Existence of a Sign-Changing Weak Solution to Doubly Nonlinear Parabolic Equations

Abstract

In this paper, assuming the initial-boundary datum belonging to suitable Sobolev and Lebesgue spaces, we prove the global existence result for a (possibly sign changing) weak solution to the Cauchy–Dirichlet problem for doubly nonlinear parabolic equations of the form

$$\begin{aligned} \partial _t\left( |u|^{q-1}u\right) -\Delta _p u=0\quad \text {in}\,\,\,\Omega _\infty , \end{aligned}$$

where $p>1$ and $q>0$. This is a fair improvement of the preceding result by authors (Nonlinear Anal 175C :157–172, 2018). The key tools we employ are energy estimates for approximate equations of Rothe type and the integral strong convergence of gradients of approximate solutions.

1 Introduction

Let $p > 1$ and $q > 0$ be given exponents and $\Omega$ be a bounded domain in ${\mathbb {R}}^n$ with $n \geqq 2$ with smooth boundary. The aim of this paper is to establish the existence result of solutions of the Cauchy–Dirichlet problem for the doubly nonlinear parabolic type equation of the form

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

(1.1)

Here, the unknown function $u = u (x, t)$ is a real-valued function defined on $\Omega _\infty$ and $\Delta _p u :=\mathrm {div}\left( |Du|^{p-2}D u\right)$ denotes the p-Laplacian, where by $Du =(\partial _{x_i}u)_{1\leqq i \leqq n}$ we mean the spatial gradient of u with respect to x, and the initial datum $u_0$ is given in some function space. The initial condition $u=u_0$ in $\Omega \times \{0\}$ has to be understood in the $L^{q+1}$-sense, while the boundary condition $u=0$ on the lateral boundary is given in the usual trace sense; see (D3) in Definition 1 below. The doubly nonlinear parabolic equations (1.1)$_1$ are classified as fast diffusion equations in the case $q>p-1$ and slow diffusion equations in the case $q<p-1$, whose criterion depends on the expansion of support of the solutions. In the special case $q=p-1$, (1.1)$_1$ covers the Trudinger equation, which has first been studied by Trudinger [32].

Prior to stating our main results, we shall give a brief outlook of the literatures and a motive for (1.1). The existence problem for (1.1) has been already studied in some preceding results. The general form equations of which the prototype equation being (1.1) was firstly treated for the case that $q \geqq 1$ in Alt & Luckhaus [2]. By a variational method the existence of non-negative solutions is succeeded by Bögelein, Duzaar, Marcellini & Scheven [4]. Their approach enables us to extend the result to non-negative variational solutions for more general parabolic equations of the type

$$\begin{aligned} \partial _t b(u)-\mathrm {div} D_\xi f(x,u,Du)=-D_uf(x,u,Du) \quad \text {in}\,\,\,\Omega _T, \end{aligned}$$

(1.2)

where $f:\Omega \times {\mathbb {R}}\times {\mathbb {R}}^n \rightarrow (-\infty ,\infty ]$ is a Carathéodory function satisfying certain convexity and coercivity assumptions. The prototype of the integrand f is of the form $f=f(x,Du)=\alpha (x) |Du|^p+\beta (x)|Du|^q$ with $1<p<q$ and $\alpha ,\,\beta$ being non-negative coefficients such that $\alpha (x)+\beta (x) \geqq \nu >0$. The historical notes and the overview of regularity theory for the elliptic equation associated with (1.2) with the nonstandard growth condition similarly as above are addressed in Mingione & Rădulescu [25] and references therein. Besides, the existence of variational solutions to the Cauchy–Dirichlet problem with time-dependent boundary values in the case $q>1$ is established by Schätzler [30]. We would also like to state that the Hölder regularity of sign-changing weak solutions has been shown in [8] in the case $q=p-1$. The regularity issue for a weak solution to the doubly nonlinear equation (1.1) will be the challenging problem in our near future. In our previous result by the authors [27], the existence of a solution in the energy class is studied in the case that $q+1= \frac{n p}{n - p}$ with $n \geqq 3$ and $2 \leqq p<n$. The definition of a weak solution is stated in Definition 1. Our main aim in this paper is to extend and refine [27], showing the global existence of a weak solution of (1.1) for any initial datum of the energy class in the general case that $q > 0$ and $p > 1$. We shall show the existence of a weak solution satisfying the energy inequality, of which the proof relies on the same Rothe-type approximation as in [27]. The new content gained here is the strong convergence of the gradients of approximating solutions (Lemma 4.8). The proof uses the exponential mollification (see Lemma 2.1), which is an effective measure to control the power nonlinearity in time derivative, and the so-called boundary integral, devised in [4]. It may have an interest of its own self. See formulae (2.1) and (2.2) for the definition of mollification and boundary integral.

The doubly nonlinear parabolic equations appear in a model of some physical phenomena like plasma physics or turbulent filtration of liquids (see e.g., [31] and references therein for a detailed description) and also describe the gradient flow associated with the Sobolev inequality and is related to the so-called Yamabe flow in differential geometry in the case that $p = 2$. Also see [28] and references therein for the nonlinear parabolic equations on Riemannian manifolds. Besides, the gradient flow of the p-elastic energy is one of geometric aspects of our motive, see [9] for a detailed description. Our motive for studying the equation (1.1) is the gradient flow associated with the Sobolev inequality in the $L^p$-setting, referred to the p-Sobolev flow. In the Sobolev critical case that $q + 1 = \frac{n p}{n - p}$ with $n \geqq 3$ and $2 \leqq p < n$, the global existence of a positive weak solution with the gradient regularity is shown in [19, 20], where the so-called intrinsic scaling method peculiar to the doubly nonlinear equation (1.1) is devised to construct the global solution of the p-Sobolev flow. The results and methods presented here will enable us to show the global existence and regularity of the p-Sobolev flow in the general case that $q > 0$ and $p > 1$ in our near future work.

Our main result reads as follows:

Theorem 1.1

Let $p > 1$, $q >0$ be given and assume that the initial datum $u_0$ belongs to $W^{1,p}_0(\Omega ) \cap L^{q+1}(\Omega )$. Then there exists a global in time weak solution to (1.1) in the sense of Definition 1 fulfilling the following energy structures and regularity:

$$\begin{aligned}&\sup _{0<t<\infty }\int _\Omega |u(t)|^{q+1}\,dx+\frac{q+1}{q}\iint _{\Omega _\infty }|Du|^p\,dxdt \leqq \int _\Omega |u_0|^{q+1}\,dx, \end{aligned}$$

(1.3)

$$\begin{aligned}&\sup _{0<t<\infty }\int _\Omega |Du(t)|^{p}\,dx\leqq \int _\Omega |Du_0|^{p}\,dx, \end{aligned}$$

(1.4)

and

$$\begin{aligned} \iint _{\Omega _\infty }\left| \partial _t\left( |u|^{\frac{q-1}{2}}u\right) \right| ^2\,dxdt \leqq c \Vert Du_0\Vert _{L^p(\Omega )}^p \end{aligned}$$

(1.5)

with a constant $c \equiv c(p,q)$.

Our method developed here can be applied for Cauchy–Dirichlet problem (1.1) for doubly nonlinear parabolic equations of more general form, having the variational structure with controllable growth on gradients. Here, unless otherwise specified, we shall deal with the prototype equation (1.1)$_1$ and present the transparent details of proof.

A synopsis of this paper is in the following: In the next section, we display the notation and summarize auxiliary material used throughout the paper. In Sect. 3, we construct approximate solutions of (1.1). Sect. 4 contains the results of the convergence of approximate solutions of (1.1). In Sect. 5, we complete the proof of our Theorem 1.1 via the weak compactness method. Appendix A is devoted to proving Lemma 3.1 by the Direct Method in the Calculus of Variations. In Appendix B, we prove the finite integration by parts formula (Lemma 4.9). Appendix C is devoted to the proof of the needed lemma for Sect. 5 (Lemma 5.1). In the final Appendix D, we show the initial regularity (5.6).

2 Preliminaries

We shall split this section in three parts: first, we display our notation, then we list some technical tools. Finally, we present the definition of weak solutions to Eq. (1.1).

2.1 Notation

In the course of the paper, we work in the Euclidean space ${\mathbb {R}}^n$ in a fixed dimension $n \geqq 2$ and $\Omega \subset {\mathbb {R}}^n$ denotes a bounded domain with the smooth boundary. For $T \in (0,\infty )$, let $\Omega _T:=\Omega \times (0,T)$ be a space-time cylinder. As is customary,

$$\begin{aligned} B_\varrho (x_0):=\left\{ x \in {\mathbb {R}}^n: |x-x_0|<\varrho \right\} \end{aligned}$$

denotes the usual Euclidean ball with radius $\varrho >0$, centered at $x_0 \in {\mathbb {R}}^n$. We adopt the convention of writing $B_\varrho$ instead of $B_\varrho (x_0)$, when the center is origin 0 of ${\mathbb {R}}^n$, or when the center is clear from the context.

Given a measurable set ${\mathcal {O}} \subset {\mathbb {R}}^k$ and an integrable map $w: {\mathcal {O}}\rightarrow {\mathbb {R}}^k$,  $k \in {\mathbb {N}}$, we write

for its mean-value on ${\mathcal {O}}$, provided $0< |{\mathcal {O}}|< \infty$. With ${\mathcal {O}} \subset {\mathbb {R}}^{k}$ being a measurable set and w being a measurable function defined on ${\mathcal {O}}$, we shall use the shorthand notation

$$\begin{aligned} {\mathcal {O}} \cap \{w>k\}&:=\left\{ z\in {\mathcal {O}}: w(z)>k\right\} , \\ {\mathcal {O}} \cap \{w<k\}&:=\left\{ z \in {\mathcal {O}}: w(z)<k\right\} . \end{aligned}$$

Finally, we will list the general notation. Let us denote by c, $c_1$, $c_2, \cdots$ different positive constants in a given context. Relevant dependencies on parameters will be emphasized using parentheses, e.g., $c\equiv c(n,\gamma , p)$ means that c depends on $n, \gamma$, and p. For the sake of readability, the dependencies of the constants will be often omitted within the chains of estimates. Furthermore, by $(\,\cdot \,)_\ell$, we mean the $\ell$-th line of the Eq. $(\,\cdot \,)$.

2.2 Exponential Mollification in Time

In this subsection, we introduce the technique of the exponential mollification in time, which appears in the paper by Landes [23]. Kinnunen & Lindqvist [18] contributed the validity of this technique for the doubly nonlinear PDEs theory. This mollification enables us to overcome the lack of regularity in the time variable. The properties of this mollification are applied to the variant doubly nonlinear equations, as seen in the literatures [3, 6,7,8].

Let $\Omega \subset {\mathbb {R}}^n$ be a bounded domain and $0<T<\infty$. For $v \in L^1(\Omega _T)$, $v_0 \in L^1(\Omega )$ and $h \in (0,T)$, we define

$$\begin{aligned}{}[v]_h(x,t):=e^{-\frac{t}{h}}v_0+\dfrac{1}{h}\int _0^te^{\frac{s-t}{h}}v(x,s)\,ds,\quad (x,t) \in \Omega \times [0,T]. \end{aligned}$$

(2.1)

Analogously, the reversed version of $[v]_h$ is defined by

$$\begin{aligned}{}[v]_{{\bar{h}}}(x,t):=\dfrac{1}{h}\int _t^Te^{\frac{t-s}{h}}v(x,s)\,ds,\quad (x,t) \in \Omega \times [0,T]. \end{aligned}$$

In this setting, we list the properties of $[v]_h$ and $[v]_{{\bar{h}}}$ displayed below, whose detailed proof can be seen in the literatures [18, Lemma 2.2], [5, Appendix B], and [31, Lemma 2.9].

Lemma 2.1

Assume that $v\in L^1(\Omega _T)$ and $p \in [1,\infty )$. Then the mollifications $[v]_{h}$ and $[v]_{{\bar{h}}}$ have the following properties:

• (i) If $v \in L^p(\Omega _T)$, then $[v]_h \in L^p(\Omega _T)$ and the following quantitative estimate holds true:

  $$\begin{aligned} \Vert [v]_h\Vert _{L^p(\Omega _T)} \leqq \Vert v\Vert _{L^p(\Omega _T)}+h^{\frac{1}{p}}\Vert v_0\Vert _{L^p(\Omega )}. \end{aligned}$$

  Furthermore, $[v]_h \rightarrow v$ strongly in $L^p(\Omega _T)$ as $h \searrow 0$. A same statement for $[v]_{{\bar{h}}}$ holds true.

• (ii) If $v \in L^p(\Omega _T)$, $v_0 \in L^p(\Omega )$, $Dv \in L^p(\Omega _T, {\mathbb {R}}^n)$, and $Dv_0 \in L^p(\Omega , {\mathbb {R}}^n)$, then $D[v]_h=[Dv]_h$. Moreover, the statement (i) holds true with v, $v_0$ and $[v]_h$ replaced by Dv, $Dv_0$, and $D[v]_h$, respectively.

• (iii) If $v_0 \in W^{1,p}_0(\Omega _T)$ and $v \in L^p(0,T\,;W^{1,p}_0(\Omega ))$, then $[v]_h \in L^p(0,T\,;W^{1,p}_0(\Omega ))$. Furthermore, there holds that

  $$\begin{aligned}{}[v]_h \rightarrow v \quad \text {strongly in}\,\,\, L^p(0,T\,;W^{1,p}_0(\Omega )) \quad \text {as}\,\,\, h \searrow 0. \end{aligned}$$

  The same statement holds true for $[v]_{{\bar{h}}}$

• (iv) If $v \in L^p(0,T\,; L^p(\Omega ))$, then $[v]_h$ and $[v]_{{\bar{h}}}$ belong to $C([0,T]\,; L^p(\Omega ))$.

• (v) If $v \in L^p(\Omega _T)$, then $[v]_h$ and $[v]_{{\bar{h}}}$ have weak time derivatives being in $L^p(\Omega _T)$ and solve the ODE:

  $$\begin{aligned} \partial _t[v]_h&=-\frac{1}{h}\left( [v]_h-v\right) \end{aligned}$$

  with the initial condition $[v]_h(\cdot ,0)=v_0$ and

  $$\begin{aligned} \partial _t[v]_{{\bar{h}}}&=\frac{1}{h}\left( [v]_{{\bar{h}}}-v\right) . \end{aligned}$$

2.3 Some Preliminary Material

In this subsection, we collect the auxiliary material used throughout the paper.

We shall use the boundary integral term, devised by Bögelein et. al [4]: For $u,k \in {\mathbb {R}}$

$$\begin{aligned} {\mathbf {B}}[u,k]:=\dfrac{1}{q+1}\left( |k|^{q+1}-|u|^{q+1}\right) -|u|^{q-1}u(k-u). \end{aligned}$$

(2.2)

This boundary term is a crucial quantity for our argument. We state the estimates for the boundary term ${\mathbf {B}}$, whose precise proof is presented in [4, Lemma 2.5] and also in [7, Lemma 3.4].

Lemma 2.2

Let $q>0$ and let ${\mathbf {B}}[u,k]$ be given by (2.2) for $u, k \in {\mathbb {R}}$. Then, there exists a positive constant $c\equiv c(q)$ such that the following estimates hold:

$$\begin{aligned} {\mathbf {B}}[u,k] \leqq \left( |k|^{q-1}k-|u|^{q-1}u\right) \left( k-u\right) \end{aligned}$$

(2.3)

and

$$\begin{aligned} c^{-1} \left| |u|^{\frac{q-1}{2}}u-|k|^{\frac{q-1}{2}}k\right| ^2 \leqq {\mathbf {B}}[u,k] \leqq c \left| |u|^{\frac{q-1}{2}}u-|k|^{\frac{q-1}{2}}k\right| ^2. \end{aligned}$$

(2.4)

We also recall the algebraic inequality retrieved from [14, Lemma 8.3] and [12, Lemma 4.4–Chap. I]. See the proof in [1, Lemma 2.2] in the case $1<\alpha <2$ and in [13, inequality (2.4)] in the case $\alpha >2$.

Lemma 2.3

(Algebraic inequality) For all $\alpha \in (1,\infty )$ there are positive constants $c_1(\alpha )$ and $c_2(\alpha )$ such that, for all $A,\,B \in {\mathbb {R}}^k, k \geqq 1$,

$$\begin{aligned}&\big ||A|^{\alpha -2}A-|B|^{\alpha -2}B\big | \leqq c_1(|A|+|B|)^{\alpha -2}|A-B|, \end{aligned}$$

(2.5)

$$\begin{aligned}&(|A|^{\alpha -2}A-|B|^{\alpha -2}B)\cdot (A-B) \geqq c_2 (|A|+|B|)^{\alpha -2}|A-B|^2 \end{aligned}$$

(2.6)

and, in particular, when $\alpha \geqq 2$

$$\begin{aligned} (|A|^{\alpha -2}A-|B|^{\alpha -2}B)\cdot (A-B) \geqq c_2 |A-B|^\alpha , \end{aligned}$$

(2.7)

where the dot $\cdot$ denotes the usual inner product in ${\mathbb {R}}^k$.

We next list the following integral convergence lemma that will be used later, whose proof can be seen in [16, Lemma 2.2].

