frac{1}{p}\left( \delta (Te^{-s})^{-\frac{2}{q+1-p}}+|\nabla V_\varepsilon |^2 \right) ^{\frac{p}{2}}\,dx\right] _{s=0}^{{\bar{S}}}\nonumber \\&\qquad -\frac{1}{q+1-p}\iint _{\Omega _{{\bar{S}}}}\delta \left( Te^{-s}\right) ^{-\frac{2}{q+1-p}}\left( \delta (Te^{-s})^{-\frac{2}{q+1-p}}+|\nabla V_\varepsilon |^2 \right) ^{\frac{p-2}{2}}\,dxds \nonumber \\&\quad \ge \left[ \int _\Omega \frac{1}{p}\left( \delta (Te^{-s})^{-\frac{2}{q+1-p}}+|\nabla V_\varepsilon |^2 \right) ^{\frac{p}{2}}\,dx\right] _{s=0}^{{\bar{S}}} \nonumber \\&\quad \quad -\frac{1}{q+1-p}\iint _{\Omega _{{\bar{S}}}}\left( \delta (Te^{-s})^{-\frac{2}{q+1-p}}+|\nabla V_\varepsilon |^2 \right) ^{\frac{p}{2}}\,dxds. \end{aligned}$$

(C.7)

Merging the preceding estimates in (C.4), we obtain

$$\begin{aligned}&\frac{4q}{(q+1)^2}\iint _{\Omega _{{\bar{S}}}}\left( \partial _s V_\varepsilon ^{\frac{q+1}{2}}\right) ^2\,dxds+\int _\Omega \frac{1}{p}\left( \delta (Te^{-{{\bar{S}}}})^{-\frac{2}{q+1-p}}+|\nabla V_\varepsilon ({\bar{S}})|^2 \right) ^{\frac{p}{2}}\,dx \\&\quad \quad \le \int _\Omega \frac{1}{p}\left( \delta T^{-\frac{2}{q+1-p}}+|\nabla V_\varepsilon (0)|^2 \right) ^{\frac{p}{2}}\,dx\\&\qquad \quad +\frac{1}{q+1-p}\iint _{\Omega _{{\bar{S}}}}\left( \delta (Te^{-s})^{-\frac{2}{q+1-p}}+|\nabla V_\varepsilon |^2 \right) ^{\frac{p}{2}}\,dxds \\&\qquad \quad +\frac{\lambda _{p,q}}{q+1}T^{-\frac{q+1}{q+1-p}}\left( \int _\Omega (u_0+\varepsilon )^{q+1}\,dx\right) e^{\frac{q+1}{q}\lambda _{p,q}S}. \end{aligned}$$

Gromwall’s inequality implies that

$$\begin{aligned}&\frac{1}{q+1-p}\iint _{\Omega _{{\bar{S}}}}\left( \delta (Te^{-s})^{-\frac{2}{q+1-p}}+|\nabla V_\varepsilon |^2 \right) ^{\frac{p}{2}}\,dxds \\&\quad \le \left( e^{\frac{p}{q+1-p}{\bar{S}}}-1\right) \left[ \int _\Omega \frac{1}{p}\left( \delta T^{-\frac{2}{q+1-p}} +|\nabla V_\varepsilon (0)|^2 \right) ^{\frac{p}{2}}\,dx \right. \\&\qquad \left. +\frac{\lambda _{p,q}}{q+1}T^{-\frac{q+1}{q+1-p}}\left( \int _\Omega (u_0+\varepsilon )^{q+1}\,dx\right) e^{\frac{q+1}{q}\lambda _{p,q}S}\right] \end{aligned}$$

and therefore, plugging this into the above display, we have

$$\begin{aligned}&\frac{4q}{(q+1)^2}\iint _{\Omega _{{\bar{S}}}}\left( \partial _s V_\varepsilon ^{\frac{q+1}{2}}\right) ^2\,dxds+\int _\Omega \frac{1}{p}\left( \delta (Te^{-{{\bar{S}}}})^{-\frac{2}{q+1-p}}+|\nabla V_\varepsilon ({\bar{S}})|^2 \right) ^{\frac{p}{2}}\,dx \\&\quad \le e^{\frac{p}{q+1-p}{\bar{S}}}\left[ \int _\Omega \frac{1}{p}\left( \delta T^{-\frac{2}{q+1-p}} +|\nabla V_\varepsilon (0)|^2 \right) ^{\frac{p}{2}}\,dx \right. \\&\qquad \left. +\frac{\lambda _{p,q}}{q+1}T^{-\frac{q+1}{q+1-p}}\left( \int _\Omega (u_0+\varepsilon )^{q+1}\,dx\right) e^{\frac{q+1}{q}\lambda _{p,q}S}\right] \\&\quad \le e^{\frac{p}{q+1-p}S}\left[ T^{-\frac{p}{q+1-p}}\int _\Omega \frac{1}{p}\left( \delta +|\nabla u_0|^2 \right) ^{\frac{p}{2}}\,dx \right. \\&\qquad \left. +\frac{\lambda _{p,q}}{q+1}T^{-\frac{q+1}{q+1-p}}\left( \int _\Omega (u_0+\varepsilon )^{q+1}\,dx\right) e^{\frac{q+1}{q}\lambda _{p,q}S}\right] . \end{aligned}$$

Finally, taking the supremum over ${\bar{S}} \in (0,S)$ in both side of the above inequality concludes the desired inequality (C.2). Therefore the proof is complete. $\square$

1.2 C.2. Passing to the limit in (B.3)

Taking the limits $\varepsilon , \delta \searrow 0$ in Lemma C.1, we also deduce the convergence result for a transformed solution to (B.3). The proof is similar as Lemma B.8.