Lemma 2.4

(Integral convergence lemma) Let $a \in (1,\infty )$ and $a^\prime$ satisfy $\frac{1}{a}+\frac{1}{a^\prime }=1$. Assume $(f_h)_{h>0}$ is bounded in $L^a(\Omega _T)$ and $f_h \rightarrow f$ almost everywhere in $\Omega _T$. Then

$$\begin{aligned} \iint _{\Omega _T} f_h \psi \,dxdt \rightarrow \iint _{\Omega _T} f\psi \,dxdt \qquad \forall \psi \in L^{a^\prime }(\Omega _T) \end{aligned}$$

holds true. Specifically, if $(f_h)_{h > 0}$ is bounded in $L^a (\Omega _T)$ and $f_h \rightarrow 0$ almost everywhere in $\Omega _T$, then for every $b \in [1, a)$,

$$\begin{aligned} \iint _{\Omega _T} |f_h|^b \,dxdt \rightarrow 0 \quad \text {as}\,\, \,h \searrow 0. \end{aligned}$$

(2.8)

Finally, we summarize an application of the classical Lebesgue dominated convergence theorem.

Lemma 2.5

(Variant Lebesgue dominated convergence theorem) Let $(f_h)_{h>0}$ be a family of measurable functions on $\Omega _T$ fulfilling the following three conditions: As $h \searrow 0$,

• (1) $f_h \rightarrow f$ almost everywhere in $\Omega _T$.

• (2) there exists $g_h \in L^1(\Omega _T)$ such that $|f_h| \leqq g_h$ almost everywhere in $\Omega _T$.

• (3) there is a function $g \in L^1(\Omega _T)$ such that

  • (3-1) $g_h \rightarrow g$ almost everywhere in $\Omega _T$,

  • (3-2) $\displaystyle \lim _{h \searrow 0}\iint _{\Omega _T}g_h\,dxdt =\iint _{\Omega _T}g\,dxdt$.

Then there holds that

$$\begin{aligned} \displaystyle \lim _{h \searrow 0}\iint _{\Omega _T}f_h\,dxdt =\iint _{\Omega _T}f\,dxdt. \end{aligned}$$

2.4 Weak Solution

We make precise the notion of solution to (1.1) that we will use.

Definition 1

(Weak solution) Assume that the initial datum $u_0$ is in the class $W^{1,p}_0(\Omega ) \cap L^{q+1}(\Omega )$. We identify a measurable function $u:\Omega _\infty \rightarrow {\mathbb {R}}$ as a weak solution to the Cauchy–Dirichlet problem (1.1) if and only if the following conditions (D1)–(D3) are fulfilled:

• (D1) $u \in L^p(0,\infty \,; W^{1,p}(\Omega )) \cap C([0,\infty )\,; L^{q+1}(\Omega ))$

• (D2) For every testing function $\phi \in C^\infty _0(\Omega _\infty )$, there holds that

  $$\begin{aligned}&\iint _{\Omega _\infty }\left( -|u|^{q-1}u\,\partial _t\phi +|Du|^{p-2}Du\cdot D\phi \right) \,dxdt=0 \end{aligned}$$

  (2.9)

• (D3) u attains the prescribed initial condition $u(0)=u_0$ in the $L^{q+1}$ -sense, that is,

  $$\begin{aligned} \lim _{t\searrow 0}\Vert u(t)-u_0\Vert _{L^{q+1}(\Omega )}=0 \end{aligned}$$

  and u satisfies the boundary condition in the following sense:

  $$\begin{aligned} u(t) \in W_0^{1,p}(\Omega )\quad \text {for}\,\,\text {a.e.}\,\,t \in (0,\infty ). \end{aligned}$$

3 Approximate Solutions

In this section, we will construct approximate solutions of (1.1) satisfying certain energy estimates. Following [27, Sect. 3], we define a family of elliptic equations of Rothe type (as seen in [29]).

Let $h \in (0,1]$ be a positive number, sent to zero later. Starting with the initial datum $u_0$ belonging to $W_0^{1,p}(\Omega ) \cap L^{q+1}(\Omega )$, we inductively construct a sequence $u_i \in W_0^{1,p}(\Omega ) \cap L^{q+1}(\Omega )$ of solutions to the following elliptic equations for every $i \in {\mathbb {N}}$:

$$\begin{aligned} {\left\{ \begin{array}{ll} \dfrac{|u_i|^{q-1}u_i-|u_{i-1}|^{q-1}u_{i-1}}{h}-\Delta _pu_i=0 \quad &{}\mathrm {in}\,\,\Omega \\ u_i=0\quad &{}\mathrm {on}\,\,\partial \Omega . \end{array}\right. } \end{aligned}$$

(3.1)

The existence of the weak solution $u_i$ to (3.1) is guaranteed by the following lemma. The proof is postponed and will be given in Appendix A.

Lemma 3.1

Suppose that Eq. (3.1) has solutions $u_j$ inductively for $j=1,\ldots ,i-1$. Then there exists a weak solution $u_i \in W_0^{1,p}(\Omega ) \cap L^{q+1}(\Omega )$ to (3.1) in the following sense:

$$\begin{aligned} \int _\Omega \left[ \dfrac{|u_i|^{q-1}u_i-|u_{i-1}|^{q-1}u_{i-1}}{h}\xi +|Du_i|^{p-2}Du_i\cdot D\xi \right] \,dx=0 \end{aligned}$$

(3.2)

holds whenever $\xi \in W^{1,p}_0(\Omega ) \cap L^{q+1}(\Omega )$.

Again, let $h<1$ be a fixed positive number, sent to zero later. Set $t_i:=ih$ for any non-negative integer i. Following the scheme as presented in [16, 27], we construct the six functions ${\bar{u}}_h,\,u_h,\,{\bar{v}}_h,\,v_h,\,{\bar{w}}_h$, and $w_h$ on $\Omega \times [-h,\infty )$. For the solutions $(u_i)_{i \in {\mathbb {N}}}$ to (3.1) and $i = 1, 2, \ldots$, we define the piecewise constant functions

$$\begin{aligned} {\left\{ \begin{array}{ll} {\bar{u}}_h(x,t):= {\left\{ \begin{array}{ll} u_0(x), &{}\quad (x,t) \in \Omega \times [-h, 0], \\ u_i(x), &{}\quad (x,t)\in \Omega \times (t_{i-1}, t_i], \end{array}\right. } \\ {\bar{v}}_h(x,t):=|{\bar{u}}_h(x,t)|^{q-1}{\bar{u}}_h(x,t), \qquad (x,t) \times \Omega \in [-h,\infty ), \\ {\bar{w}}_h(x,t):=|{\bar{u}}_h(x,t)|^{\frac{q-1}{2}}{\bar{u}}_h(x,t), \qquad (x,t) \in \Omega \times [-h,\infty ). \end{array}\right. } \end{aligned}$$

(3.3)

For $(x,t) \in \Omega \times [t_{i-1},t_i]$ with $i = 1, 2,\ldots$, we also define the piecewise linear functions

$$\begin{aligned} {\left\{ \begin{array}{ll} u_h(x,t):=\dfrac{t-t_{i-1}}{h}u_i(x)+\dfrac{t_i-t}{h}u_ {i-1}(x), \\ v_h(x,t):=\dfrac{t-t_{i-1}}{h}|u_i(x)|^{q-1}u_i(x)+\dfrac{t_i-t}{h}|u_{i-1}(x)|^{q-1}u_{i-1}(x), \\ w_h(x,t):=\dfrac{t-t_{i-1}}{h}|u_i(x)|^{\frac{q-1}{2}}u_i(x)+\dfrac{t_i-t}{h}|u_{i-1}(x)|^{\frac{q-1}{2}}u_{i-1}(x). \end{array}\right. } \end{aligned}$$

(3.4)

We call all of six functions ${\bar{u}}_h$, ${\bar{v}}_h$, ${\bar{w}}_h$, $u_h$ , $v_h$, and $w_h$ as approximate solutions of (1.1). At this stage, (3.1) is rewritten as

$$\begin{aligned} \partial _t v_h-\Delta _p{\bar{u}}_h=0 \quad \text {in}\,\,\,\Omega _\infty , \end{aligned}$$

equivalently, in the sense of distribution,

$$\begin{aligned}&\iint _{\Omega _T} \left( \partial _tv_h\cdot \varphi + |D{\bar{u}}_h|^{p-2}D{\bar{u}}_h\cdot D\varphi \right) \,dxdt=0 \end{aligned}$$

(3.5)

holds for any positive $T<\infty$ and all testing functions $\varphi \in L^1 (0, T ; W^{1,p}_0 (\Omega ) \cap L^{q+1} (\Omega ))$.

In the following, we shall make some integral estimates for the approximate solutions. Here, we always suppose the setting as follows : For any positive number $T < \infty$ and any step size $h \in (0,1]$, let k be the positive integer fulfilling $(k - 1) h \leqq T < k h$.

Lemma 3.2

(Energy estimates) Let ${\bar{u}}_h$ and $u_h$ be the approximate solutions of (1.1) defined as in (3.3)–(3.4). The following energy estimates hold for any positive $T < \infty$ :

$$\begin{aligned} \sup _{0<t<T}\int _\Omega |{\bar{u}}_h(t)|^{q+1}\,dx+\frac{q+1}{q}\iint _{\Omega _T} |D{\bar{u}}_h(t)|^p\,dxdt\leqq \int _\Omega |u_0|^{q+1}\,dx \end{aligned}$$

(3.6)

and

$$\begin{aligned} \iint _{\Omega _T}I_h(t)|\partial _tu_h|^2\,dxdt+\frac{1}{p}\sup _{0<t<T}\int _\Omega |D{\bar{u}}_h(t)|^p\,dx \leqq \frac{1}{p}\int _\Omega |Du_0|^p\,dx, \end{aligned}$$

(3.7)

where $I_h(t):=\displaystyle \int _0^1\left| \theta {\bar{u}}_h(t)+(1-\theta ){\bar{u}}_h(t-h)\right| ^{q-1}\,d\theta$.

Proof

The proof can be achieved by the following arguments in [27, Lemma 3.2]; therefore, we will only sketch it. Testing $\xi =u_i$ in (3.2) and summing up on $i=1,\ldots ,k$, subsequently, we use Young’s inequality to obtain (3.6). Similarly,  (3.7) is verified by choosing $\xi =\frac{u_i-u_{i-1}}{h}$ in (3.2). This finishes the proof. $\square$

The statement of Lemma 3.2 also holds for the linear interpolation $u_h$ as follows.

Lemma 3.3

(Energy estimates) Let $u_h$ be the approximate solution of (1.1) defined by (3.4)$_1$. Then, the following energy estimates hold true for any positive $T < \infty$ :

$$\begin{aligned} \sup _{0<t<T}\int _\Omega |u_h(t)|^{q+1}\,dx\leqq \int _\Omega |u_0|^{q+1}\,dx \end{aligned}$$

(3.8)

and

$$\begin{aligned} \sup _{0<t<T}\int _\Omega |Du_h(t)|^p\,dx \leqq \int _\Omega |Du_0|^p\,dx. \end{aligned}$$

(3.9)

We next deduce the integral bounds of time derivative.

Lemma 3.4

(Time-derivative estimates) Let $v_h$ and $w_h$ be the approximate solutions of (1.1) defined by (3.4). Then, the time derivatives of approximating solutions $(\partial _t w_h)_{h>0}$ are bounded in $L^2 (\Omega _T)$ for any positive $T < \infty$:

$$\begin{aligned} \iint _{\Omega _T} |\partial _t w_h|^2\,dxdt \leqq c \Vert Du_0\Vert _{L^p(\Omega )}^p \end{aligned}$$

(3.10)

and, furthermore, if $q \geqq 1$ the time derivatives of approximating solutions $(\partial _t v_h)_{h>0}$ are bounded in $L^1 (\Omega _T)$ for any positive $T < \infty$,

$$\begin{aligned} \iint _{\Omega _T} |\partial _t v_h|\,dxdt \leqq c|\Omega |^{\frac{1}{q+1}}T^{\frac{1}{2}}\Vert u_0\Vert _{L^{q+1}(\Omega )}^{\frac{q-1}{2}}\Vert Du_0\Vert _{L^p(\Omega )}^{\frac{p}{2}} \end{aligned}$$

(3.11)

with a positive constant $c\equiv c(p,q)$.

Proof

As before, we choose a positive integer k to satisfy $t_{k-1} \leqq T < t_k$. By (2.5) in Lemma 2.3 with $\alpha = \frac{q+3}{2}$

$$\begin{aligned} \left| |u_i|^{\frac{q-1}{2}}u_i-|u_{i-1}|^{\frac{q-1}{2}}u_{i-1} \right| ^2 \leqq c\big (|u_i| + |u_{i-1}|\big )^{q - 1} |u_i - u_{i-1}|^2. \end{aligned}$$

(3.12)

We observe that by the very definition (3.4)$_3$, (3.12), and (3.7) from Lemma 3.2

$$\begin{aligned}&\iint _{\Omega _T} |\partial _tw_h|^2\,dx \\&\quad =\sum _{i=1}^{k-1} \int _{t_{i-1}}^{t_i} \int _\Omega |\partial _tw_h|^2\,dx+\int _{t_{k-1}}^T\int _\Omega |\partial _tw_h|^2\,dx \\&\quad =\sum _{i=1}^{k-1}h\int _\Omega \left| \frac{|u_i|^{\frac{q-1}{2}}u_i-|u_{i-1}|^{\frac{q-1}{2}}u_{i-1}}{h} \right| ^2\,dx\\&\qquad +\int _{t_{k-1}}^T\int _\Omega \left| \frac{|u_k|^{\frac{q-1}{2}}u_k-|u_{k-1}|^{\frac{q-1}{2}}u_{k-1}}{h} \right| ^2\,dx\\&\quad {\mathop {\leqq }\limits ^{(3.12)}} c\sum _{i=1}^{k-1} h \int _\Omega \big (|u_i| + |u_{i-1}|\big )^{q-1} \left| \frac{u_i-u_{i-1}}{h} \right| ^2\,dx + c\int _{t_{k-1}}^T\\&\qquad \int _\Omega \big (|u_k| + |u_{k-1}|\big )^{q-1} \left| \frac{u_k-u_{k-1}}{h} \right| ^2\,dx \\&\quad =c\iint _{\Omega _T} \big (|{\bar{u}}_h (x,t)| + |{\bar{u}}_h (x,t-h)|\big )^{q - 1} |\partial _tu_h(x,t)|^2\,dxdt \\&\quad {\mathop {\leqq }\limits ^{(3.7)}} c\Vert Du_0\Vert _{L^p(\Omega _T)}^p, \end{aligned}$$

which implies (3.10).

As we are considering the case $q \geqq 1$,  (2.5) in Lemma 2.3 with $\alpha = q+1$ implies

$$\begin{aligned} \left| |u_i|^{q-1}u_i-|u_{i-1}|^{q-1}u_{i-1} \right| \leqq c\big (|u_i| + |u_{i-1}|\big )^{q - 1} |u_i - u_{i-1}|, \end{aligned}$$

(3.13)

and this combined with Hölder’s inequality twice with pair of exponents $\left( 2, 2\right)$ and $\left( \frac{q+1}{q-1},\,\frac{q+1}{2}\right)$, and (3.6)–(3.7) in Lemma 3.2 yields that

$$\begin{aligned} \iint _{\Omega _T} |\partial _tv_h|\,dx&{\mathop {\leqq }\limits ^{(3.13)}} c\sum _{i=1}^{k-1} h\int _\Omega \big (|u_i| + |u_{i-1}|\big )^{q-1} \left| \frac{u_i-u_{i-1}}{h} \right| \,dx \\&\quad +c \int _{t_{k-1}}^T\int _\Omega \big (|u_k| + |u_{k-1}|\big )^{q-1} \left| \frac{u_k-u_{k-1}}{h} \right| \,dx \\&=c \iint _{\Omega _T} \Big (|{\bar{u}}_h(t)|+|{\bar{u}}_h(t-h)|\Big )^{q-1}|\partial _t u_h|\,dxdt \\&\leqq c\left( \iint _{\Omega _T}\Big (|{\bar{u}}_h(t)|+|{\bar{u}}_h(t-h)|\Big )^{q-1}\,dxdt \right) ^{\frac{1}{2}} \\&\quad \times \left( \iint _{\Omega _T}\Big (|{\bar{u}}_h(t)|+|{\bar{u}}_h(t-h)|\Big )^{q-1}|\partial _t u_h|^2\,dxdt \right) ^{\frac{1}{2}} \\&\leqq c|\Omega _T|^{\frac{1}{q+1}}\left( \iint _{\Omega _T}\Big (|{\bar{u}}_h(t)|+|{\bar{u}}_h(t-h)|\Big )^{q+1}\,dxdt \right) ^{\frac{q-1}{2(q+1)}}\\&\quad \times \left( \iint _{\Omega _T}\Big (|{\bar{u}}_h(t)|+|{\bar{u}}_h(t-h)|\Big )^{q-1}|\partial _t u_h|^2\,dxdt \right) ^{\frac{1}{2}} \\&{\mathop {\leqq }\limits ^{~(3.6),~(3.7)}} c|\Omega |^{\frac{1}{q+1}}\underbrace{T^{\frac{1}{q+1}+\frac{q-1}{2(q+1)}}}_{=T^{\frac{1}{2}}}\Vert u_0\Vert _{L^{q+1}(\Omega )}^{\frac{q-1}{2}}\Vert Du_0\Vert _{L^p(\Omega )}^{\frac{p}{2}}, \end{aligned}$$

finishing the proof of (3.11). $\square$

4 Convergence of Approximate Solutions

This section is devoted to deriving the convergence of approximate solution of (1.1) defined in (3.3)–(3.4) by means of the truncated function stated below. This scheme is necessary for the strong convergence of approximate solutions in the space-time Sobolev space; see Lemma 4.4. In the sequel, we assume the initial datum $u_0$ belongs to $W_0^{1,p}(\Omega ) \cap L^{q+1}(\Omega )$.

We introduce the following truncated function: For the solution $(u_i(x))_{i\in {\mathbb {N}}}$ to (3.1) and $\ell \in {\mathbb {N}}_{\geqq 2}$, let us define as

$$\begin{aligned} (u_i(x))_\pm ^{(\ell )}:=\min \left\{ \max \left\{ (u_i(x))_\pm ,\,\dfrac{1}{\ell }\right\} ,\,\ell \right\} , \end{aligned}$$

where $(u_i(x))_+:=\max \{u_i(x),0\}$ and $(u_i(x))_-:=(-u_i(x))_+$. Under the notation above, we define the piecewise constant function $({\bar{u}}_h)_\pm ^{(\ell )}$ by

$$\begin{aligned} ({\bar{u}}_h)_\pm ^{(\ell )}(x,t):=(u_i(x))_\pm ^{(\ell )},\quad \text {for}\,\, (x,t) \in \Omega \times (t_{i-1},t_i]\,\,\text {with}\,\,i=1,2,\ldots . \end{aligned}$$

Further, we define a function $(u_h)_{\pm }^{(\ell )}$ by linearly interpolating $(u_i(x))_{\pm }^{(\ell )}$ and $(u_{i-1}(x))_{\pm }^{(\ell )}$ on the interval $[t_{i-1},t_i]$ with $i=1,2\ldots$

$$\begin{aligned} (u_h)_{\pm }^{(\ell )}(x,t):=\dfrac{t-t_{i-1}}{h}(u_i(x))_{\pm }^{(\ell )}+\dfrac{t_i-t}{h}(u_{i-1}(x))_{\pm }^{(\ell )} \quad \text {for}\,\, (x,t) \in \Omega \times [t_{i-1},t_i].\nonumber \\ \end{aligned}$$

(4.1)

From the definition above,

$$\begin{aligned} (u_h)_{\pm }^{(\ell )}(x,t) \rightarrow \dfrac{t-t_{i-1}}{h}(u_i(x))_{\pm }+\dfrac{t_i-t}{h}(u_{i-1}(x))_{\pm }\quad \text {for} \,\,(x, t) \in \Omega \times [t_{i-1}, t_i] \end{aligned}$$

in the limit $\ell \rightarrow \infty$. On the time interval $t \in (t_{i-1}, t_i)$ with $i=1,2,\ldots$, the time derivative of $(u_h)_{\pm }^{(\ell )}$ is computed as

$$\begin{aligned} \partial _t(u_h)_{\pm }^{(\ell )}=\dfrac{(u_i(x))_{\pm }^{(\ell )}-(u_{i-1}(x))_{\pm }^{(\ell )}}{h}. \end{aligned}$$

4.1 Energy Estimates for the Truncated Solution

Firstly, we deduce certain energy estimate of the truncated solution defined in (4.1) above.

Lemma 4.1

Let $(u_h)_\pm ^{(\ell )}$ be the function defined by (4.1). Then there holds for any positive $T < \infty$

$$\begin{aligned} \iint _{\Omega _T} |\partial _t (u_h)_\pm ^{(\ell )}|^2\,dxdt \leqq c\,\ell ^{q-1}\Vert Du_0\Vert _{L^p(\Omega )}^p \end{aligned}$$

(4.2)

whenever $q \geqq 1$ and,

$$\begin{aligned} \min \left\{ \iint _{\Omega _T} |\partial _t (u_h)_\pm ^{(\ell )}|^2\,dxdt, \,\iint _{\Omega _T} |\partial _t (u_h)_\pm ^{(\ell )}|^{q+1}\,dxdt \right\} \leqq c\,\ell ^{1-q}\Vert Du_0\Vert _{L^p(\Omega )}^p\nonumber \\ \end{aligned}$$

(4.3)

whenever $0<q<1$, where $c \equiv c(p,q)$.

Proof

We shall follow almost the same argument as in [16, Lemma 4.1] although, for the sake of completeness, we shall nevertheless give the full proof. We confine the case $(u_h)_+^{(\ell )}$ only since the other case is similarly treated. As before, let a positive integer k to satisfy $t_{k-1} \leqq T<t_k$. To begin, we see by energy estimate (3.7) in Lemma 3.2 that

$$\begin{aligned} \sum \limits _{i=1}^k h\int _\Omega \big (|u_i| + |u_{i-1}|\big )^{q - 1}\bigg |\frac{u_i-u_{i-1}}{h} \bigg |^2 \,dx \leqq c \Vert Du_0\Vert _{L^p(\Omega )}^p \end{aligned}$$

(4.4)

for a constant $c\equiv c(p,q)$. In the sequel, set $[u_i \geqq L] \equiv \Omega \cap \{u_i \geqq L\}$ for short; other symbols are also abbreviated similarly. We shall distinguish between the cases $q \geqq 1$ and $0<q<1$.

In the first case $q \geqq 1$, for any $i=1,\ldots ,k$,

$$\begin{aligned}&\int _\Omega \big (|u_i| + |u_{i- 1}|\big )^{q - 1}\bigg |\frac{u_i-u_{i-1}}{h} \bigg |^2 \,dx\\&\quad \geqq \int _\Omega \big (|u_i| + |u_{i - 1}|\big )^{q - 1}\bigg |\frac{(u_i)_+^{(\ell )}-(u_{i-1})_+^{(\ell )}}{h} \bigg |^2 \,dx \\&\quad =\int _{[u_i>\frac{1}{\ell }]\,\cup \,[u_{i-1}>\frac{1}{\ell }]}(\ldots )\,dx+\int _{[u_i \leqq \frac{1}{\ell }]\,\cap \,[u_{i-1} \leqq \frac{1}{\ell }]}(\ldots )\,dx \\&\quad \geqq \int _{[u_i>\frac{1}{\ell }]\,\cup \,[u_{i-1}>\frac{1}{\ell }]}\left( \tfrac{1}{\ell }\right) ^{q-1}\Big |\tfrac{(u_i)_+^{(\ell )}-(u_{i-1})_+^{(\ell )}}{h} \Big |^2\,dx \\&\quad =\int _{[u_i>\frac{1}{\ell }]\,\cup \,[u_{i-1}>\frac{1}{\ell }]}\left( \tfrac{1}{\ell }\right) ^{q-1}\Big |\tfrac{(u_i)_+^{(\ell )}-(u_{i-1})_+^{(\ell )}}{h} \Big |^2\,dx \\&\qquad +\int _{[u_i \leqq \frac{1}{\ell }]\,\cap \,[u_{i-1} \leqq \frac{1}{\ell }]}\left( \tfrac{1}{\ell }\right) ^{q-1}\Big |\tfrac{(u_i)_+^{(\ell )}-(u_{i-1})_+^{(\ell )}}{h} \Big |^2\,dx\\&\quad =\left( \tfrac{1}{\ell }\right) ^{q-1}\int _\Omega \Big |\tfrac{(u_i)_+^{(\ell )}-(u_{i-1})_+^{(\ell )}}{h} \Big |^2\,dx \\&\quad =\left( \tfrac{1}{\ell }\right) ^{q-1}\int _\Omega \big |\partial _t(u_h)_+^{(\ell )}\big |^2\,dx, \end{aligned}$$

where, in the second line, we discarded the non-negative second term on the right-hand side and, further, in the fourth line, we used the fact that $\frac{(u_i)_+^{(\ell )}-(u_{i-1})_+^{(\ell )}}{h}=0$ on the set $[u_i \leqq \frac{1}{\ell }]\,\cap \,[u_{i-1} \leqq \frac{1}{\ell }]$. This together with (4.4) implies the first desired estimate (4.2).

Hereafter, it only remains to consider the latter case $0<q<1$. For this, we need the following quantitative estimate:

$$\begin{aligned}&\big (|u_i| + |u_{i-1}|\big )^{q - 1}\Big |\tfrac{u_i-u_{i-1}}{h} \Big |^2\nonumber \\&\quad \geqq 3^{q-1}\min \left\{ \ell ^{q-1}\Big |\tfrac{(u_i)_+^{(\ell )}-(u_{i-1})_+^{(\ell )}}{h} \Big |^2,\,\Big |\tfrac{(u_i)_+^{(\ell )}-(u_{i-1})_+^{(\ell )}}{h} \Big |^{q+1} \right\} . \end{aligned}$$

(4.5)

In order to verify (4.5), a distinction must be made among the cases in the following: Firstly, on the set $\Omega _1:=[u_i \leqq \frac{1}{\ell }] \cap [u_{i-1} \leqq \frac{1}{\ell }]$ or $\Omega _2:=[u_i> \ell ] \cap [u_{i-1} >\ell ]$, it holds $\frac{(u_i)_+^{(\ell )}-(u_{i-1})_+^{(\ell )}}{h} =0$ and thus, (4.5) is clearly valid. On the set $\Omega \setminus (\Omega _1 \cup \Omega _2)$, we distinguish among the four cases (I)–(IV).

Case (I)   $[u_i \leqq \frac{1}{\ell }] \cap [\frac{1}{\ell } <u_{i-1} \leqq \ell ]$: In this region, we further consider the two cases separately: In the case $u_i>0$, since $u_i-u_{i-1}<0$, we have

$$\begin{aligned} |u_i|+|u_{i-1}|=u_i+u_{i-1}&=(u_i-u_{i-1})+2u_{i-1} \\&\leqq 2u_{i-1} \leqq 2\ell <3\ell . \end{aligned}$$

In the remaining case $u_i \leqq 0$, we plainly get

$$\begin{aligned} |u_i|+|u_{i-1}|=-u_i+u_{i-1} \leqq |u_i-u_{i-1}|. \end{aligned}$$

Case (II)   $[u_i \leqq \frac{1}{\ell }] \cap [u_{i-1}>\ell ]$: Also, we distinguish between the two cases $u_i>0$ and $u_i \leqq 0$. We start with the first case. Since $u_{i-1}-u_i \geqq \ell -\frac{1}{\ell } \geqq 2-\frac{1}{2} \geqq \frac{1}{\ell }$, we find

$$\begin{aligned} |u_i|+|u_{i-1}|=(u_{i-1}-u_i)+2u_i&\leqq (u_{i-1}-u_i)+\frac{2}{\ell } \\&\leqq (u_{i-1}-u_i)+2(u_{i-1}-u_i) \\&=3(u_{i-1}-u_i) \\&\leqq 3|u_i-u_{i-1}|. \end{aligned}$$

In the latter case $u_i \leqq 0$, we have again,

$$\begin{aligned} |u_i|+|u_{i-1}|=-u_i+u_{i-1} \leqq |u_i-u_{i-1}|. \end{aligned}$$

Case (III)   $[\frac{1}{\ell }<u_i \leqq \ell ] \cap [\frac{1}{\ell }<u_{i-1}\leqq \ell ]$: In this region, it in turn holds that

$$\begin{aligned} |u_i|+|u_{i-1}|=u_i+u_{i-1} \leqq 2\ell . \end{aligned}$$

Case (IV)   $[\frac{1}{\ell }<u_i \leqq \ell ] \cap [u_{i-1}>\ell ]$: In this region, we further distinguish between the two cases $u_{i-1}-u_i \leqq \ell$ and $u_{i-1}-u_i >\ell$. In the first case $u_{i-1}-u_i \leqq \ell$

$$\begin{aligned} |u_i|+|u_{i-1}|=(u_{i-1}-u_i)+2u_i \leqq \ell +2\ell =3\ell . \end{aligned}$$

In the latter case $u_{i-1}-u_i >\ell$, we have

$$\begin{aligned} |u_i|+|u_{i-1}|=(u_{i-1}-u_i)+2u_i&\leqq (u_{i-1}-u_i)+2\ell \\&\leqq (u_{i-1}-u_i)+2(u_{i-1}-u_i) \\&=3(u_{i-1}-u_i) \\&=3|u_i-u_{i-1}|. \end{aligned}$$

By interchanging of the role of $u_i(x)$ and $u_{i-1}(x)$, we also gain the same estimates as Cases (I), (II), and (IV) in the regions $[\frac{1}{\ell }<u_i\leqq \ell ] \cap [u_{i-1} \leqq \frac{1}{\ell }]$, $[u_i>\ell ] \cap [u_{i-1} \leqq \frac{1}{\ell }]$ and $[u_i> \ell ] \cap [\frac{1}{\ell }<u_{i-1} \leqq \ell ]$, respectively. Collecting all cases, we finally arrive at

$$\begin{aligned} |u_i|+|u_{i-1}| \leqq 3\max \left\{ \ell , \,|u_i-u_{i-1}| \right\} \quad \mathrm {in} \quad \Omega \setminus (\Omega _1 \cup \Omega _2), \end{aligned}$$

that is,

$$\begin{aligned} \big (|u_i|+|u_{i-1}|\big )^{q-1} \geqq 3^{q-1}\min \left\{ \ell ^{q-1}, \,|u_i-u_{i-1}|^{q-1} \right\} \quad \mathrm {in} \quad \Omega \setminus (\Omega _1 \cup \Omega _2). \end{aligned}$$

This together with the fact that $h^{q-1} \geqq 1$ implies that

$$\begin{aligned}&\big (|u_m| + |u_{i - 1}|\big )^{q - 1}\Big |\tfrac{u_i-u_{i-1}}{h} \Big |^2\\&\quad \geqq 3^{q-1}\min \left\{ \ell ^{q-1}\left| \tfrac{u_i-u_{i-1}}{h}\right| ^2, \,h^{q-1}\left| \tfrac{u_i-u_{i-1}}{h}\right| ^{q+1} \right\} \\&\quad \geqq 3^{q-1}\min \left\{ \ell ^{q-1}\Big |\tfrac{(u_i)_+^{(\ell )}-(u_{i-1})_+^{(\ell )}}{h} \Big |^2,\,\Big |\tfrac{(u_i)_+^{(\ell )}-(u_{i-1})_+^{(\ell )}}{h} \Big |^{q+1} \right\} . \end{aligned}$$

holds true on $\Omega \setminus (\Omega _1 \cup \Omega _2)$. Thereby, we have established (4.5) in the whole region $\Omega$ and, the preceding estimates (4.4) and (4.5) yield  (4.3). This completes the proof of Lemma 4.1. $\square$

4.2 Convergence Results for the Truncated Solution

In the subsequent lemma, we state the convergence result on the truncated solution in (4.1).

Lemma 4.2

Let $(u_h)_\pm ^{(\ell )}$ be the truncated function defined by (4.1). Then there exists a subsequence $((u_h)_\pm ^{(\ell )})_{h>0}$ (which we still denote by $(u_h)_\pm ^{(\ell )}$) and limit functions $(\omega _\ell )_{\pm }$ such that for any positive $T < \infty$ and every $\gamma$ with $1\leqq \gamma < \frac{n+1}{n}$,

$$\begin{aligned} (u_h)_+^{(\ell )} \rightarrow (\omega _\ell )_{\pm }\quad \mathrm {strongly\,\,in}\quad L^\gamma (\Omega _T) \quad \text {as}\,\,\,h \searrow 0. \end{aligned}$$

Proof

We prove the implication for $((u_h)_+^{(\ell )})_h$ only, as the other case is similar. By (3.8) and (3.9) in Lemma 3.3, $\left( (u_h)_+^{(\ell )} \right) _{h > 0}$ and $\left( D(u_h)_+^{(\ell )}\right) _{h>0}$ are bounded in $L^\infty (0,T\,; L^{q+1} (\Omega ))$ and $L^\infty (0,T\,; L^p(\Omega ; {\mathbb {R}}^{n}))$, respectively. Furthermore, in view of Lemma 4.1, the time derivative $\left( \partial _t(u_h)_+^{(\ell )}\right) _{h>0}$ is bounded in $L^1(\Omega _T)$. According to the Rellich–Kondrachov compactness theorem, there exist a subsequence (which we still denote by $u_h$) and the limit function $(\omega _\ell )_+ \in L^\gamma (\Omega _T)$ depending on $\ell$, such that

$$\begin{aligned} (u_h)_+^{(\ell )} \rightarrow (\omega _\ell )_+\quad \text {strongly}\,\,\text {in}\,\,L^\gamma (\Omega _T)\quad \text {as} \,\,h \searrow 0 \end{aligned}$$

for every $\gamma \in \left[ 1, \frac{(n+1)}{(n+1)-1}\right)$. This finishes the proof. $\square$

The next lemma allows us to estimate the $L^\gamma$-norm of $(u_h)_\pm -(u_{h'})_\pm$.