Lemma C.2

Let $V_\varepsilon =v_{\varepsilon , \delta }+\varepsilon$ be a weak solution to (B.3) with $T=t^*$, where $t^*$ is the finite extinction time for a solution u to (3.6). Then there exist a subsequence $\{v_{\varepsilon , \delta }\}$ (still denoted same notation) and a limit function $v_\infty \in L^\infty (0,S; W_0^{1,p}(\Omega ))$ for any positive $S<\infty$ the following convergence results hold true in the limit $\varepsilon , \delta \searrow 0$:

$$\begin{aligned} v_{\varepsilon ,\delta } \rightarrow v_\infty \quad&*\text {-weakly}\,\,in\,\,L^\infty (0,S\,; W_0^{1,p}(\Omega ))\,\,; \\ v_{\varepsilon ,\delta } \rightarrow v_\infty \quad&strongly\,\,in\,\,L^r(\Omega _S)\,\,\text {for}\,\,any \,\,r \ge 1\,\,; \\ \nabla v_{\varepsilon ,\delta } \rightarrow \nabla v_\infty \quad&strongly\,\,in \,\,L^r(\Omega _S)\,\,\text {for}\,\,any\,\,1 \le r <p\,\,; \\ \partial _sv_{\varepsilon , \delta }^{\frac{q+1}{2}} \rightarrow \partial _s v_\infty ^{\frac{q+1}{2}} \quad&weakly \,\,in\,\,L^2(\Omega _S)\,\,; \\ \partial _sv_{\varepsilon , \delta }^q \rightarrow \partial _s v_\infty ^q \quad&weakly \,\,in\,\,L^2(\Omega _S). \end{aligned}$$

In addition, $v_\infty$ is a weak solution of (5.2) and, for every positive $S < \infty$, $v_\infty$ satisfies the energy inequalities:

$$\begin{aligned}&\sup _{0<s<S}\frac{q}{q+1}\int _\Omega v_\infty (s)^{q+1}\,dx +\iint _{\Omega _S}|\nabla v_\infty |^p\,dxds \nonumber \\&\quad \le \frac{2q}{q+1}(t^*)^{-\frac{q+1}{q+1-p}}\left[ \int _\Omega u_0^{q+1}\,dx\right] e^{\frac{q+1}{q}\lambda _{p,q}S}, \end{aligned}$$

(C.8)

and

$$\begin{aligned}&\frac{4q}{(q+1)^2}\iint _{\Omega _S}\left( \partial _s v_\infty ^{\frac{q+1}{2}} \right) ^2\,dxds+\frac{1}{p}\sup _{0<s<S}\int _\Omega |\nabla v_\infty (s)|^p\,dx \nonumber \\&\quad \le 2e^{\frac{p}{q+1-p}S}\left[ (t^*)^{-\frac{p}{q+1-p}}\int _\Omega \frac{1}{p}|\nabla u_0|^p\,dx \right. \nonumber \\&\qquad \left. +\frac{\lambda _{p,q}}{q+1}(t^*)^{-\frac{q+1}{q+1-p}}\left( \int _\Omega u_0^{q+1}\,dx\right) e^{\frac{q+1}{q}\lambda _{p,q}S}\right] . \end{aligned}$$

(C.9)

Remark C.3

It is worth remarking that, in the proof of Lemma C.2, we use the diagonal method to choose a subsequence $\{v_{\varepsilon ,\delta }\}$ and a limit function $v_\infty$. Let $\{S_\lambda \}$ with $0 < \lambda \nearrow \infty$ be a sequence of times such that $S_\lambda \nearrow \infty$ as $\lambda \nearrow \infty$. For each $S_\lambda$, we subtract a subsequence, satisfying, in a time-interval $(0, S_\lambda )$, the same convergences to a limit as the statement of Lemma C.2, those may depend on $S_\lambda$. Finally, we take out the diagonal sequence of the sequences above.

Finally, we arrive at the following conclusion.

Lemma C.4

Let $v_\infty$ be a weak solution to (5.2), obtained in Lemma C.2 and v be a weak solution to (5.2) in Proposition 5.2, defined by (5.1). Then, there holds that $v_\infty \equiv v$ in $\Omega _\infty$ and thus, the solution v satisfies the energy inequality

$$\begin{aligned}{} & {} \iint _{\Omega _\infty }\left( \partial _s v^{\frac{q+1}{2}} \right) ^2\,dxds\\{} & {} \quad \le \frac{(q+1)^2}{4q}\left( (t^*)^{-\frac{p}{q+1-p}}\int _\Omega \frac{1}{p}|\nabla u_0|^p\,dx+\frac{\lambda _{p,q}}{q+1}\left( \frac{R(u_0)^{q+1}}{\lambda _{p,q}}\right) ^{\frac{p}{q+1-p}}\right) \end{aligned}$$

and $\partial _sv^q \in L^2(\Omega _S)$ for any positive $S<\infty$.

Proof

First, by the uniqueness of (5.2), Theorem A.1, we find that $v_\infty = v$ in $\Omega _S$ for any positive $S < \infty$. Here we use the following squared integrability in space-time of the time-derivative: Since by $q \ge 1$ the function ${\mathbb {R}}_+ \ni w \mapsto f(w)=w^{\frac{2q}{q+1}}$ is locally Lipschitz continuous, we obtain from (C.9) in Lemma C.2 that, for any positive $S < \infty$

$$\begin{aligned} \partial _sv_\infty ^{q}=\partial _s\left( v_\infty ^{\frac{q+1}{2}}\right) ^{\frac{2q}{q+1}}=\tfrac{2q}{q+1}\left( v_\infty ^{\frac{q+1}{2}}\right) ^{\frac{q-1}{q+1}}\partial _sv_\infty ^{\frac{q+1}{2}}=\tfrac{2q}{q+1}v_\infty ^{\frac{q-1}{2}}\partial _sv_\infty ^{\frac{q+1}{2}} \in L^2(\Omega _S), \end{aligned}$$

and therefore, the squared integrability on $\Omega _S$ of the time-derivative of $v_\infty ^q$ is actually verified for any positive $S<\infty$. Consequently, the solution $v_\infty = v$ possesses (5.3) and (5.4) in Proposition 5.2.

Next we shall observe the integrability of time-derivative in $\Omega _\infty$. From (C.6)$_1$ and (C.7)$_{1,2}$ in the proof of Lemma C.1 with $T=t^*$, we also deduce that, for every positive $S<\infty$,

$$\begin{aligned}&\frac{4q}{(q+1)^2}\iint _{\Omega _{S}}\left( \partial _s V_\varepsilon ^{\frac{q+1}{2}}\right) ^2\,dxds+\int _\Omega \frac{1}{p}\left( \delta (t^*e^{-S})^{-\frac{2}{q+1-p}}+|\nabla V_\varepsilon (S)|^2 \right) ^{\frac{p}{2}}\,dx \\&\quad \quad \le \int _\Omega \frac{1}{p}\left( \delta (t^*)^{-\frac{2}{q+1-p}}+|\nabla V_\varepsilon (0)|^2 \right) ^{\frac{p}{2}}\,dx \\&\qquad \quad +\frac{\delta }{q+1-p}\iint _{\Omega _{S}}(t^*e^{-s})^{-\frac{2}{q+1-p}}\left( \delta (t^*e^{-s})^{-\frac{2}{q+1-p}}+|\nabla V_\varepsilon |^2 \right) ^{\frac{p-2}{2}}\,dxds \\&\qquad \quad +\frac{\lambda _{p,q}}{q+1}\int _\Omega V_\varepsilon (S)^{q+1}\,dx. \end{aligned}$$

Again, passing to the limit $\varepsilon ,\delta \searrow 0$ and using Lemma C.2 implies that the limit function $v_\infty$ obtained in Lemma C.2 fulfills the following inequality too:

$$\begin{aligned}&\frac{4q}{(q+1)^2}\iint _{\Omega _{S}}\left( \partial _s v_\infty ^{\frac{q+1}{2}}\right) ^2\,dxds\\&\quad \le \int _\Omega \frac{1}{p}|\nabla v_\infty (0)|^p\,dx+\frac{\lambda _{p,q}}{q+1}\int _\Omega v_\infty (S)^{q+1}\,dx \\&\quad =(t^*)^{-\frac{p}{q+1-p}}\int _\Omega \frac{1}{p}|\nabla u_0|^p\,dx+\frac{\lambda _{p,q}}{q+1}\int _\Omega v_\infty (S)^{q+1}\,dx, \end{aligned}$$

which together with estimates (5.3) and (5.4) yields that

$$\begin{aligned}{} & {} \iint _{\Omega _S}\left( \partial _s v_\infty ^{\frac{q+1}{2}} \right) ^2\,dxds\\{} & {} \quad \le \frac{(q+1)^2}{4q}\left( (t^*)^{-\frac{p}{q+1-p}}\int _\Omega \frac{1}{p}|\nabla u_0|^p\,dx+\frac{\lambda _{p,q}}{q+1}\left( \frac{R(u_0)^{q+1}}{\lambda _{p,q}}\right) ^{\frac{p}{q+1-p}}\right) . \end{aligned}$$

Since the right-hand side is independent of S, letting $S \rightarrow \infty$ implies that

$$\begin{aligned}{} & {} \iint _{\Omega _\infty }\left( \partial _s v_\infty ^{\frac{q+1}{2}} \right) ^2\,dxds\\{} & {} \quad \le \frac{(q+1)^2}{4q}\left( (t^*)^{-\frac{p}{q+1-p}}\int _\Omega \frac{1}{p}|\nabla u_0|^p\,dx+\frac{\lambda _{p,q}}{q+1}\left( \frac{R(u_0)^{q+1}}{\lambda _{p,q}}\right) ^{\frac{p}{q+1-p}}\right) , \end{aligned}$$

as desired. $\square$

Appendix D: Proof of Lemma 4.1

In this appendix, we shall prove Lemma 4.1 in terms of approximating solution as in Appendix B.

Proof of Lemma 4.1

Let $U_{\varepsilon , \delta }$ be the family of weak solutions to (B.2) and define the the approximating Rayleigh quotient by

$$\begin{aligned} \texttt{R}_{\varepsilon , \delta }(t):=\frac{\displaystyle \int _{\Omega }\left( \delta +|\nabla U_{\varepsilon , \delta }(t)|^2\right) ^{\frac{p}{2}}\,dx}{\left( \displaystyle \int _{\Omega }U_{\varepsilon , \delta }(t)^{q+1}\,dx\right) ^{\frac{p}{q+1}}}. \end{aligned}$$

From the energy identities (B.21)–(B.23) in Lemma B.7, it readily follows that

$$\begin{aligned}&\dfrac{d}{dt}\left[ \frac{q}{q+1}\int _\Omega U_{\varepsilon , \delta }^{q+1}\,dx -\varepsilon \int _\Omega U_{\varepsilon , \delta }^q\,dx \right] +\int _{\Omega }\left( \delta +|\nabla U_{\varepsilon , \delta }|^2 \right) ^{\frac{p-2}{2}}|\nabla u_{\varepsilon ,\delta }|^2\,dx=0, \end{aligned}$$