Lemma 4.3

Let $(u_i)_{i \in {\mathbb {N}}}$ be the solution to (3.1). Then, for every $\ell \geqq 2$ and $\gamma \in [1, r)$, we have

$$\begin{aligned} \left| \Omega \cap \{(u_i)_+ \geqq \ell \} \right| \leqq \dfrac{c}{\ell ^r}\Vert Du_0\Vert _{L^p(\Omega )}^r \end{aligned}$$

(4.6)

and

$$\begin{aligned} \left\| (u_i)_+\right\| _{L^\gamma (\Omega \,\cap \, \{(u_i)_+ \geqq \,\ell \})} \leqq \dfrac{c}{\ell ^{\frac{r}{\gamma }-1}}\Vert Du_0\Vert _{L^p(\Omega )}^{\frac{r}{\gamma }} \end{aligned}$$

(4.7)

for a constant $c\equiv c(n,p,r,\Omega )$, where $r \in \left[ 1,\frac{np}{n-p}\right)$ whenever $1 \leqq p<n$ and $r \in [1,\infty )$ whenever $n \leqq p<\infty$.

Proof

With r defined above, the Sobolev-Gagliardo-Nirenberg inequality

$$\begin{aligned} \left( \int _\Omega |v(x)|^r\,dx\right) ^{\frac{1}{r}}\leqq c(n,p,r)|\Omega |^{\frac{1}{n}-\frac{1}{p}+\frac{1}{r}}\left( \int _\Omega |Dv(x)|^p\,dx\right) ^{\frac{1}{p}} \end{aligned}$$

holds true for $v \in W^{1,p}_0(\Omega )$. This together with (3.7) in Lemma 3.2 leads to

$$\begin{aligned} \ell ^{r}\left| \Omega \cap \{(u_i)_+ \geqq \ell \} \right| \leqq \int _\Omega (u_i)_+^r\,dx&\leqq \Vert u_i\Vert _{L^r(\Omega )}^{r} \leqq c(n,p,r,\Omega )\Vert Du_i\Vert _{L^p(\Omega )}^r \\&{\mathop {\leqq }\limits ^{(3.7)}} c(n,p,r,\Omega )\Vert Du_0\Vert _{L^p(\Omega )}^r, \end{aligned}$$

which in turn implies (4.6). The remaining estimate (4.7) now follows from the Hölder inequality and the first estimate (4.6). $\square$

We shall prove that $\left( (u_h)_\pm \right) _{h>0}$ is a Cauchy sequence in $L^\gamma (\Omega _T)$ for all $\gamma$ with $1 \leqq \gamma <\min \left\{ r,\frac{n+1}{n}\right\}$ in the subsequent lemma.

Lemma 4.4

Let $r \in \left[ 1,\frac{np}{n-p}\right)$ for $1 \leqq p<n$ and $r \in [1,\infty )$ for $n \leqq p<\infty$. Then, $\left( (u_h)_\pm \right) _{h>0}$ is a Cauchy sequence in $L^\gamma (\Omega _T)$ for any positive $T < \infty$ and all $\gamma$ with $1 \leqq \gamma <\min \left\{ r,\frac{n+1}{n}\right\}$.

Proof

We prove the statement for $\left( (u_h)_+\right) _{h>0}$ only, as the other case is similarly treated. With $\gamma$ being $1\leqq \gamma <\min \left\{ r,\frac{n+1}{n}\right\}$, we split into three terms:

$$\begin{aligned}&\left\| (u_h)_+-(u_{h'})_+\right\| _{L^\gamma (\Omega _T)} \nonumber \\&\quad \leqq \left\| (u_h)_+-(u_h)_+^{(\ell )}\right\| _{L^\gamma (\Omega _T)} +\left\| (u_h)_+^{(\ell )}-(u_{h'})_+^{(\ell )}\right\| _{L^\gamma (\Omega _T)} \nonumber \\&\qquad +\left\| (u_{h'})_+^{(\ell )}-(u_{h'})_+\right\| _{L^\gamma (\Omega _T)} \nonumber \\&\quad :={\mathbf {I}}_{h,\ell }+{\mathbf {I}}_{h,h',\ell }+{\mathbf {I}}_{h',\ell }, \end{aligned}$$

(4.8)

where the meanings of ${\mathbf {I}}_{h,\ell }$, ${\mathbf {I}}_{h,h',\ell }$, and ${\mathbf {I}}_{h',\ell }$ are clear from the context. Since

$$\begin{aligned}&(u_h)_+(x,t)-(u_h)_+^{(\ell )}(x,t) \\&\quad =\dfrac{t-t_{i-1}}{h}\left( (u_i)_+(x)-(u_i)_+^{(\ell )}(x) \right) +\dfrac{t_i-t}{h}\left( (u_{i-1})_+(x)-(u_{i-1})_+^{(\ell )}(x)\right) \end{aligned}$$

holds for $(x,t) \in \Omega \times (t_{i-1},t_i]$ and $i=1,\ldots , k$, the Minkowski inequality yields that

$$\begin{aligned} {\mathbf {I}}_{h,\ell }&\leqq \left( h \sum _{i = 1}^k \left\| (u_i)_+ - (u_i)_+^{(\ell )}\right\| _{L^\gamma (\Omega )}^\gamma \right) ^{\frac{1}{\gamma }} + \left( h \sum _{i = 1}^k \left\| (u_{i- 1})_+ - (u_{i - 1})_+^{(\ell )} \right\| _{L^\gamma (\Omega )}^\gamma \right) ^{\frac{1}{\gamma }}\\&=:{\mathbf {I}}_1+{\mathbf {I}}_2. \end{aligned}$$

We now estimate ${\mathbf {I}}_1$ since ${\mathbf {I}}_2$ is proved as well as ${\mathbf {I}}_1$. Since

$$\begin{aligned} (u_i)_+-(u_i)_+^{(\ell )}=0\quad \text {in}\quad \Omega \cap \left\{ \frac{1}{\ell }<(u_i)_+<\ell \right\} \end{aligned}$$

for $i = 1, \ldots , k$, we obtain from  (4.6)–(4.7) in Lemma 4.3 that

$$\begin{aligned}&\left\| (u_i)_+-(u_i)_+^{(\ell )}\right\| _{L^\gamma (\Omega )}\\&\quad \leqq \left\| (u_i)_+-(u_i)_+^{(\ell )}\right\| _{L^\gamma (\Omega \,\cap \, \{(u_i)_+ \leqq \frac{1}{\ell }\})}\\&\qquad +\left\| (u_i)_+-(u_i)_+^{(\ell )}\right\| _{L^\gamma (\Omega \,\cap \, \{(u_i)_+\geqq \ell \})} \\&\quad \leqq \left\| (u_i)_+\right\| _{L^\gamma (\Omega \,\cap \, \{(u_i)_+ \leqq \frac{1}{\ell }\})}+\Big \Vert \underbrace{(u_i)_+^{(\ell )}}_{\equiv \, \frac{1}{\ell }}\Big \Vert _{L^\gamma ( \Omega \,\cap \, \{(u_i)_+ \leqq \frac{1}{\ell }\})} \\&\qquad +\left\| (u_i)_+\right\| _{L^\gamma (\Omega \,\cap \, \{(u_i)_+\geqq \ell \})}+\Big \Vert \underbrace{(u_i)_+^{(\ell )}}_{\equiv \,\ell }\Big \Vert _{L^\gamma (\Omega \,\cap \, \{(u_i)_+\geqq \ell \})} \\&\quad \leqq \dfrac{1}{\ell } \left| \Omega \cap \left\{ (u_i)_+ \leqq \frac{1}{\ell }\right\} \right| ^{\frac{1}{\gamma }}+\dfrac{1}{\ell } \left| \Omega \cap \left\{ (u_i)_+ \leqq \frac{1}{\ell } \right\} \right| ^{\frac{1}{\gamma }}\\&\qquad +\left\| (u_i)_+\right\| _{L^\gamma (\Omega \,\cap \, \{(u_i)_+\geqq \ell \})}+\ell \left| \Omega \cap \left\{ (u_i)_+ \geqq \ell \right\} \right| ^{\frac{1}{\gamma }} \\&\quad {\mathop {\leqq }\limits ^{(4.6),(4.7)}} \dfrac{2}{\ell }\left| \Omega \right| ^{\frac{1}{\gamma }}+\frac{c}{\ell ^{\frac{r}{\gamma }-1}}\Vert Du_0\Vert _{L^p(\Omega )}^{\frac{r}{\gamma }}+\ell \left( \frac{c}{\ell ^r}\Vert Du_0\Vert _{L^p(\Omega )}^{r}\right) ^{\frac{1}{\gamma }} \\&\quad =c\left( \frac{1}{\ell }\left| \Omega \right| ^{\frac{1}{\gamma }}+\frac{\Vert Du_0\Vert _{L^p(\Omega )}^{\frac{r}{\gamma }}}{\ell ^{\frac{r}{\gamma }-1}}\right) . \end{aligned}$$

Hence,

$$\begin{aligned} {\mathbf {I}}_1\leqq cT^{\frac{1}{\gamma }}\left( \frac{1}{\ell }\left| \Omega \right| ^{\frac{1}{\gamma }}+\frac{\Vert Du_0\Vert _{L^p(\Omega )}^{\frac{r}{\gamma }}}{\ell ^{\frac{r}{\gamma }-1}}\right) \end{aligned}$$

that is,

$$\begin{aligned} {\mathbf {I}}_{h,\ell }\leqq cT^{\frac{1}{\gamma }}\left( \frac{1}{\ell }\left| \Omega \right| ^{\frac{1}{\gamma }}+\frac{\Vert Du_0\Vert _{L^p(\Omega )}^{\frac{r}{\gamma }}}{\ell ^{\frac{r}{\gamma }-1}}\right) =:\zeta (\ell ). \end{aligned}$$

Clearly, ${\mathbf {I}}_{h^\prime ,\ell }$ is estimated similarly as ${\mathbf {I}}_{h,\ell }$. Inserting the preceding estimates into (4.8) and employing Lemma 4.2, we arrive at

$$\begin{aligned} \limsup _{h,h'\searrow 0}\left\| (u_h)_+-(u_{h'})_+\right\| _{L^\gamma (\Omega _T)}&\leqq \limsup _{h\searrow 0} {\mathbf {I}}_{h,\ell }+\limsup _{h,h'\searrow 0} {\mathbf {I}}_{h,h',\ell }+\limsup _{h'\searrow 0} {\mathbf {I}}_{h',\ell } \\&\leqq 2\zeta (\ell ). \end{aligned}$$

Since by $\gamma <r$ $\lim \limits _{\ell \rightarrow \infty }\zeta (\ell )=0$, we finally pass to the limit $\ell \rightarrow \infty$ in the above display to get

$$\begin{aligned} \limsup _{h,h'\searrow 0}\left\| (u_h)_+-(u_{h'})_+\right\| _{L^\gamma (\Omega _T)}=0. \end{aligned}$$

Analogously, the same statement holds for $(u_h)_--(u_{h'})_-$, which finishes the proof. $\square$

4.3 Convergence of Approximate Solutions I

Here, we shall present the convergence of the approximate solutions $u_h$ and ${\bar{u}}_h$.

Lemma 4.5

(Convergence of approximate solutions I) Let $r \in \left[ 1,\frac{np}{n-p}\right)$ for $1 \leqq p<n$ and $r \in [1,\infty )$ for $n \leqq p<\infty$. Then, for any positive $T<\infty$ and all $\gamma \in \left[ 1, \min \{r,\frac{n+1}{n}\}\right)$, there exist a subsequence $\left( u_h\right) _{h > 0}$, $\left( {{{\bar{u}}}}_h\right) _{h > 0}$ (still denoted by the same notation) and a limit function $u \in L^\gamma (\Omega _T)$ such that

$$\begin{aligned} u_h,\,{\bar{u}}_h\rightarrow u \quad \text {a.e.}\,\,\text {in}\,\,\Omega _T \quad \text {as}\,\,\,h \searrow 0. \end{aligned}$$

(4.9)

Proof

Since $u_h=(u_h)_+-(u_h)_-$ Lemma 4.4 implies that there exists a limit function $u \in L^\gamma (\Omega _T)$ such that

$$\begin{aligned} u_h \rightarrow u \quad \text {strongly}\,\,\text {in}\,\, L^\gamma (\Omega _T)\quad \text {as}\,\,\,h \searrow 0 \end{aligned}$$

(4.10)

whenever $\gamma \in \left[ 1, \min \{r,\frac{n+1}{n}\}\right)$. Hence, passing to a subsequence (also labeled with h) we in turn obtain that $u_h \rightarrow u$ almost everywhere in $\Omega _T$, that is the first statement of (4.9).

In order to prove the second statement of (4.9), we employ Lemma 2.4 and the truncated argument, as described in Sect. 4.2. For this, we shall handle the case $({\bar{u}}_h)_+$ only, as the other case is similar. Again, by the Minkowski inequality, we have

$$\begin{aligned}&\left\| ({\bar{u}}_h)_+-(u_h)_+ \right\| _{L^1(\Omega _T)}\nonumber \\&\quad \leqq \left\| ({\bar{u}}_h)_+-({\bar{u}}_h)_+^{(\ell )} \right\| _{L^1(\Omega _T)}+\left\| ({\bar{u}}_h)_+^{(\ell )}-(u_h)_+^{(\ell )} \right\| _{L^1(\Omega _T)}\nonumber \\&\qquad +\left\| (u_h)_+^{(\ell )}-(u_h)_+ \right\| _{L^1(\Omega _T)} \nonumber \\&\quad =:{\mathbf {I}}_{h,\ell }+\mathbf {II}_{h,\ell }+\mathbf {III}_{h,\ell }. \end{aligned}$$

(4.11)

As described in the proof of Lemma 4.4, we infer that

$$\begin{aligned} {\mathbf {I}}_{h,\ell }, \,\,\mathbf {III}_{h,\ell } \leqq cT^{\frac{1}{\gamma }}\left( \frac{1}{\ell }\left| \Omega \right| ^{\frac{1}{\gamma }}+\frac{\Vert Du_0\Vert _{L^p(\Omega )}^{\frac{r}{\gamma }}}{\ell ^{\frac{r}{\gamma }-1}}\right) =:\zeta (\ell ), \end{aligned}$$

(4.12)

where $\zeta (\ell ) \rightarrow 0$ as $\ell \rightarrow \infty$. Since

$$\begin{aligned} \left| ({\bar{u}}_h)_+^{(\ell )}-(u_h)_+^{(\ell )} \right| \leqq h \left| \partial _t (u_h)_+^{(\ell )} \right| \end{aligned}$$

we have

$$\begin{aligned} \mathbf {II}_{h,\ell }=\left\| ({\bar{u}}_h)_+^{(\ell )}-(u_h)_+^{(\ell )} \right\| _{L^1\Omega _T)} \leqq h \left\| \partial _t (u_h)_+^{(\ell )} \right\| _{L^1(\Omega _T)}, \end{aligned}$$

which in conjunction with (4.2) and  (4.3) in Lemma 4.1 implies

$$\begin{aligned} \lim _{h \searrow 0}\mathbf {II}_{h,\ell }=0. \end{aligned}$$

Collecting the preceding estimates yields that

$$\begin{aligned} \limsup _{h \searrow 0}\left\| ({\bar{u}}_h)_+-(u_h)_+ \right\| _{L^1(\Omega _T)}&\leqq \limsup _{h \searrow 0}{\mathbf {I}}_{h,\ell }+\limsup _{h \searrow 0}\mathbf {II}_{h,\ell }+\limsup _{h \searrow 0}\mathbf {III}_{h,\ell } \\&\leqq 2\zeta (\ell ) \end{aligned}$$

and, subsequently, taking the limit $\ell \rightarrow \infty$ gives

$$\begin{aligned} \lim _{h \searrow 0}\left\| ({\bar{u}}_h)_+-(u_h)_+ \right\| _{L^1(\Omega _T)}=0. \end{aligned}$$

(4.13)

Furthermore, (4.10) and (4.13) imply that

$$\begin{aligned} \Vert ({\bar{u}}_h)_+ - u_+\Vert _{L^1 (\Omega _T)} \leqq \Vert ({\bar{u}}_h)_+ - (u_h)_+\Vert _{L^1 (\Omega _T)} + \Vert (u_h)_+ - u_+\Vert _{L^1 (\Omega _T)} \rightarrow 0 \end{aligned}$$

as $h \searrow 0$. Hence, passing to a further subsequence (also labeled with h), we get $({\bar{u}}_h)_+ \rightarrow u_+$ almost everywhere in $\Omega _T$ in the limit $h \searrow 0$. In an analogous way as done above, the almost convergence $({\bar{u}}_h)_- \rightarrow u_-$ in $\Omega _T$ is verified. Hence, the second assertion of (4.9) is concluded, finishing the proof. $\square$

4.4 Convergence of Approximate Solutions II

One of the main tools in this paper is the convergence results summarized in the following Lemmas 4.6, 4.7 and 4.8 .