(D.1)

$$\begin{aligned}&q\int _{\Omega }U_{\varepsilon , \delta }^{q-1}(\partial _t U_{\varepsilon , \delta })^2\,dx+\dfrac{d}{dt}\int _\Omega \frac{1}{p}\left( \delta +|\nabla U_{\varepsilon , \delta }(t)|^2 \right) ^{\frac{p}{2}}\,dx=0, \end{aligned}$$

(D.2)

and

$$\begin{aligned} \frac{4q}{(q+1)^2}\int _{\Omega }\left( \partial _t (U_{\varepsilon , \delta })^{\frac{q+1}{2}} \right) ^2\,dx+\dfrac{d}{dt}\int _\Omega \frac{1}{p}\left( \delta +|\nabla U_{\varepsilon ,\delta }(t)|^2 \right) ^{\frac{p}{2}}\,dx=0. \end{aligned}$$

(D.3)

Using the above energy identities (D.1) and (D.2), it is

$$\begin{aligned} \frac{d}{dt}{} \texttt{R}_{\varepsilon , \delta }(t)&=\frac{d}{dt}\left[ \frac{\displaystyle \int _{\Omega }\left( \delta +|\nabla U_{\varepsilon , \delta }(t)|^2\right) ^{\frac{p}{2}}\,dx}{\left( \displaystyle \int _{\Omega }U_{\varepsilon , \delta }(t)^{q+1}\,dx\right) ^{\frac{p}{q+1}}}\right] \nonumber \\&=\Vert u_{\varepsilon , \delta }\Vert _{q+1}^{-p}\frac{d}{dt}\int _{\Omega }(\delta +|\nabla U_{\varepsilon , \delta }|^2)^{\frac{p}{2}}\,dx\nonumber \\&\quad +\int _{\Omega }(\delta +|\nabla U_{\varepsilon , \delta }|^2)^{\frac{p}{2}}\,dx\,\frac{d}{dt}\left( \Vert U_{\varepsilon , \delta }\Vert _{q+1}^{q+1}\right) ^{-\frac{p}{q+1}}\nonumber \\&\!\!\!{\mathop {=}\limits ^{(D.2)}}\Vert U_{\varepsilon , \delta }\Vert _{q+1}^{-p}\,\left( -pq\,\int _{\Omega }(U_{\varepsilon , \delta })^{q-1}(\partial _tU_{\varepsilon , \delta })^2\,dx\right) \nonumber \\&\quad +\int _{\Omega }(\delta +|\nabla U_{\varepsilon , \delta }|^2)^{\frac{p}{2}}\,dx \left( -\frac{p}{q+1}\right) \Vert U_{\varepsilon , \delta }\Vert _{q+1}^{-p-q-1}\frac{d}{dt}\Vert U_{\varepsilon , \delta }\Vert _{q+1}^{q+1}\nonumber \\&\!\!\!{\mathop {=}\limits ^{(D.1)}}-pq\Vert U_{\varepsilon , \delta }\Vert _{q+1}^{-p}\,\int _{\Omega }(U_{\varepsilon , \delta })^{q-1}(\partial _t U_{\varepsilon , \delta })^2\,dx\nonumber \\&\quad +\frac{p}{q}\Vert U_{\varepsilon , \delta }\Vert _{q+1}^{-p-q-1}\int _{\Omega }(\delta +|\nabla U_{\varepsilon , \delta }|^2)^{\frac{p}{2}}\,dx \nonumber \\&\quad \cdot \left\{ -\varepsilon \frac{d}{dt}\Vert U_{\varepsilon , \delta }\Vert _{q}^{q} + \int _{\Omega }(\delta +|\nabla U_{\varepsilon , \delta }|^2)^{\frac{p-2}{2}} |\nabla U_{\varepsilon , \delta }|^2\,dx\right\} . \end{aligned}$$

(D.4)

As already observed in (B.4) in Proposition B.1, $U_{\varepsilon , \delta } \ge \varepsilon$ in $\Omega _\infty$ by the comparison principle and $U_{\varepsilon , \delta } = \varepsilon$ on $(\partial \Omega )_\infty$ and so, it is

$$\begin{aligned} \nu \cdot \nabla U_{\varepsilon , \delta } \le 0\quad \text {on the boundary} \,\,\,\partial \Omega \end{aligned}$$

with $\nu$ being the outward unit normal on $\partial \Omega$, where the gradient of $U_{\varepsilon , \delta }$ is continuous up to the boundary $\partial \Omega$ by the regularity of u in Proposition B.6. Hence,

$$\begin{aligned} \frac{d}{dt}\Vert U_{\varepsilon , \delta }\Vert _{q}^{q}&=\int _{\Omega }\textrm{div} \left( (\delta +|\nabla U_{\varepsilon , \delta }|^2)^{\frac{p-2}{2}}\nabla U_{\varepsilon ,\delta } \right) \,dx \nonumber \\&=-\int _{\partial \Omega }(\delta +|\nabla U_{\varepsilon , \delta }|^2)^{\frac{p-2}{2}}(\nu \cdot \nabla U_{\varepsilon , \delta })\,dS_x\ge 0. \end{aligned}$$

This together with (D.4) yields

$$\begin{aligned} \frac{d}{dt}{} \texttt{R}_{\varepsilon , \delta }(t)&\le -pq\Vert v_{\varepsilon , \delta }\Vert _{q+1}^{-p}\,\int _{\Omega }(U_{\varepsilon , \delta })^{q-1}(\partial _t U_{\varepsilon , \delta })^2\,dx\nonumber \\&\quad +\left( \frac{p}{q}\Vert U_{\varepsilon , \delta }\Vert _{q+1}^{-p-q-1}\int _{\Omega }(\delta +|\nabla U_{\varepsilon , \delta }|^2)^{\frac{p}{2}}\,dx\right) \nonumber \\&\quad \cdot \left( \int _{\Omega }(\delta +|\nabla U_{\varepsilon , \delta }|^2)^{\frac{p-2}{2}} |\nabla U_{\varepsilon , \delta }|^2\,dx\right) . \end{aligned}$$

(D.5)

Note that $U_{\varepsilon , \delta } \ge u_{\varepsilon , \delta }$, $\nabla U_{\varepsilon , \delta }= \nabla u_{\varepsilon , \delta }$ in $\Omega _\infty$ and $u_{\varepsilon , \delta }=0$ on $\partial \Omega \times (0,\infty )$. Keeping these facts in mind and using Cauchy-Schwarz inequality, we gain

$$\begin{aligned} \int _{\Omega }(\delta +|\nabla U_{\varepsilon , \delta }|^2)^{\frac{p-2}{2}} |\nabla U_{\varepsilon , \delta } |^2\,dx&=\int _{\Omega }(\delta +|\nabla U_{\varepsilon , \delta }|^2)^{\frac{p-2}{2}} \nabla U_{\varepsilon , \delta }\cdot \nabla u_{\varepsilon , \delta }\,dx\nonumber \\&=-\int _{\Omega }\textrm{div}\left( \left( \delta +|\nabla U_{\varepsilon , \delta }|^2\right) ^{\frac{p-2}{2}} \nabla U_{\varepsilon , \delta }\right) u_{\varepsilon , \delta }\,dx \nonumber \\&\le \int _{\Omega }\left| \Delta _p^\delta U_{\varepsilon , \delta }\right| U_{\varepsilon , \delta }\,dx \nonumber \\&\le \left( \int _{\Omega }\frac{(\Delta _p^\delta U_{\varepsilon , \delta })^2}{U_{\varepsilon , \delta }^{q-1}}\,dx\right) ^{1/2}\left( \int _{\Omega }U_{\varepsilon , \delta }^{q+1}\,dx\right) ^{1/2}, \end{aligned}$$

(D.6)

where we shortened $\Delta _p^\delta U_{\varepsilon , \delta }:=\textrm{div}\left( \left( \delta +|\nabla U_{\varepsilon , \delta }|^2\right) ^{\frac{p-2}{2}} \nabla U_{\varepsilon , \delta }\right)$. Since by (B.2)$_1$ and (B.5)$_3$ in Proposition B.1 it is

$$\begin{aligned} \Delta _p^\delta U_{\varepsilon , \delta }=\partial _tU_{\varepsilon , \delta }^q=qU_{\varepsilon , \delta }^{q-1}\partial _tU_{\varepsilon , \delta } \in L^2(\Omega _\infty ), \end{aligned}$$

this together with the fact that $U_{\varepsilon , \delta } \ge \varepsilon$ on $({\bar{\Omega }})_\infty$ concludes

$$\begin{aligned} \displaystyle \int _{\Omega }\frac{(\Delta _p^\delta U_{\varepsilon , \delta })^2}{U_{\varepsilon , \delta }^{q-1}}\,dx <\infty , \end{aligned}$$

therefore, we get from (D.6) that

$$\begin{aligned}{} & {} \int _{\Omega }(\delta +|\nabla U_{\varepsilon , \delta }|^2)^{\frac{p-2}{2}} |\nabla U_{\varepsilon , \delta } |^2\,dx\nonumber \\{} & {} \quad \le \left( q^2\int _{\Omega }U_{\varepsilon , \delta }^{q-1}(\partial _t U_{\varepsilon , \delta })^2\,dx\right) ^{1/2}\left( \int _{\Omega }U_{\varepsilon , \delta }^{q+1}\,dx\right) ^{1/2}. \end{aligned}$$

(D.7)

Simply,

$$\begin{aligned} \int _{\Omega }(\delta +|\nabla U_{\varepsilon , \delta }|^2)^{\frac{p}{2}}\,dx&=\int _{\Omega }(\delta +|\nabla U_{\varepsilon , \delta }|^2)^{\frac{p-2}{2}} |\nabla U_{\varepsilon , \delta }|^2\,dx\nonumber \\&\quad +\delta \int _{\Omega }(\delta +|\nabla U_{\varepsilon , \delta }|^2)^{\frac{p-2}{2}}\,dx \end{aligned}$$

(D.8)