Lemma 4.6

Let $u_h$, ${\bar{u}}_h$, ${\bar{v}}_h$, $v_h$, $w_h$, and ${\bar{w}}_h$ be the approximate solutions of (1.1) defined by (3.3)–(3.4). Then there exist subsequences of them (denoted by the same symbol unless otherwise stated) and a limit function $u \in L^\infty \left( 0,\infty \,;L^{q+1}(\Omega ) \cap W^{1,p}(\Omega )\right)$ such that, the following convergences hold true for any positive $T < \infty$:

$$\begin{aligned} ({\bar{u}}_h, D{\bar{u}}_h) \rightarrow (u, Du)\quad&\text {weakly}\,\,\text {in}\,\,L^p(0, \infty \,; L^p(\Omega \,; {\mathbb {R}}^{n+1})), \end{aligned}$$

(4.14)

$$\begin{aligned} {\bar{u}}_h \rightarrow u \quad \quad \quad \,\,\,&\text {weakly}^*\,\,\text {in}\,\,L^\infty (0,\infty \,; L^{q+1}(\Omega )), \end{aligned}$$

(4.15)

$$\begin{aligned} ({\bar{u}}_h, D{\bar{u}}_h) \rightarrow (u, Du)\quad&\text {weakly}^*\,\,\text {in}\,\,L^\infty (0, \infty \,; L^p(\Omega \,; {\mathbb {R}}^{n+1})), \end{aligned}$$

(4.16)

$$\begin{aligned} {\bar{w}}_h, \,w_h \rightarrow |u|^{\frac{q - 1}{2}} u \quad&\text {strongly}\,\,\text {in}\,\,L^\alpha (\Omega _T), \quad \forall \alpha \in [1,2), \end{aligned}$$

(4.17)

$$\begin{aligned} {\bar{v}}_h,\,v_h \rightarrow |u|^{q-1}u \quad&\text {strongly}\,\,\text {in}\,\,L^\beta (\Omega _T),\quad \forall \beta \in \left[ 1,\tfrac{q+1}{q}\right) \end{aligned}$$

(4.18)

as $h\searrow 0$.

Proof

The proof is based on the energy estimates in Lemmata 3.2, 3.3, and 3.4 and we split the proof into several steps.

Step 1: The proof of (4.14). The Poincaré inequality and estimate (3.6) in Lemma 3.2 imply that $({\bar{u}}_h, D{\bar{u}}_h)_{h>0}$ is uniformly bounded in $L^p(0, \infty \,; L^{p}(\Omega \,; {\mathbb {R}}^{n+1}))$. Therefore, there exist a (not re-labeled) subsequence and a function $u \in L^p(0, \infty \,; L^{p}(\Omega \,; {\mathbb {R}}^{n+1}))$ such that

$$\begin{aligned} \left( {\bar{u}}_h ,D{\bar{u}}_h\right) \rightarrow \left( u, Du\right) \quad \text {weakly}\,\,\text {in}\,\,L^p(0, \infty \,;L^{p}(\Omega \,; {\mathbb {R}}^{n+1})), \end{aligned}$$

proving the claim (4.14), where, as usual, we use the definition of weak derivatives and by the diagonal argument, this limit function u is equal to that obtained in Lemma  4.5.

Step 2: The proof of (4.15)–(4.16). From (3.6) and (3.7) in Lemma 3.2, we infer that $({\bar{u}}_h)_{h>0}$ is uniformly bounded (with respect to h) in $L^{\infty }(0,\infty \,; L^{q+1}(\Omega ))$ and so $({\bar{u}}_h, D{\bar{u}}_h)_{h>0}$ is in $L^\infty (0, \infty \,;L^{p}(\Omega \,; {\mathbb {R}}^{n+1}))$. Then after passing a subsequence $({\bar{u}}_h)$ (denoted by the same symbol), there exists a limit function v satisfying

$$\begin{aligned} v \in L^\infty (0, \infty \,; L^{q + 1} (\Omega )), \quad (v,Dv) \in L^{\infty }(0,\infty \,; L^{p}(\Omega \,;{\mathbb {R}}^{n+1})) \end{aligned}$$

such that

$$\begin{aligned} {\left\{ \begin{array}{ll} {\bar{u}}_h \rightarrow v &{}\text {weakly}^*\,\,\text {in}\,\,L^\infty (0,\infty \,; L^{q+1}(\Omega )),\\ \left( {\bar{u}}_h, D{\bar{u}}_h\right) \rightarrow \left( v, Dv\right) &{}\text {weakly}^*\,\,\text {in}\,\,L^\infty (0,\infty \,; L^p(\Omega \,; {\mathbb {R}}^{n+1})). \end{array}\right. } \end{aligned}$$

This together with (4.14) in turn implies the implications (4.15)–(4.16).

Step 3: The proof of (4.17). We further divide this step into two steps:

At first, we shall prove the strong convergence ${\bar{w}}_h\equiv |{\bar{u}}_{h}|^{\frac{q-1}{2}}{\bar{u}}_{h}$ to $|u|^{\frac{q-1}{2}}u$ in $L^\alpha (\Omega _T)$ for all $\alpha \in [1, 2)$. Since by the energy estimate (3.6) in Lemma 3.2, $\left( {\bar{w}}_h\right) _{h > 0}$ is bounded in $L^2 (\Omega _T)$ and, by  (4.9) in Lemma 4.5, ${\bar{w}}_h - |u|^{\frac{q - 1}{2}} u \rightarrow 0$ almost everywhere in $\Omega _T$, where we use algebraic inequality (2.5) in Lemma 2.3 with $\alpha =\frac{q+3}{2}$. This allows us to use (2.8) in Lemma 2.4 to get

$$\begin{aligned} {\bar{w}}_h \rightarrow |u|^{\frac{q-1}{2}}u\quad \text {strongly}\,\,\text {in}\,\, L^\alpha (\Omega _T), \end{aligned}$$

(4.19)

which proves the first statement of (4.17).

Next, the difference between ${\bar{w}}_h$ and w is estimated as

$$\begin{aligned} \left| w_h-{\bar{w}}_h\right| \leqq h\left| \partial _t w_h\right| \quad \text {for}\quad (x,t) \in \Omega _T \end{aligned}$$

and thus, the energy estimate (3.10) in Lemma 3.4 gives

$$\begin{aligned} \left\| w_h-{\bar{w}}_h \right\| _{L^2(\Omega _T)}&\leqq h \left\| \partial _t w_h\right\| _{L^2(\Omega _T)} \leqq ch\Vert Du_0\Vert _{L^p(\Omega )}^{\frac{p}{2}}, \end{aligned}$$

which converges to zero as $h \searrow 0$. This together with (4.19) finishes the proof of (4.17).

Step 4: The proof of (4.18). Now we proceed exactly same as in Step 2 and split this stage into two steps.

We again use the energy estimate (3.6) in Lemma 3.2 to see that $\left( {\bar{v}}_h\right) _{h > 0}$ is bounded in $L^{\frac{q+1}{q}} (\Omega _T)$. From (4.9) in Lemma 4.5, as $h \searrow 0$, ${\bar{v}}_h-|u|^{q - 1} u \rightarrow 0$ almost everywhere in $\Omega _T$ follows, where we use algebraic inequality (2.5) in Lemma 2.3 with $\alpha =q+1$. Hence, with the help of (2.8) in Lemma 2.4, we gain

$$\begin{aligned} {{{\bar{v}}}}_ h \rightarrow |u|^{q - 1} u\quad \text {strongly}\,\,\text {in}\,\,\, L^\beta (\Omega _T),\quad \forall \beta \in \left[ 1,\tfrac{q+1}{q}\right) \end{aligned}$$

and therefore the first statement of (4.18) is actually verified.

To conclude the proof, we shall consider the difference between $v_h$ and $|u|^{q-1}u$. For this, we now distinguish two cases between $q \geqq 1$ and the opposite case $0<q<1$. In the first case $q\geqq 1$, since

$$\begin{aligned} \left| |{\bar{u}}_h|^{q-1}{\bar{u}}_h-v_h\right| =\left| {\bar{v}}_h-v_h\right| \leqq h \left| \partial _t v_h \right| \end{aligned}$$

the energy estimate (3.11) in Lemma 3.4 in turn gives

$$\begin{aligned} \left\| {\bar{v}}_h-v_h\right\| _{L^1(\Omega _T)}&\leqq h\left\| \partial _t v_h \right\| _{L^1(\Omega _T)} \nonumber \\&\leqq chT^{\frac{1}{2}}\left| \Omega \right| ^{\frac{1}{q+1}}\Vert u_0\Vert _{L^{q+1}(\Omega )}^{\frac{q-1}{2}}\Vert Du_0\Vert _{L^p(\Omega )}^{\frac{p}{2}}. \end{aligned}$$

(4.20)

In the latter case $0<q<1$, in view of (2.5) in Lemma 2.3$|v_h - {{{\bar{v}}}}_h|$ is estimated as

$$\begin{aligned} |v_h - {{{\bar{v}}}}_h|&\leqq c \left( |{\bar{u}}_h (t)|+ |{\bar{u}}_h (t - h)|\right) ^{q - 1}\left| {\bar{u}}_h(t)-{\bar{u}}_h (t-h)\right| \\&\leqq c \left| {\bar{u}}_h (t) - {\bar{u}}_h (t - h)\right| ^q\\&= c h^q \left( |{\bar{u}}_h (t)| + |{\bar{u}}_h (t - h)|\right) ^{\frac{q (1- q)}{2}}\left( |{\bar{u}}_h (t)| + |{\bar{u}}_h (t - h)|\right) ^{\frac{q (q - 1)}{2}}|\partial _t u_h|^q \end{aligned}$$

and, subsequently, exploiting Hölder’s inequality twice with pair of exponents

$$\begin{aligned} \left( \tfrac{2}{q}, \tfrac{2}{2 - q}\right) \quad \text {and} \quad \left( \tfrac{(2-q)(q+1)}{q(1-q)},\,\tfrac{(2-q)(q+1)}{2}\right) \end{aligned}$$

and using the energy estimates (3.6)–(3.7) in Lemma 3.2, we infer that

$$\begin{aligned} \iint _{\Omega _T}|v_h - {\bar{v}}_h|\,dxdt&\leqq h^q\left( \,\iint _{\Omega _T}\left( |{\bar{u}}_h (t)|+ |{\bar{u}}_h (t - h)|\right) ^{q - 1}|\partial _t u_h|^2\,dxdt\right) ^{\frac{q}{2}} \nonumber \\&\quad \times \left( \,\iint _{\Omega _T}\left( |{\bar{u}}_h (t)|+ |{\bar{u}}_h (t - h)|\right) ^{\frac{q (1 - q)}{2- q}}\,dxdt\right) ^{\frac{2 - q}{2}} \nonumber \\&\leqq h^q\left( \,\iint _{\Omega _T}\left( |{\bar{u}}_h (t)|+ |{\bar{u}}_h (t - h)|\right) ^{q - 1}|\partial _t u_h|^2\,dxdt\right) ^{\frac{q}{2}} \nonumber \\&\quad \times \left( \,\iint _{\Omega _T}\left( |{\bar{u}}_h (t)|+ |{\bar{u}}_h (t - h)|\right) ^{q+1}\,dxdt\right) ^{\frac{q(1-q)}{2(q+1)}}\left| \Omega _T\right| ^{\frac{1}{q+1}} \nonumber \\&\leqq ch^q T^{\frac{q(1-q)+2}{2(q+1)}}\left| \Omega \right| ^{\frac{1}{q+1}} \Vert u_0\Vert _{L^{q + 1} (\Omega )}^{\frac{q(1-q)}{2}}||Du_0||_{L^p(\Omega )}^{{pq}/{2}}. \end{aligned}$$

(4.21)

Passing to the limit $h \searrow 0$ in the preceding estimates (4.20)–(4.21) together with (4.18)$_1$, we conclude that the convergence $v_h \rightarrow |u|^{q - 1} u$ in $L^1 (\Omega _T)$, which in turn implies $v_ h \rightarrow |u|^{q - 1} u$ almost everywhere in $\Omega _T$.

This time,  (3.8) in Lemma 3.3 tells us that $\left( v_h\right) _{h>0}$ is bounded in $L^{\frac{q + 1}{q}} (\Omega _T)$. We therefore apply again  (2.8) in Lemma 2.4 to conclude the second statement of (4.18).

Finally, the proof of Lemma 4.6 is complete. $\square$

From Lemma 2.4, we easily deduce the following convergence, used later in Sect. 5.

Lemma 4.7

For every test function $\varphi \in C^\infty _0(\Omega _\infty )$, the following convergences hold true:

$$\begin{aligned} -\iint _{\Omega _\infty } w_h \, \partial _t \varphi \,dxdt \rightarrow -\iint _{\Omega _\infty } |u|^{\frac{q-1}{2}}u \,\partial _t \varphi \,dxdt \end{aligned}$$

(4.22)

and

$$\begin{aligned} -\iint _{\Omega _\infty } v_h \, \partial _t \varphi \,dxdt \rightarrow -\iint _{\Omega _\infty } |u|^{q-1}u \,\partial _t \varphi \,dxdt \end{aligned}$$

(4.23)

as $h \searrow 0$.

Proof

We confine the proof of (4.23) only, as the other case (4.22) is similarly treated. We may suppose the support of $\varphi$ is contained in $\Omega _T$ for some positive $T < \infty$. Since by (3.8) in Lemma 3.3$\left( v_h\right) _{h>0}$ is bounded in $L^{\frac{q+1}{q}}(\Omega _T)$ and by  (4.18) in Lemma 4.6

$$\begin{aligned} v_h\,\partial _t\varphi - |u|^{q-1}u\,\partial _t\varphi \rightarrow 0\quad \text {a.e.}\,\,\text {in}\,\,\Omega _T. \end{aligned}$$

Thus, (2.8) in Lemma 2.4 yields

$$\begin{aligned} \iint _{\Omega _T} v_h \, \partial _t \varphi \,dxdt \rightarrow \iint _{\Omega _T} |u|^{q-1}u \, \partial _t \varphi \,dxdt \end{aligned}$$

as $h \searrow 0$, and therefore, the proof is complete. $\square$

In the subsequent lemma, we state the strong convergence of spatial gradient, which is the main output in the paper.

Lemma 4.8

Let ${\bar{u}}_h$ and u be as in Lemma 4.6. Then,

$$\begin{aligned} D{\bar{u}}_h \rightarrow Du \quad \text {strongly}\,\,\text {in}\,\,L^p(\Omega _T)\quad \text {as}\,\,h \searrow 0 \end{aligned}$$

holds true for any positive $T < \infty$.

Proof

The $L^1(\Omega _T)$-boundedness of $\partial _t v_h$ is an essential requirement for the p-harmonic map type approximation argument given in [10, Theorem 2.1] in the case $p \geqq 2$ (see also [27, Lemma 4.2]) and [24, Theorem 3.2] in the case $1<p<2$, where the time-space integral boundedness of the time derivative is required for the strong convergence of gradients. Here, we shall prove the strong convergence of gradients by the combination of compactness argument for the difference quotient with respect to time [4, Lemma 2.1] and the exponential time mollification technique. The strong convergence of gradients is the new result in the weak compactness method based on the approximation of Rothe type.

Before proceeding, let us give some definitions: For $h \ne 0$, we denote the differential quotient of a function v with respect to time by

$$\begin{aligned} \Delta _hv(t):=\dfrac{v(t+h)-v(t)}{h}. \end{aligned}$$

By this notation, $\partial _tv_h$ is written as

$$\begin{aligned} \partial _t v_h=\Delta _{-h}\left( |{\bar{u}}_h|^{q-1}{\bar{u}}_h\right) . \end{aligned}$$

Thus, the weak formulation (3.5) is rewritten as

$$\begin{aligned} \iint _{\Omega _T} \left[ \Delta _{-h}\left( |{\bar{u}}_h|^{q-1}{\bar{u}}_h\right) \,\varphi + |D{\bar{u}}_h|^{p-2}D{\bar{u}}_h\cdot D\varphi \right] \,dxdt=0 \end{aligned}$$

and testing $\varphi ={\bar{u}}_h-[u]_\varepsilon$ in the above formula, we have

$$\begin{aligned}&-\iint _{\Omega _T} \Delta _{-h}\left( |{\bar{u}}_h|^{q-1}{\bar{u}}_h\right) \,\left( {\bar{u}}_h-[u]_\varepsilon \right) \,dxdt\nonumber \\&\quad =\iint _{\Omega _T}|D{\bar{u}}_h|^{p-2}D{\bar{u}}_h\cdot D\left( {\bar{u}}_h-[u]_\varepsilon \right) \,dxdt. \end{aligned}$$

(4.24)

In the subsequent lemma, we abbreviate ${\bar{u}}_h\equiv {\bar{u}}_h(t)$ and $[u]_\varepsilon \equiv [u]_\varepsilon (t)$. Here, $[u]_\varepsilon$ is defined by (2.1) with $h=\varepsilon$ and $v_0=u_0$. In this setting, we deduce the following finite integration by parts formula (see also [4, Lemma 2.10]).

Lemma 4.9

Assume the initial datum $u_0 \in L^{q+1}(\Omega ) \cap W^{1,p}_0(\Omega )$. Let ${\bar{u}}_h$ be as in (3.3) and u be as in Lemma 4.6, where u is extended to negative times by $u(t):=u_0$ for $t<0$. Then, the following quantitative estimate holds true:

$$\begin{aligned}&-\iint _{\Omega _T}\Delta _{-h}\left( |{\bar{u}}_h|^{q-1}{\bar{u}}_h\right) \left( {\bar{u}}_h-[u]_\varepsilon \right) \,dxdt\nonumber \\&\quad \leqq \iint _{\Omega _T}\Delta _h[u]_\varepsilon \left( |[u]_\varepsilon |^{q-1}[u]_\varepsilon -|{\bar{u}}_h|^{q-1}{\bar{u}}_h\right) \,dxdt \nonumber \\&\qquad -\frac{1}{h}\iint _{\Omega \times (T-h,T)}{\mathbf {B}}\left[ {\bar{u}}_h,[u]_\varepsilon (t+h)\right] \,dxdt \nonumber \\&\qquad +\frac{1}{h}\iint _{\Omega \times (-h,0)}{\mathbf {B}}\left[ {\bar{u}}_h,[u]_\varepsilon \right] \,dxdt \nonumber \\&\qquad +\varvec{\Delta }^{(\varepsilon )}_1(h)+\varvec{\Delta }_2^{(\varepsilon )}(h), \end{aligned}$$

(4.25)

where the error terms are given by

$$\begin{aligned} \varvec{\Delta }^{(\varepsilon )}_1(h)&:=\frac{1}{h}\iint _{\Omega _T}{\mathbf {B}}\left[ [u]_\varepsilon ,\,[u]_{\varepsilon }(t+h)\right] \,dxdt, \\ \varvec{\Delta }^{(\varepsilon )}_2(h)&:=\iint _{\Omega \times (-h,0)}\Delta _h[u]_\varepsilon \Big (\left| [u]_\varepsilon (t+h)\right| ^{q-1}[u]_\varepsilon (t+h)-\left| {\bar{u}}_h\right| ^{q-1}{\bar{u}}_h \Big )\,dxdt. \end{aligned}$$

In particular, for any $\varepsilon >0$

$$\begin{aligned} \lim _{h \searrow 0}\varvec{\Delta }_1^{(\varepsilon )}(h)=\lim _{ h \searrow 0}\varvec{\Delta }_2^{(\varepsilon )}(h)=0. \end{aligned}$$

The proof of this lemma is postponed, and will be presented in Appendix B.

Since, by (2.4) in Lemma 2.2, ${\mathbf {B}}[\cdot , \cdot ]$ is non-negative, we cancel the second term in the right-hand side of (4.25) in Lemma 4.9 to get

$$\begin{aligned}&-\iint _{\Omega _T}\Delta _{-h}\left( |{\bar{u}}_h|^{q-1}{\bar{u}}_h\right) \left( {\bar{u}}_h-[u]_\varepsilon \right) \,dxdt \nonumber \\&\quad \leqq \iint _{\Omega _T}\Delta _h[u]_\varepsilon \left( |[u]_\varepsilon |^{q-1}[u]_\varepsilon -|{\bar{u}}_h|^{q-1}{\bar{u}}_h\right) \,dxdt\nonumber \\&\qquad +\frac{1}{h}\iint _{\Omega \times (-h,0)}{\mathbf {B}}\left[ {\bar{u}}_h,[u]_\varepsilon \right] \,dxdt \nonumber \\&\qquad +\varvec{\Delta }^{(\varepsilon )}_1(h)+\varvec{\Delta }_2^{(\varepsilon )}(h). \end{aligned}$$

(4.26)

For the first term of (4.26), we use the following convergences : By the integrability of u in Lemma 4.6 and Lemma 2.1-(v),

$$\begin{aligned} \Delta _h [u]_\varepsilon \rightarrow \partial _t [u]_\varepsilon \quad \text {in} \quad L^{q + 1} (\Omega _T) \quad \text {as} \quad h \searrow 0 \end{aligned}$$

for any positive $\varepsilon < 1$ and, $\left| {\bar{u}}_h\right| ^{q-1}{\bar{u}}_h \rightarrow |u|^{q-1}u$ almost everywhere on $\Omega _T$ by (4.9) in Lemma 4.5. These convergences together with Lemma 2.1-(iii) and (2.5) in Lemma 2.3 yield that

$$\begin{aligned}&\lim _{h \searrow 0}\iint _{\Omega _T}\Delta _h[u]_\varepsilon \left( |[u]_\varepsilon |^{q-1}[u]_\varepsilon -|{\bar{u}}_h|^{q-1}{\bar{u}}_h\right) \,dxdt \nonumber \\&\quad =\iint _{\Omega _T}\partial _t[u]_\varepsilon \left( |[u]_\varepsilon |^{q-1}[u]_\varepsilon -|u|^{q-1}u\right) \,dxdt \nonumber \\&\quad =\iint _{\Omega _T}\tfrac{1}{\varepsilon }\left( u-[u]_\varepsilon \right) \left( |[u]_\varepsilon |^{q-1}[u]_\varepsilon -|u|^{q-1}u\right) \,dxdt \leqq 0. \end{aligned}$$

(4.27)

The second term on the right-hand side of (4.26) is computed as

$$\begin{aligned} \frac{1}{h}\iint _{\Omega \times (-h,0)}{\mathbf {B}}\left[ {\bar{u}}_h,[u]_\varepsilon \right] \,dxdt= & {} \frac{1}{h}\iint _{\Omega \times (-h,0)}{\mathbf {B}}\left[ u_0,u_0 \right] \,dxdt\\= & {} \int _\Omega {\mathbf {B}}\left[ u_0,u_0 \right] \,dx=0 \end{aligned}$$

since ${\bar{u}}_h(t)=u_0$ and $[u]_\varepsilon (t)=u_0$ for $t \in (-h,0)$. Combining this with (4.26)–(4.27) and using formula (4.24), we infer that

$$\begin{aligned}&\limsup _{h\searrow 0}\iint _{\Omega _T} \left| D{\bar{u}}_h\right| ^{p-2}D{\bar{u}}_h\cdot D\left( {\bar{u}}_h-[u]_\varepsilon \right) \,dxdt \nonumber \\&\quad {\mathop {=}\limits ^{(4.24)}}\limsup _{h\searrow 0}\left( -\iint _{\Omega _T}\Delta _{-h}\left( |{\bar{u}}_h|^{q-1}{\bar{u}}_h\right) \left( {\bar{u}}_h-[u]_\varepsilon \right) \,dxdt\right) \leqq 0. \end{aligned}$$

(4.28)

On the other hand, it follows from (3.6) in Lemma 3.2 and Hölder’s inequality that

$$\begin{aligned}&\limsup _{h\searrow 0}\iint _{\Omega _T}\left| D{\bar{u}}_h\right| ^{p-2}D{\bar{u}}_h\cdot D\left( [u]_\varepsilon -u\right) \,dxdt \nonumber \\&\quad \leqq \limsup _{h\searrow 0}\left( \iint _{\Omega _T}\left| D{\bar{u}}_h\right| ^p\,dxdt\right) ^{\frac{p-1}{p}}\left( \iint _{\Omega _T}\left| D[u]_\varepsilon -Du\right| ^p\,dxdt \right) ^{\frac{1}{p}} \nonumber \\&\quad \leqq c(q)\left( \int _{\Omega }\left| u_0\right| ^{q+1}\,dx\right) ^{\frac{p-1}{p}}\left( \iint _{\Omega _T}\left| [Du]_\varepsilon -Du\right| ^p\,dxdt \right) ^{\frac{1}{p}}. \end{aligned}$$

(4.29)

Due to the weak convergence (4.14) in Lemma 4.6

$$\begin{aligned} \lim _{h \searrow 0}\iint _{\Omega _T}\left| Du\right| ^{p-2}Du\cdot D\left( u-{\bar{u}}_h\right) \,dxdt=0. \end{aligned}$$

(4.30)

At this point, we need to distinguish cases between $p \geqq 2$ and $1<p<2$. Note that all estimations (4.28)–(4.30) are valid for all $1<p<\infty$.

In the first case $p \geqq 2$, the estimations (4.28)–(4.30) together with (2.7) in Lemma 2.3 imply that

$$\begin{aligned} 0&\leqq c(p) \limsup _{h \searrow 0}\iint _{\Omega _T}\left| D{\bar{u}}_h-Du\right| ^p\,dxdt \nonumber \\&\leqq \limsup _{h \searrow 0}\iint _{\Omega _T}\left( \left| D{\bar{u}}_h\right| ^{p-2}D{\bar{u}}_h-\left| Du\right| ^{p-2}Du\right) \cdot \left( D{\bar{u}}_h-Du\right) \,dxdt \nonumber \\&\leqq \limsup _{h\searrow 0}\iint _{\Omega _T} \left| D{\bar{u}}_h\right| ^{p-2}D{\bar{u}}_h\cdot D\left( {\bar{u}}_h-[u]_\varepsilon \right) \,dxdt \nonumber \\&\quad +\limsup _{h\searrow 0}\iint _{\Omega _T}\left| D{\bar{u}}_h\right| ^{p-2}D{\bar{u}}_h\cdot D\left( [u]_\varepsilon -u\right) \,dxdt \nonumber \\&\quad +\limsup _{h \searrow 0}\iint _{\Omega _T}\left| Du\right| ^{p-2}Du\cdot D\left( u-{\bar{u}}_h\right) \,dxdt \nonumber \\&\leqq c(q)\left( \int _{\Omega }\left| u_0\right| ^{q+1}\,dx\right) ^{\frac{p-1}{p}}\left( \iint _{\Omega _T}\left| [Du]_\varepsilon -Du\right| ^p\,dxdt \right) ^{\frac{1}{p}}. \end{aligned}$$

(4.31)

Since $[Du]_\varepsilon \rightarrow Du$ in $L^p(\Omega _T)$ as $\varepsilon \searrow 0$ by Lemma 2.1-(ii), passing to the limit as $\varepsilon \searrow 0$ in the above display, we get

$$\begin{aligned} \limsup _{h \searrow 0}\iint _{\Omega _T}\left| D{\bar{u}}_h-Du\right| ^p\,dxdt=0, \end{aligned}$$

proving the claim in the first case $p \geqq 2$.

We are considering the latter case $1<p<2$, Hölder’s inequality, inequality (2.6) in Lemma 2.3 and estimate (3.6) in Lemma 3.2 yield

$$\begin{aligned}&\iint _{\Omega _T}\left| D{\bar{u}}_h-Du\right| ^p\,dxdt \\&\quad \leqq \left( \iint _{\Omega _T}\left| D{\bar{u}}_h-Du\right| ^2\left( |D{\bar{u}}_h|+|Du|\right) ^{p-2}\,dxdt\right) ^{\frac{p}{2}} \\&\qquad \times \left( \iint _{\Omega _T}\left( |D{\bar{u}}_h|+|Du|\right) ^{p}\,dxdt\right) ^{\frac{2-p}{2}} \\&\quad \leqq c\left( \iint _{\Omega _T}\left( \left| D{\bar{u}}_h\right| ^{p-2}D{\bar{u}}_h-\left| Du\right| ^{p-2}Du\right) \cdot \left( D{\bar{u}}_h-Du\right) \,dxdt\right) ^{\frac{p}{2}}\\&\qquad \times \Big (\Vert u_0\Vert _{L^{q+1}(\Omega )}^{(q+1)\frac{2-p}{2}}+\Vert Du\Vert _{L^p(\Omega _T)}^{p\frac{2-p}{2}}\Big ). \end{aligned}$$

Taking into account the fact that (4.31)$_2$–(4.31)$_6$ do not require the condition $p\geqq 2$, we infer that

$$\begin{aligned} 0&\leqq \limsup _{h\searrow 0}\iint _{\Omega _T}\left| D{\bar{u}}_h-Du\right| ^p\,dxdt\\&\leqq c \Big (\Vert u_0\Vert _{L^{q+1}(\Omega )}^{(q+1)\frac{2-p}{2}}+\Vert Du\Vert _{L^p(\Omega _T)}^{p\frac{2-p}{2}}\Big ) \\&\quad \quad \times \left( \int _{\Omega }\left| u_0\right| ^{q+1}\,dx\right) ^{\frac{p-1}{2}}\left( \iint _{\Omega _T}\left| [Du]_\varepsilon -Du\right| ^p\,dxdt \right) ^{\frac{1}{2}} \end{aligned}$$

and letting $\varepsilon \searrow 0$ in the above display implies that

$$\begin{aligned} \limsup _{h \searrow 0}\iint _{\Omega _T}\left| D{\bar{u}}_h-Du\right| ^p\,dxdt=0 \end{aligned}$$

holds for $1<p <2$. This completes the proof of Lemma 4.8. $\square$

5 Proof of Theorem 1.1

We are now ready for Theorem 1.1 since we have all prerequisites at hand.

Proof of Theorem 1.1

We will prove that the limit function u obtained in Lemma 4.6 is a weak solution of  (1.1) in the sense of Definition 1. For this we split the proof into several steps. For the sake of readability, we describe the flow chart of whole steps:

$$\begin{aligned}&\text {(D1)}_1 \Longrightarrow \text {(D2)} \Longrightarrow \text {Regularity~(1.5)} \\ \Longrightarrow&\text {(D1)}_2 \Longrightarrow \text {(D3)} \Longrightarrow \text {Energy structures~(1.3)--(1.4)}. \end{aligned}$$

Step 1 : The condition (D1), first part. Let T be any finite positive number. From convergences (4.14)–(4.16) in Lemma 4.6, energy estimates (3.6)–(3.7) in lemma 3.2 and the lower semi-continuity of norms with respect to the weak and $^*$-weak convergence, we infer that the limit u obtained in Lemma 4.6 belongs to $L^p (0, \infty \,;W^{1,p} (\Omega )) \cap L^\infty (0, \infty \,;L^{q+1} (\Omega ))$. We will give the proof of the implication $u \in C([0,T]\,; L^{q+1}(\Omega ))$ in Step 4.

Step 2 : The condition (D2). By using (4.23) in Lemma 4.7 and Lemma 4.8, we pass to the limit as $h \searrow 0$ in the identity (3.5) and obtain

$$\begin{aligned} \iint _{\Omega _T}\Big (-|u|^{q-1}u \,\partial _t\varphi +|Du|^{p-2}Du\cdot D\varphi \Big )\,dxdt=0\qquad \forall \varphi \in C_0^\infty (\Omega _T), \end{aligned}$$

which ensures (2.9). Here, we notice that the class of test functions $C^\infty _0 (\Omega _T)$ is a subspace of that in (3.5). The function u also satisfies the following identity:

Lemma 5.1

Let u be obtained in Step 1. Then

$$\begin{aligned} \iint _{\Omega _T}\Big [-|u|^{q-1}u \,\partial _t\varphi +|Du|^{p-2}Du\cdot D\varphi \Big ]\,dxdt=\int _\Omega |u_0|^{q-1}u_0\varphi (0)\,dx. \end{aligned}$$

(5.1)

holds whenever $\varphi \in C^\infty \left( \Omega \times [0,T)\right)$ with support contained in $\Omega \times [0,T)$.

We will postpone the proof to Appendix C.

Step 3 : The regularity part. Since by (3.10) in Lemma 3.4$(\partial _tw_h)_{h>0}$ is bounded in $L^2(\Omega _T)$, there are a subsequence, still denoted by $\partial _tw_h$, and a limit function $\omega \in L^2(\Omega _T)$ such that

$$\begin{aligned} \partial _t w_h \rightarrow \omega \quad \text {weakly}\,\,\text {in}\,\,\,L^2(\Omega _T) \end{aligned}$$

(5.2)

as $h \searrow 0$. Using convergence (4.22) in Lemma 4.7 and passing to the limit as $h \searrow 0$ in the identity

$$\begin{aligned} \iint _{\Omega _T}\partial _t w_h\cdot \varphi \,dxdt =-\iint _{\Omega _T}w_h\,\partial _t \varphi \,dxdt, \quad \forall \varphi \in C^\infty _0(\Omega _T) \end{aligned}$$

give us that

$$\begin{aligned} \omega =\partial _t(|u|^{\frac{q-1}{2}}u)\quad \text {in}\,\,L^2(\Omega _T). \end{aligned}$$

(5.3)

Accordingly, it follows from (5.2) and (5.3) that

$$\begin{aligned} \partial _t w_h \rightarrow \partial _t\left( |u|^{\frac{q-1}{2}}u\right) \quad \text {weakly}\,\,\text {in}\,\,\,L^2(\Omega _T) \end{aligned}$$

and thus, the lower semi-continuity with respect to this weak convergence and (3.10) in Lemma 3.4 render that

$$\begin{aligned} \left\| \partial _t \left( |u|^{\frac{q-1}{2}}u\right) \right\| _{L^2(\Omega _T)}^2 \leqq c\Vert Du_0\Vert _{L^p(\Omega )}^p. \end{aligned}$$

(5.4)

Since the right-hand side is independent of T, sending $T \nearrow \infty$ concludes the desired regularity (1.5).