and merging (D.7) and (D.8) in (D.5) renders

$$\begin{aligned} \frac{d}{dt}{} \texttt{R}_{\varepsilon , \delta }(t)&{\mathop {\le }\limits ^{(D.8)}} -pq\Vert v_{\varepsilon , \delta }\Vert _{q+1}^{-p}\,\int _{\Omega }(U_{\varepsilon , \delta })^{q-1}(\partial _t U_{\varepsilon , \delta })^2\,dx\nonumber \\&\quad \quad +\frac{p}{q} \Vert U_{\varepsilon , \delta }\Vert _{q+1}^{-p-q-1}\left( \int _{\Omega }(\delta +|\nabla U_{\varepsilon , \delta }|^2)^{\frac{p-2}{2}} |\nabla U_{\varepsilon , \delta }|^2\,dx\right) ^2 \nonumber \\&\quad \quad +\frac{p}{q} \Vert U_{\varepsilon , \delta }\Vert _{q+1}^{-p-q-1}\delta \int _{\Omega }(\delta +|\nabla U_{\varepsilon , \delta }|^2)^{\frac{p-2}{2}}\,dx \nonumber \\&\quad \quad \times \left( \int _{\Omega }(\delta +|\nabla U_{\varepsilon , \delta }|^2)^{\frac{p}{2}}\,dx-\delta \int _{\Omega }(\delta +|\nabla U_{\varepsilon , \delta }|^2)^{\frac{p-2}{2}}\,dx\right) \nonumber \\&{\mathop {\le }\limits ^{(D.7)}}-pq\Vert U_{\varepsilon , \delta }\Vert _{q+1}^{-p}\,\int _{\Omega }(U_{\varepsilon , \delta })^{q-1}(\partial _t U_{\varepsilon , \delta })^2\,dx\nonumber \\&\quad \quad +pq\Vert U_{\varepsilon , \delta }\Vert _{q+1}^{-p}\,\int _{\Omega }(U_{\varepsilon , \delta })^{q-1}(\partial _t U_{\varepsilon , \delta })^2\,dx\nonumber \\&\quad \quad +\frac{p}{q} \Vert U_{\varepsilon , \delta }\Vert _{q+1}^{-p-q-1}\delta \nonumber \\&\quad \left( \int _{\Omega }(\delta +|\nabla U_{\varepsilon , \delta }|^2)^{\frac{p-2}{2}}\,dx\right) \left( \int _{\Omega }(\delta +|\nabla U_{\varepsilon , \delta }|^2)^{\frac{p}{2}}\,dx\right) \nonumber \\&\le {\left\{ \begin{array}{ll} \dfrac{p}{q} \Vert U_{\varepsilon , \delta }\Vert _{q+1}^{-p-q-1}\delta |\Omega |^{\frac{2}{p}}\left( \displaystyle \int _{\Omega }(\delta +|\nabla U_{\varepsilon , \delta }|^2)^{\frac{p}{2}}\,dx\right) ^{\frac{2p-2}{p}}\quad &{}\text {if}\quad p \ge 2,\\ \dfrac{p}{q} \Vert U_{\varepsilon , \delta }\Vert _{q+1}^{-p-q-1}\delta ^{\frac{p}{2}}|\Omega | \left( \displaystyle \int _{\Omega }(\delta +|\nabla U_{\varepsilon , \delta }|^2)^{\frac{p}{2}}\,dx\right) \quad &{}\text {if}\quad 1<p<2 \end{array}\right. } \end{aligned}$$

(D.9)

where, in the penultimate inequality, we discarded

$$\begin{aligned} \displaystyle \frac{p}{q} \Vert U_{\varepsilon , \delta }\Vert _{q+1}^{-p-q-1}\left( \delta \int _{\Omega }(\delta +|\nabla U_{\varepsilon , \delta }|^2)^{\frac{p-2}{2}}\,dx\right) ^2 \end{aligned}$$

and, in the last inequality, we used Hölder’s inequality in the case $p \ge 2$ only.

We now need the positivity of solutions $u_\infty$ and $u_{\varepsilon , \delta }$ before the extinction time, that is the key to our energy estimates (refer to [23, Lemma 7.4]).

Lemma D.1

( [23, Lemma 7.4]) Let $u_{\varepsilon , \delta }$ be a weak solution to (B.1). Let $u_\infty$ be a weak solution to the prototype equations (3.6), obtained in Lemma B.8. Let $t^\star$ be a finite extinction time of $u_\infty$, presented in Proposition 3.5. Then, the following statements hold true:

• (i) Let $t_0<t^*$ be any positive number. Then, there exists a positive number $c_0$ such that, for every nonnegative $t \le t_0$,

  $$\begin{aligned} \Vert u_\infty (t)\Vert _{q+1} \ge c_0. \end{aligned}$$

  (D.10)

• (ii) For every $\nu >0$, there are positive constants $\varepsilon _0=\varepsilon _0(t_0,c_0)$ and $\delta _0=\delta _0(t_0,c_0)$ such that, for every positive numbers $\varepsilon \le \varepsilon _0$ and $\delta \le \delta _0$,

  $$\begin{aligned} \bigg | \Vert u_{\varepsilon , \delta }(t)+\varepsilon \Vert _{q+1}- \Vert u_\infty (t)\Vert _{q+1} \bigg |<\nu \end{aligned}$$

  (D.11)

  and, in particular,

  $$\begin{aligned} \Vert U_{\varepsilon , \delta }(t)\Vert _{q+1}=\Vert u_{\varepsilon , \delta }(t)+\varepsilon \Vert _{q+1} > \frac{c_0}{2}. \end{aligned}$$

  (D.12)

  hold for every nonnegative $t\le t_0$.

Proof of Lemma D.1

We note the outline on the proof. By Theorem 1.1 the solution $u_\infty$ is positive before the extinction time in Proposition 3.5. By the convergence (B.26) in Lemma B.8, the approximation $u_{\varepsilon ,\delta }$ is close to the limit $u_\infty$ in $L^{q+1} (\Omega )$.

We back to the proof of Lemma 4.1. Combining (D.12) in Lemma D.1 and the energy inequality (B.31) with (D.9), we arrive at, for every nonnegative $t\le t_0$,

$$\begin{aligned} \texttt{R}_{\varepsilon , \delta }(t)-\texttt{R}_{\varepsilon , \delta }(0) \le C\delta t^*\left( \frac{c_0}{2} \right) ^{-p-q-1}\left\| \left( \delta +|\nabla u_0 |^2 \right) ^{\frac{1}{2}} \right\| _p^p. \end{aligned}$$

(D.13)

From the strong convergence (B.27) in Lemma B.8, there exists a subsequence $\{u_{\varepsilon , \delta }\}$, denoted by same notation, such that

$$\begin{aligned} \nabla v_{\varepsilon ,\delta }=\nabla u_{\varepsilon ,\delta } \rightarrow \nabla u_\infty \quad \text {a.e.}\,\,\text {in}\,\,\Omega _T. \end{aligned}$$

(D.14)