Step 4 : The condition (D1), second part (time continuity of solution). We start with taking $t_1,\,t_2 \in [0,T]$ arbitrarily. By Hölder’s inequality and (5.4), we infer that

$$\begin{aligned} \left\| |u|^{\frac{q-1}{2}}u(t_1)-|u|^{\frac{q-1}{2}}u(t_2)\right\| _{L^2(\Omega )}^2&=\int _\Omega \left| \int _{t_2}^{t_1} \partial _t \left( |u|^{\frac{q-1}{2}}u\right) \,dt\right| ^2\,dx \nonumber \\&\leqq |t_1-t_2|\int _{t_1}^{t_2}\int _\Omega \left| \partial _t \left( |u|^{\frac{q-1}{2}}u\right) \right| ^2\,dxdt \nonumber \\&{\mathop {\leqq }\limits ^{(5.4)}}c\Vert Du_0\Vert _{L^p(\Omega )}^p|t_1-t_2|. \end{aligned}$$

(5.5)

We now distinguish the two cases between $q \geqq 1$ and $0<q<1$. When $q \geqq 1$, we use (2.7) with $\alpha =\tfrac{q+3}{2}$ to infer that

$$\begin{aligned} \Big | u(t_1)-u(t_2)\Big |^{\frac{q+1}{2}} \leqq c \Big ||u|^{\frac{q-1}{2}}u(t_1)-|u|^{\frac{q-1}{2}}u(t_2)\Big | \end{aligned}$$

and thus, this together with (5.5) in turn implies

$$\begin{aligned} \left\| u(t_1)-u(t_2)\right\| _{L^{q+1}(\Omega )}^{q+1}&\leqq \left\| |u|^{\frac{q-1}{2}}u(t_1)-|u|^{\frac{q-1}{2}}u(t_2)\right\| _{L^2(\Omega )}^2\\&\leqq c\Vert Du_0\Vert _{L^p(\Omega )}^p|t_1-t_2|, \end{aligned}$$

showing our implication $u \in C([0,T]\,; L^{q+1}(\Omega ))$ in the case $q \geqq 1$. In the latter case $0<q<1$, we apply  (2.6) with $\alpha =\tfrac{q+3}{2}$ to get

$$\begin{aligned} \Big (|u(t_1)|+|u(t_2)|\Big )^{\frac{q-1}{2}}\left| u(t_1)-u(t_2)\right| \leqq c \Big ||u|^{\frac{q-1}{2}}u(t_1)-|u|^{\frac{q-1}{2}}u(t_2)\Big |, \end{aligned}$$

which together with (5.5) implies that

$$\begin{aligned} \int _\Omega \big (|u(t_1)|+|u(t_2)|\big )^{q-1}\left| u(t_1)-u(t_2)\right| ^2\,dx\leqq c\Vert Du_0\Vert _{L^p(\Omega )}^p|t_1-t_2|. \end{aligned}$$

Using this estimate and Hölder’s inequality with the exponent $\left( \tfrac{2}{q+1},\tfrac{2}{1-q}\right)$ (note that $0<q<1 \iff \frac{2}{q+1}>1$), we have

$$\begin{aligned} \int _\Omega \left| u(t_1)-u(t_2)\right| ^{q+1}\,dx&\leqq \left( \int _\Omega \big (|u(t_1)|+|u(t_2)|\big )^{q-1}\left| u(t_1)-u(t_2)\right| ^2\,dx\right) ^{\frac{q+1}{2}} \\&\quad \quad \times \left( \int _\Omega \big (|u(t_1)|+|u(t_2)|\big )^{q+1}\,dx\right) ^{\frac{1-q}{2}} \\&\leqq c\Vert Du_0\Vert _{L^p(\Omega )}^{p\frac{q+1}{2}}|t_1-t_2|^{\frac{q+1}{2}}\left( \int _\Omega \big (|u(t_1)|+|u(t_2)|\big )^{q+1}\,dx\right) ^{\frac{1-q}{2}}. \end{aligned}$$

Furthermore, the lower semicontinuous with respect to the $^*$-weak convergence (4.15) in Lemma 4.6 and estimate (3.6) in Lemma 3.2 imply that

$$\begin{aligned} \int _\Omega |u(t_i)|^{q+1}\,dx \leqq \liminf _{h \searrow 0}\int _{\Omega }|{\bar{u}}_h(t_i)|^{q+1}\,dx \leqq \int _\Omega |u_0|^{q+1}\,dx \end{aligned}$$

holds true for all $i=1,2$. Collecting the preceding estimates, we infer that, for all $t_1,\,t_2 \in [0,T]$

$$\begin{aligned} \int _\Omega \left| u(t_1)-u(t_2)\right| ^{q+1}\,dx \leqq c\Vert Du_0\Vert _{L^p(\Omega )}^{\frac{p(q+1)}{2}}\Vert u_0\Vert _{L^{q+1}(\Omega )}^{\frac{(q+1)(1-q)}{2}}|t_1-t_2|^{\frac{q+1}{2}}, \end{aligned}$$

which also proves the claim in the case $0<q<1$. Therefore, the condition (D1) is finally verified.

Step 5 : The condition (D3). In this step, we first show the validity of the initial condition (D3). Since we have proven that $u \in C\left( [0,T]\,; L^{q+1}(\Omega )\right)$ in the previous step, it is sufficient to show $u(0)=u_0$.

Let $\psi =\psi (x) \in C^\infty _0(\Omega )$ and, for any $\varepsilon >0$ set

$$\begin{aligned} \chi _\varepsilon (t):={\left\{ \begin{array}{ll} 1-\dfrac{t}{\varepsilon } &{}\quad \text {if}\quad 0 \leqq t \leqq \varepsilon , \\ 0 &{}\quad \text {if}\quad t >\varepsilon . \end{array}\right. } \end{aligned}$$

We test (5.1) with $\psi (x)\chi _\varepsilon (t)$ to get

Passing to the limit $\varepsilon \searrow 0$ in the above display in turn implies

Since $u \in C\left( [0,T]\,; L^{q+1}(\Omega ) \right)$ as showed in Step 3, we infer that

Since $\psi$ is arbitrary, we merge the previous two displays to gain

$$\begin{aligned} |u_0|^{q-1}u_0=|u(0)|^{q-1}u(0), \end{aligned}$$

which together with the monotonicity of $w \mapsto |w|^{q-1}w$ concludes that $u_0=u(0)$, and therefore, the initial condition (D3) is confirmed.

Next, by definition (3.3), ${\bar{u}}_h(t) \in W_0^{1,p}(\Omega )$ holds for almost every $t \in (0,T)$. Since $W_0^{1,p}(\Omega )$ is closed and convex in $W^{1,p}(\Omega )$, Mazur’s theorem and (4.15) in Lemma 4.6 imply that $u \in L^\infty (0,T\,;W_0^{1,p}(\Omega ))$, proving the the boundary condition described as in (D3).

Step 6 : The energy structures. Using convergences (4.14) and (4.15) in Lemma 4.6 and the lower semi-continuity of norms with respect to the weak and $^*$-weak convergence, we pass to the limit $h \searrow 0$ in energy estimate (3.6) in Lemma 3.2 to deduce (1.3). Inequality (1.4) similarly follows from convergence (4.16) in Lemma 4.6 and energy boundedness (3.7) in Lemma 3.2.

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

Remark 5.2

It is worth remaking that, we have the further initial regularity

$$\begin{aligned} \lim _{t \searrow 0}\Vert u(t)-u_0\Vert _{W^{1,p}(\Omega )}=0. \end{aligned}$$

(5.6)

We shall present the full proof of this identity in Appendix D.

Appendices

Appendix A: Proof of Lemma 3.1

In this appendix, we give the proof of Lemma 3.1. The proof relies on the Direct Method in the Calculus of Variations. In the course of this appendix, fix $p>1$, $q>0$ and a step size $h \in (0,1]$ and set ${\mathfrak {X}}:=W^{1,p}_0(\Omega ) \cap L^{q+1}(\Omega )$.

Proof of Lemma 3.1

Given $u_{i-1} \in {\mathfrak {X}}$, we define the variational functional

$$\begin{aligned} {\mathfrak {X}} \ni w\quad \longmapsto \quad {\mathcal {F}}[w]:= & {} \frac{1}{p}\int _\Omega |Dw|^p\,dx\\&+\frac{1}{h}\int _\Omega \left[ \tfrac{1}{q+1}|w|^{q+1}-|u_{i-1}|^{q-1}u_{i-1}w\right] \,dx. \end{aligned}$$

The proof is twofold: first, we prove the existence of a minimizer of this variational functional ${\mathcal {F}}$, then we manipulate the first variation of ${\mathcal {F}}$, proving the claim.

By Young’s inequality

$$\begin{aligned} {\mathcal {F}}[w]&=\frac{1}{p}\int _\Omega |Dw|^p\,dx+\frac{1}{h}\int _\Omega \left[ \tfrac{1}{q+1}|w|^{q+1}-|u_{i-1}|^{q-1}u_{i-1}w\right] \,dx \nonumber \\&\geqq \frac{1}{p}\int _\Omega |Dw|^p\,dx+\frac{1}{h}\int _\Omega \left[ \tfrac{1}{2(q+1)}|w|^{q+1}-c(q)|u_{i-1}|^{q+1}\right] \,dx \nonumber \\&>-\infty \end{aligned}$$

(A.1)

holds for any $w \in {\mathfrak {X}}$. Let $(u_i)_{i \in {\mathbb {N}}}$ be a minimizing sequence satisfying

$$\begin{aligned} m\leqq {\mathcal {F}}[u_i]<m+\frac{1}{i} \end{aligned}$$

for all $i \in {\mathbb {N}}$, where $m:=\inf \limits _{w \in {\mathfrak {X}}}{\mathcal {F}}[w]$. We observe from (A.1) that

$$\begin{aligned} -\infty<-c(q) \Vert u_{i-1}\Vert _{L^{q+1}(\Omega )}^{q+1}/h\leqq m \leqq {\mathcal {F}}[w] <\infty \end{aligned}$$

holds for every $w\in {\mathfrak {X}}$. Additionally, $\sup \limits _{i \in {\mathbb {N}}}{\mathcal {F}}[u_i] \leqq m+1$ holds and we obtain from (A.1) that

$$\begin{aligned} \frac{1}{p}\int _\Omega |Du_i|^p\,dx+\frac{1}{h}\int _\Omega \tfrac{1}{2(q+1)}|u_i|^{q+1}\,dx\leqq m+1+\frac{c(q)}{h}\int _\Omega |u_{i-1}|^{q+1}\,dx, \end{aligned}$$

which implies that the sequence $(u_i)_{i \in {\mathbb {N}}}$ is uniformly bounded (with respect to $i \in {\mathbb {N}}$) in $W_0^{1,p}(\Omega )\cap L^{q+1}(\Omega )$. Thus, there exist a subsequence, still denoted by $(u_i)_{i \in {\mathbb {N}}}$, and a weak limit $u \in W^{1,p}_0(\Omega ) \cap L^{q+1}(\Omega )$ such that

$$\begin{aligned} {\left\{ \begin{array}{ll} u_i \rightarrow u\quad \text {weakly}\quad \text {in}\,\,W_0^{1,p}(\Omega )\\ u_i \rightarrow u\quad \text {weakly}\quad \text {in}\,\,L^{q+1}(\Omega ) \end{array}\right. } \end{aligned}$$

as $i \rightarrow \infty$. Since ${\mathcal {F}}$ is convex, ${\mathcal {F}}$ is lower semicontinuous with respect to the weak convergence in ${\mathfrak {X}}\equiv W^{1,p}(\Omega ) \cap L^{q+1}(\Omega )$ and thus, we arrive at

$$\begin{aligned} {\mathcal {F}}[u]\leqq \liminf _{i\rightarrow \infty }{\mathcal {F}}[u_i]=m\equiv \inf \limits _{w \in {\mathfrak {X}}}{\mathcal {F}}[w], \end{aligned}$$

which ensures the existence of the minimizer of ${\mathcal {F}}$.

Next, we compute the first variation of ${\mathcal {F}}$. Let $u_i \in {\mathfrak {X}}$ be a minimizer of ${\mathcal {F}}$. For $s \in (-1,1)$ and every $\xi \in {\mathfrak {X}}$, it holds that

$$\begin{aligned} \dfrac{{\mathcal {F}}[u_i+s\xi ]-{\mathcal {F}}[u_i]}{s}&=\frac{1}{p}\int _\Omega \dfrac{|D(u_i+s\xi )|^p-|Du_i|^p}{s}\,dx\nonumber \\&\quad +\frac{1}{h(q+1)}\int _\Omega \dfrac{|u_i+s\xi |^{q+1}-|u_i|^{q+1}}{s}\,dx \nonumber \\&\quad -\frac{1}{h}\int _\Omega \dfrac{|u_{i-1}|^{q-1}u_{i-1}(u_i+s\xi )-|u_{i-1}|^{q-1}u_{i-1}u_i}{s}\,dx \nonumber \\&=:{\mathbf {I}}_1+{\mathbf {I}}_2+{\mathbf {I}}_3, \end{aligned}$$

(A.2)

where the definitions of ${\mathbf {I}}_1$–${\mathbf {I}}_3$ are obvious in the context. In turn, it sees that

$$\begin{aligned} {\mathbf {I}}_3=-\frac{1}{h}\int _\Omega |u_{i-1}|^{q-1}u_i\xi \,dx \end{aligned}$$

while, due to the fundamental theorem of calculus, there holds that

$$\begin{aligned} {\mathbf {I}}_1&=\frac{1}{hp}\int _\Omega \dfrac{1}{s}\left( \int _0^1\dfrac{d}{d\theta }|Du_i+\theta sD\xi |^{p}\,d\theta \right) \,dx \\&=\frac{1}{h}\int _\Omega \left( \int _0^1|Du_i+\theta sD\xi |^{p-2}(Du_i+\theta s D\xi )\,d\theta \right) \cdot D \xi \,dx \end{aligned}$$

and, by Young’s inequality, the integrant of the integral on the last line is estimated as

$$\begin{aligned}&\left| \left( \int _0^1|Du_i+\theta sD\xi |^{p-2}(Du_i+\theta s D\xi )\,d\theta \right) \cdot D \xi \right| \\&\quad \leqq 2^{p-1}\left( |Du_i|^{p-1}+|D\xi |^{p-1}\right) |D\xi | \\&\quad \leqq 2^{p-1}\left( \tfrac{p-1}{p}|Du_i|^p+\tfrac{p+1}{p}|D\xi |^p\right) \in L^1(\Omega ). \end{aligned}$$

Thus, Lebesgue’s dominated convergence theorem renders that

$$\begin{aligned} {\mathbf {I}}_1 \rightarrow \frac{1}{h}\int _\Omega |Du_i|^{p-2}Du_i\cdot D\xi \,dx \end{aligned}$$

as $s \rightarrow 0$. Similarly, we deduce

$$\begin{aligned} {\mathbf {I}}_2&=\frac{1}{h(q+1)}\int _\Omega \dfrac{1}{s}\left( \int _0^1\dfrac{d}{d\theta }|u_i+\theta s\xi |^{q+1}\,d\theta \right) \,dx \\&=\frac{1}{h}\int _\Omega \left( \int _0^1|u_i+\theta s\xi |^{q-1}(u_i+\theta s \xi )\,d\theta \right) \xi \,dx \rightarrow \frac{1}{h}\int _\Omega |u_i|^{q-1}u_i\xi \,dx \end{aligned}$$

in the limit $s\rightarrow 0$. Combining the three preceding formulae and passing to the limit $s \rightarrow 0$ in (A.2), we arrive at

$$\begin{aligned} 0&=\lim _{s\rightarrow 0}\dfrac{{\mathcal {F}}[u_i+s\xi ]-{\mathcal {F}}[u_i]}{s}\\&=\frac{1}{h}\int _\Omega |Du_i|^{p-2}Du_i\cdot D\xi \,dx+\frac{1}{h}\int _\Omega \left( |u_i|^{q-1}u_i-|u_{i-1}|^{q-1}u_{i-1}\right) \,\xi \,dx \end{aligned}$$

for every $\xi \in {\mathfrak {X}}$. This finishes the proof of the lemma. $\square$

Appendix B: Proof of Lemma 4.9

In this appendix, we aim at proving Lemma 4.9.

Proof of Lemma 4.9

The proof is twofold: first, we derive the quantitative estimate (4.25), then we show the convergence (4.26).