By (D.14), the norm convergence (D.11) and Fatou’s lemma, we get

$$\begin{aligned} \liminf _{\varepsilon , \delta \searrow 0} \texttt{R}_{\varepsilon , \delta }(t) \ge \dfrac{\displaystyle \int _\Omega |\nabla u_\infty (t)|^p\,dx}{\left( \displaystyle \int _\Omega u_\infty (t)^{q+1}\,dx \right) ^{\frac{p}{q+1}}}=:R(u_\infty (t))^p. \end{aligned}$$

(D.15)

By the dominated convergence theorem

$$\begin{aligned} \limsup _{\varepsilon , \delta \searrow 0} \texttt{R}_{\varepsilon , \delta }(0) \le \limsup _{\delta \searrow 0} \dfrac{\displaystyle \int _\Omega \left( \delta +|\nabla u_0|^2 \right) ^\frac{p}{2}\,dx}{\left( \displaystyle \int _\Omega u_0^{q+1}\,dx \right) ^{\frac{p}{q+1}}}=\dfrac{\displaystyle \int _\Omega |\nabla u_0|^p\,dx}{\left( \displaystyle \int _\Omega u_0^{q+1}\,dx \right) ^{\frac{p}{q+1}}}=:R(u_0)^p.\nonumber \\ \end{aligned}$$

(D.16)

Notice that the right hand side of (D.13) converges to zero as $\varepsilon , \delta \searrow 0$ and thus, combining this with (D.15) and (D.16) and passing to the limit in (D.13) as $\varepsilon , \delta \searrow 0$, we have, for any nonnegative $t \le t_0$,

$$\begin{aligned} R(u_\infty (t)) \le R(u_0). \end{aligned}$$

Therefore the proof is complete. $\square$

Appendix E: Proof of Lemma 5.3

In this appendix, we prove the strong convergence of the gradient, Lemma 5.3.

Proof of Lemma 5.3

We basically follow the argument as in [6, Theorem 2.1, pp.31–33].

Firstly, let us define, for any positive $\delta <1$ and any $k\in {\mathbb {N}}$,

$$\begin{aligned} S(\delta ,\,k):=\left\{ x \in \Omega :\,|v(x,\theta _k)-U(x)| \ge \delta \right\} . \end{aligned}$$

(E.1)

From the strong convergence of (5.11), we see that the Lebesgue measure $|S(\delta , k)|$ converges to zero as $k\rightarrow \infty$, for any positive $\delta <1$:

$$\begin{aligned} |S(\delta ,\,k)| \rightarrow 0. \end{aligned}$$

(E.2)

Firstly, from the Hölder inequality, (5.4) and (E.2) we see that for all $r \in [1, p)$, as $k \rightarrow \infty$,

$$\begin{aligned} \int _{S(\delta ,\,k)} |\nabla v(\theta _k)-\nabla U|^r\,dx&\le \left( \,\,\int _{S(\delta ,k)} |\nabla v(\theta _k)-\nabla U|^p\,dx \right) ^\frac{r}{p} |S(\delta , k)|^{1-\frac{r}{p}} \nonumber \\ {}&\le C\left( \Vert \nabla v(\theta _k)\Vert _p^r+\Vert \nabla U\Vert _p^r \right) |S(\delta , k)|^{1-\frac{r}{p}} \nonumber \\ {}&\le C \left( \left[ \frac{R(u_0)^{q+1}}{\lambda _{p,q}}\right] ^\frac{r}{q+1-p}+\Vert \nabla U\Vert _p^r \right) |S(\delta , k)|^{1-\frac{r}{p}} \rightarrow 0. \end{aligned}$$

(E.3)

Next, we will observe the convergence of the gradient $\nabla v(\theta _k)$ on the set $\Omega {\setminus } S(\delta ,\,k)$. We claim that, as $k \rightarrow \infty$ and $\delta \searrow 0$,

$$\begin{aligned} \int _{\Omega \setminus S(\delta ,\,k)}|\nabla v(\theta _k)-\nabla U|^p\,dx \rightarrow 0. \end{aligned}$$

(E.4)

In order to prove the validity of (E.4), we distinguish two cases $p \ge 2$ and $1<p<2$. When considering the case $p \ge 2$, from the fundamental algebraic inequality (2.5) in Lemma 2.1 it is

$$\begin{aligned}&C\int _{\Omega \setminus S(\delta ,\,k)}|\nabla v(\theta _k)-\nabla U|^p\,dx \nonumber \\&\quad \le \int _{\Omega \setminus S(\delta ,\,k)}\left( |\nabla v|^{p-2}\nabla v(\theta _k)-|\nabla U|^{p-2}\nabla U \right) \cdot \left( \nabla v(\theta _k)-\nabla U \right) \,dx \nonumber \\&\quad =\int _{\Omega \setminus S(\delta ,\,k)}|\nabla v|^{p-2}\nabla v(\theta _k)\cdot \left( \nabla v(\theta _k)-\nabla U \right) \,dx \nonumber \\&\quad \quad -\int _{\Omega \setminus S(\delta ,\,k)}|\nabla U|^{p-2}\nabla U \cdot \left( \nabla v(\theta _k)-\nabla U \right) \,dx. \end{aligned}$$

(E.5)

In order to estimate the above display, let us consider the following Lipschitz truncation $f_\delta : w \in {\mathbb {R}} \mapsto f_\delta (w) \in {\mathbb {R}}$ defined by

$$\begin{aligned} f_\delta (w):= {\left\{ \begin{array}{ll} w &{}\quad \text {if} \quad |w| \le \delta \\ \delta &{}\quad \text {if} \quad w>\delta \\ -\delta &{}\quad \text {if}\quad w<-\delta . \end{array}\right. } \end{aligned}$$

From now on we abbreviate $f_\delta =f_\delta (v(\theta _k)-U)$. We choose a test function as $\varphi =f_\delta$ in (5.9). Notice that

$$\begin{aligned} \nabla f_\delta = {\left\{ \begin{array}{ll} 0 \quad &{}\text {on} \quad S(\delta ,\,k) \\ \nabla v(\theta _k)-\nabla U \quad &{}\text {on} \quad \Omega \setminus S(\delta ,\,k). \end{array}\right. } \end{aligned}$$

As a consequence, it follows from (E.5) that

$$\begin{aligned}&C\int _{\Omega \setminus S(\delta ,\,k)}|\nabla v(\theta _k)-\nabla U|^p\,dx \nonumber \\&\quad \le \int _{\Omega }|\nabla v|^{p-2}\nabla v(\theta _k)\cdot \nabla f_\delta \,dx -\int _{\Omega }|\nabla U|^{p-2}\nabla U \cdot \nabla f_\delta \,dx \nonumber \\&\quad =:(\textrm{I})+(\textrm{II}). \end{aligned}$$

(E.6)

By use of the Hölder inequality, (5.9), (5.3) and (5.4), the first term $(\textrm{I})$ of the right-hand side in (E.6) is estimated as

$$\begin{aligned} |(\textrm{I})|&=\Bigg |\int _\Omega |\nabla v|^{p-2}\nabla v(\theta _k) \cdot \nabla f_\delta \,dx \Bigg | \nonumber \\&\le \lambda _{p,q}\int _\Omega v^q|f_\delta |\,dx+\int _\Omega |\partial _s v^q||f_\delta |\,dx \nonumber \\&\le \lambda _{p,q}\delta \left( \frac{R(u_0)^p}{\lambda _{p,q}}\right) ^\frac{1}{q+1-p}|\Omega |^\frac{1}{q+1}+\delta |\Omega |^\frac{1}{2}\left( \,\,\int _\Omega (\partial _s v^q)^2(\theta _k)\,dx \right) ^\frac{1}{2} \rightarrow 0 \end{aligned}$$

(E.7)

as $k \rightarrow \infty$ and $\delta \searrow 0$. By the weak convergence (5.10), the second term II of the right hand side in (E.6) converges to zero as $k \rightarrow \infty$;

$$\begin{aligned} (\textrm{II})=-\int _\Omega |\nabla U|^{p-2}\nabla U \cdot \left( \nabla v(\theta _k)-\nabla U \right) \,dx \rightarrow 0. \end{aligned}$$

(E.8)

Merging (E.6)–(E.8), we end up with the desired conclusion (E.4) in the case $p \ge 2$. At this point it remains to prove (E.4) in the case $1<p<2$. As argued before, by means of Hölder’s inequality, the algebraic inequality (2.4) and the estimate (5.3) we bound:

$$\begin{aligned}&\int _{\Omega \setminus S(\delta ,\,k)}|\nabla v(\theta _k)-\nabla U|^p\,dx \\&\quad \le \left( \int _{\Omega \setminus S(\delta ,\,k)} |\nabla v(\theta _k)-\nabla U|^2 \left( |\nabla v(\theta _k)|+|\nabla U| \right) ^{p-2} \,dx\right) ^{\frac{p}{2}} \\&\qquad \cdot \left( \int _{\Omega \setminus S(\delta ,\,k)}\left[ |\nabla v(\theta _k)|+|\nabla U| \right] ^{p}\,dx \right) ^{\frac{2-p}{2}} \\&\quad \le C \left( \int _{\Omega \setminus S(\delta ,\,k)} \left( |\nabla v(\theta _k)|^{p-2}\nabla v(\theta _k)-|\nabla U|^{p-2}\nabla U\right) \cdot \left( \nabla v(\theta _k)-\nabla U\right) \,dx \right) ^{\frac{p}{2}} \\&\qquad \cdot \left( \int _{\Omega }\left[ |\nabla v(\theta _k)|+|\nabla U| \right] ^{p}\,dx \right) ^{\frac{2-p}{2}} \\&\quad \le C \left( \int _{\Omega \setminus S(\delta ,\,k)} \left( |\nabla v(\theta _k)|^{p-2}\nabla v(\theta _k)-|\nabla U|^{p-2}\nabla U\right) \cdot \left( \nabla v(\theta _k)-\nabla U\right) \,dx \right) ^{\frac{p}{2}} \\&\qquad \cdot \left( \left[ \dfrac{R(u_0)^{q+1}}{\lambda _{p,q}}\right] ^{\frac{p(2-p)}{2(q+1-p)}}+\Vert \nabla U\Vert _p^{\frac{p(2-p)}{2}}\right) . \end{aligned}$$

Repeating the above argument in the case $p \ge 2$, we can deduce that

$$\begin{aligned} \left( \int _{\Omega \setminus S(\delta ,\,k)} \left( |\nabla v(\theta _k)|^{p-2}\nabla v(\theta _k)-|\nabla U|^{p-2}\nabla U\right) \cdot \left( \nabla v(\theta _k)-\nabla U\right) \,dx \right) ^{\frac{p}{2}} \rightarrow 0 \end{aligned}$$

as $k \rightarrow \infty$, because of the condition $p \ge 2$ is not required in the convergences (E.7) and (E.8). In this way, in the remaining case $1<p<2$, we obtain (E.4) too.

Hence we are finally led to

$$\begin{aligned}{} & {} \int _{\Omega \setminus S(\delta ,\,k)}|\nabla v(\theta _k)-\nabla U|^r\,dx\\{} & {} \quad \le \left( \,\,\int _{\Omega \setminus S(\delta ,\,k)}|\nabla v(\theta _k)-\nabla U|^p\,dx \right) ^\frac{r}{p}\,| \Omega \setminus S(\delta ,\,k) |^{1-\frac{r}{p}} \rightarrow 0. \end{aligned}$$

Combining this with (E.3), we complete the proof.