Step 1.  By Young’s inequality, we infer that

$$\begin{aligned}&\left| {\bar{u}}_h(t-h)\right| ^{q+1}-\left| {\bar{u}}_h(t)\right| ^{q+1} \\&\quad =\Big [\left| {\bar{u}}_h(t-h)\right| ^{q-1}{\bar{u}}_h(t-h)-\left| {\bar{u}}_h(t)\right| ^{q-1}{\bar{u}}_h(t)\Big ]{\bar{u}}_h(t) \\&\qquad +\left| {\bar{u}}_h(t-h)\right| ^{q+1}-\left| {\bar{u}}_h(t-h)\right| ^{q-1}{\bar{u}}_h(t-h){\bar{u}}_h(t) \\&\quad \geqq \Big [\left| {\bar{u}}_h(t-h)\right| ^{q-1}{\bar{u}}_h(t-h)-\left| {\bar{u}}_h(t)\right| ^{q-1}{\bar{u}}_h(t)\Big ]{\bar{u}}_h(t)\\&\qquad +\tfrac{1}{q+1}\Big (\left| {\bar{u}}_h(t-h)\right| ^{q+1}-\left| {\bar{u}}_h(t)\right| ^{q+1}\Big ) \end{aligned}$$

that is,

$$\begin{aligned}&\tfrac{q}{q+1}\Big (\left| {\bar{u}}_h(t-h)\right| ^{q+1}-\left| {\bar{u}}_h(t)\right| ^{q+1}\Big )\\&\quad \geqq \Big [\left| {\bar{u}}_h(t-h)\right| ^{q-1}{\bar{u}}_h(t-h)-\left| {\bar{u}}_h(t)\right| ^{q-1}{\bar{u}}_h(t)\Big ]{\bar{u}}_h(t). \end{aligned}$$

Dividing this estimate by $h>0$ leads to

$$\begin{aligned} -\tfrac{q}{q+1}\Delta _{-h}\left( |{\bar{u}}_h(t)|^{q+1}\right) \geqq -{\bar{u}}_h\Delta _{-h}\left( |{\bar{u}}_h|^{q-1}{\bar{u}}_h\right) . \end{aligned}$$

(B.1)

Next, a simple calculation gives us that

$$\begin{aligned}&\iint _{\Omega _T}\tfrac{1}{q+1}\Delta _h\left( |[u]_\varepsilon |^{q+1}\right) \,dxdt \\&\quad =\frac{1}{h}\iint _{\Omega \times (T-h,T)}\tfrac{1}{q+1}\left| [u]_\varepsilon (t+h)\right| ^{q+1}\,dxdt\\&\qquad -\frac{1}{h}\iint _{\Omega \times (-h,0)}\tfrac{1}{q+1}\left| [u]_\varepsilon (t+h)\right| ^{q+1}\,dxdt. \end{aligned}$$

This formula together with (B.1) implies that

$$\begin{aligned}&-\iint _{\Omega _T}\Delta _{-h}\left( |{\bar{u}}_h|^{q-1}{\bar{u}}_h\right) \left( {\bar{u}}_h-[u]_\varepsilon \right) \,dxdt \nonumber \\&\quad =-\iint _{\Omega _T}{\bar{u}}_h\Delta _{-h}\left( |{\bar{u}}_h|^{q-1}{\bar{u}}_h\right) \,dxdt+\iint _{\Omega _T}[u]_\varepsilon \Delta _{-h}\left( |{\bar{u}}_h|^{q-1}{\bar{u}}_h\right) \,dxdt \nonumber \\&\quad {\mathop {\leqq }\limits ^{(\mathrm{B.1})}}-\iint _{\Omega _T}\tfrac{q}{q+1}\Delta _{-h}\left( |{\bar{u}}_h|^{q+1}\right) \,dxdt+\iint _{\Omega _T}[u]_\varepsilon \Delta _{-h}\left( |{\bar{u}}_h|^{q-1}{\bar{u}}_h\right) \,dxdt \nonumber \\&\quad =\iint _{\Omega _T}\Big [ [u]_\varepsilon \Delta _{-h}\left( |{\bar{u}}_h|^{q-1}{\bar{u}}_h\right) -\tfrac{q}{q+1}\Delta _{-h}\left( \left| {\bar{u}}_h\right| ^{q+1}\right) +\tfrac{1}{q+1}\Delta _h\left( |[u]_\varepsilon |^{q+1}\right) \Big ]\,dxdt\nonumber \\&\qquad -\tfrac{1}{h}\tfrac{1}{q+1}\iint _{\Omega \times (T-h,T)}\left| [u]_\varepsilon (t+h)\right| ^{q+1}\,dxdt\nonumber \\&\qquad +\tfrac{1}{h}\tfrac{1}{q+1}\iint _{\Omega \times (-h,0)}\left| [u]_\varepsilon (t+h)\right| ^{q+1}\,dxdt. \end{aligned}$$

(B.2)

The first term on the right-hand side of (B.2) is computed as

$$\begin{aligned} \iint _{\Omega _T}[u]_\varepsilon \Delta _{-h}\left( |{\bar{u}}_h|^{q-1}{\bar{u}}_h\right) \,dxdt&=-\iint _{\Omega _T}|{\bar{u}}_h|^{q-1}{\bar{u}}_h \Delta _h[u]_\varepsilon \,dxdt \\&\quad +\tfrac{1}{h}\iint _{\Omega \times (T-h,T)}[u]_\varepsilon (t+h)|{\bar{u}}_h|^{q-1}{\bar{u}}_h\,dxdt\\&\quad -\tfrac{1}{h}\iint _{\Omega \times (-h,0)}[u]_\varepsilon (t+h)|{\bar{u}}_h|^{q-1}{\bar{u}}_h\,dxdt \end{aligned}$$

and the second term on the right-hand side of (B.2) is written as

$$\begin{aligned}&-\iint _{\Omega _T}\tfrac{q}{q+1}\Delta _{-h}\left( \left| {\bar{u}}_h\right| ^{q+1}\right) \,dxdt=-\iint _{\Omega \times (T-h,T)}\tfrac{q}{q+1}\tfrac{1}{h}\left| {\bar{u}}_h\right| ^{q+1}\,dxdt\\&\quad +\iint _{\Omega \times (-h,0)}\tfrac{q}{q+1}\tfrac{1}{h}\left| {\bar{u}}_h\right| ^{q+1}\,dxdt. \end{aligned}$$

Inserting the preceding formulae into (B.2), we have

$$\begin{aligned}&-\iint _{\Omega _T}\Delta _{-h}\left( |{\bar{u}}_h|^{q-1}{\bar{u}}_h\right) \left( {\bar{u}}_h-[u]_\varepsilon \right) \,dxdt \\&\quad \leqq \iint _{\Omega _T}\Big [-\left| {\bar{u}}_h\right| ^{q-1}{\bar{u}}_h \Delta _h[u]_\varepsilon +\tfrac{1}{q+1}\Delta _h\left( |[u]_\varepsilon |^{q+1}\right) \Big ]\,dxdt+\varvec{\Xi } \end{aligned}$$

with the remainder term

$$\begin{aligned} \varvec{\Xi }&:=\tfrac{1}{h}\iint _{\Omega \times (T-h,T)}\Big [[u]_\varepsilon (t+h)\left| {\bar{u}}_h\right| ^{q-1}{\bar{u}}_h-\tfrac{q}{q+1}\left| {\bar{u}}_h\right| ^{q+1}-\tfrac{1}{q+1}\left| [u]_\varepsilon (t+h)\right| ^{q+1}\Big ]\,dxdt\\&\quad -\tfrac{1}{h}\iint _{\Omega \times (-h,0)}\Big [[u]_\varepsilon (t+h)\left| {\bar{u}}_h\right| ^{q-1}{\bar{u}}_h-\tfrac{q}{q+1}\left| {\bar{u}}_h\right| ^{q+1}-\tfrac{1}{q+1}\left| [u]_\varepsilon (t+h)\right| ^{q+1}\Big ]\,dxdt. \end{aligned}$$

This joined to the identity

$$\begin{aligned}&|[u]_\varepsilon |^{q-1}[u]_\varepsilon \,\Delta _h[u]_\varepsilon +\tfrac{1}{h}{\mathbf {B}}\left[ [u]_\varepsilon , [u]_\varepsilon (t+h)\right] \\&\quad =\tfrac{1}{q+1}\Delta _h\left( |[u]_\varepsilon |^{q+1}\right) \end{aligned}$$

in turn implies that

$$\begin{aligned}&-\iint _{\Omega _T}\Delta _{-h}\left( |{\bar{u}}_h|^{q-1}{\bar{u}}_h\right) \left( {\bar{u}}_h-[u]_\varepsilon \right) \,dxdt \nonumber \\&\quad \leqq \iint _{\Omega _T}\Big [\Delta _h[u]_\varepsilon \left( |[u]_\varepsilon |^{q-1}[u]_\varepsilon -\left| {\bar{u}}_h\right| ^{q-1}{\bar{u}}_h \right) +\tfrac{1}{h}{\mathbf {B}}\left[ [u]_\varepsilon , [u]_\varepsilon (t+h)\right] \Big ]\,dxdt+\varvec{\Xi }. \end{aligned}$$

(B.3)

We are going to reformulate the remainder term $\varvec{\Xi }$. Using the boundary term ${\mathbf {B}}[\cdot ,\cdot ]$, the integrand of the first term of $\varvec{\Xi }$ is rewritten as

$$\begin{aligned}&\frac{1}{h}\Big [[u]_\varepsilon (t+h)\left| {\bar{u}}_h\right| ^{q-1}{\bar{u}}_h-\tfrac{q}{q+1}\left| {\bar{u}}_h\right| ^{q+1}-\tfrac{1}{q+1}\left| [u]_\varepsilon (t+h)\right| ^{q+1}\Big ]\\&\quad =-\frac{1}{h}{\mathbf {B}}\left[ {\bar{u}}_h(t),[u]_\varepsilon (t+h)\right] \end{aligned}$$

and the second one becomes

$$\begin{aligned}&\tfrac{1}{h}\Big [[u]_\varepsilon (t+h)\left| {\bar{u}}_h\right| ^{q-1}{\bar{u}}_h-\tfrac{q}{q+1}\left| {\bar{u}}_h\right| ^{q+1}-\tfrac{1}{q+1}\left| [u]_\varepsilon (t+h)\right| ^{q+1}\Big ] \\&\quad =\frac{1}{h}\Big ([u]_\varepsilon (t+h)-[u]_\varepsilon (t)\Big )\left| {\bar{u}}_h\right| ^{q-1}{\bar{u}}_h-\tfrac{1}{q+1}\frac{1}{h}\Big (\left| [u]_\varepsilon (t+h)\right| ^{q+1}-\left| [u]_\varepsilon (t)\right| ^{q+1}\Big )\\&\qquad +\frac{1}{h}\Big [[u]_\varepsilon (t)\left| {\bar{u}}_h\right| ^{q-1}{\bar{u}}_h-\tfrac{q}{q+1}\left| {\bar{u}}_h\right| ^{q+1}-\tfrac{1}{q+1}\left| [u]_\varepsilon (t)\right| ^{q+1}\Big ] \\&\quad = \left| {\bar{u}}_h\right| ^{q-1}{\bar{u}}_h\Delta _h[u]_\varepsilon -\tfrac{1}{q+1}\Delta _h\left( \left| [u]_\varepsilon \right| ^{q+1}\right) +{\mathbf {B}}\left[ {\bar{u}}_h(t),[u]_\varepsilon (t)\right] . \end{aligned}$$

Furthermore, by the convexity of a function ${\mathbb {R}}\ni x \mapsto \tfrac{1}{q+1}|x|^{q+1} \in {\mathbb {R}}$, we get

$$\begin{aligned} \tfrac{1}{q+1}\Delta _h\left( |[u]_\varepsilon |^{q+1}\right) \leqq \Delta _h[u]_\varepsilon \left| [u]_\varepsilon (t+h)\right| ^{q-1}[u]_\varepsilon (t+h). \end{aligned}$$

Merging the preceding formulae and estimate, we therefore have

$$\begin{aligned} \varvec{\Xi }&=-\frac{1}{h}\iint _{\Omega \times (T-h,T)}{\mathbf {B}}\left[ {\bar{u}}_h, {\bar{u}}_h(t+h)\right] \,dxdt+\frac{1}{h}\iint _{\Omega \times (-h,0)}{\mathbf {B}}\left[ {\bar{u}}_h, [u]_\varepsilon \right] \,dxdt\nonumber \\&\quad +\iint _{\Omega \times (-h,0)}\Big [\tfrac{1}{q+1}\Delta _h\left( |[u]_\varepsilon |^{q+1}\right) -\Delta _h[u]_\varepsilon \left| {\bar{u}}_h\right| ^{q-1}{\bar{u}}_h\Big ]\,dxdt \nonumber \\&\leqq -\frac{1}{h}\iint _{\Omega \times (T-h,T)}{\mathbf {B}}\left[ {\bar{u}}_h, {\bar{u}}_h(t+h)\right] \,dxdt+\frac{1}{h}\iint _{\Omega \times (-h,0)}{\mathbf {B}}\left[ {\bar{u}}_h, [u]_\varepsilon \right] \,dxdt \nonumber \\&\quad +\iint _{\Omega \times (-h,0)}\Delta _h[u]_\varepsilon \Big [\left| [u]_\varepsilon (t+h)\right| ^{q-1}[u]_\varepsilon (t+h)-\left| {\bar{u}}_h\right| ^{q-1}{\bar{u}}_h\Big ]\,dxdt \end{aligned}$$

(B.4)

Combining (B.3) with (B.4), we finally arrive at the desired estimate

$$\begin{aligned}&-\iint _{\Omega _T}\Delta _{-h}\left( |{\bar{u}}_h|^{q-1}{\bar{u}}_h\right) \left( {\bar{u}}_h-[u]_\varepsilon \right) \,dxdt\\&\quad \leqq \iint _{\Omega _T}\Delta _h[u]_\varepsilon \left( |[u]_\varepsilon |^{q-1}[u]_\varepsilon -|{\bar{u}}_h|^{q-1}{\bar{u}}_h\right) \,dxdt \\&\qquad -\frac{1}{h}\iint _{\Omega \times (T-h,T)}{\mathbf {B}}\left[ {\bar{u}}_h,[u]_\varepsilon (t+h)\right] \,dxdt \\&\qquad +\frac{1}{h}\iint _{\Omega \times (-h,0)}{\mathbf {B}}\left[ {\bar{u}}_h,[u]_\varepsilon \right] \,dxdt \\&\qquad +\varvec{\Delta }^{(\varepsilon )}_1(h)+\varvec{\Delta }_2^{(\varepsilon )}(h), \end{aligned}$$

where

$$\begin{aligned} \varvec{\Delta }^{(\varepsilon )}_1(h)&:=\frac{1}{h}\iint _{\Omega _T}{\mathbf {B}}\left[ [u]_\varepsilon ,\,[u]_{\varepsilon }(t+h)\right] \,dxdt \end{aligned}$$

and

$$\begin{aligned} \varvec{\Delta }^{(\varepsilon )}_2(h)&:=\iint _{\Omega \times (-h,0)}\Delta _h[u]_\varepsilon \Big [\left| [u]_\varepsilon (t+h)\right| ^{q-1}[u]_\varepsilon (t+h)-\left| {\bar{u}}_h\right| ^{q-1}{\bar{u}}_h \Big ]\,dxdt. \end{aligned}$$

Step 2. Next, we shall prove that $\lim \limits _{h \searrow 0}\varvec{\Delta }^{(\varepsilon )}_1(h)=\lim \limits _{h \searrow 0}\varvec{\Delta }^{(\varepsilon )}_2(h)=0$ for any $\varepsilon >0$. Fix any number $h_0 \in (0,1]$. By (2.3) in Lemma 2.2, we have

$$\begin{aligned} 0\leqq & {} \frac{1}{h}{\mathbf {B}}\left[ [u]_\varepsilon (t), [u]_\varepsilon (t+h)\right] \\\leqq & {} \Big (\left| [u]_\varepsilon (t+h)\right| ^{q-1}[u]_\varepsilon (t+h)-\left| [u]_\varepsilon (t)\right| ^{q-1}[u]_\varepsilon (t) \Big )\Delta _h[u]_\varepsilon \end{aligned}$$

and integrating over $\Omega _T$ leads to

$$\begin{aligned} 0&\leqq \varvec{\Delta }_1^{(\varepsilon )}(h)=\iint _{\Omega _T}\frac{1}{h}{\mathbf {B}}\left[ [u]_\varepsilon (t), [u]_\varepsilon (t+h)\right] \,dxdt \nonumber \\&\leqq \iint _{\Omega _T}\left| [u]_\varepsilon (t+h)\right| ^{q-1}[u]_\varepsilon (t+h)\Delta _h[u]_\varepsilon \,dxdt\nonumber \\&\quad -\iint _{\Omega _T}\left| [u]_\varepsilon \right| ^{q-1}[u]_\varepsilon \Delta _h[u]_\varepsilon \,dxdt \nonumber \\&=:{\mathbf {I}}_h-\mathbf {II}_h, \end{aligned}$$

(B.5)

where the definition of ${\mathbf {I}}_h$ and $\mathbf {II}_h$ are clear from the context. For the integrand of ${\mathbf {I}}_h$

$$\begin{aligned} \left| [u]_\varepsilon (t+h)\right| ^{q-1}[u]_\varepsilon (t+h)\Delta _h[u]_\varepsilon \rightarrow \left| [u]_\varepsilon \right| ^{q-1}[u]_\varepsilon \partial _t[u]_\varepsilon \quad \text {a.e.} \,\,\text {in}\,\,\Omega _T \end{aligned}$$

holds true as $h \searrow 0$ because $\partial _t[u]_\varepsilon \in L^{q+1}(\Omega _{T+h_0})$. Since by Hölder’s inequality

holds almost everywhere in $\Omega _T$ for every $h \in (0,h_0]$, Young’s inequality yields that

in the limit $h \searrow 0$. Consequently, Lemma 2.5 implies that

$$\begin{aligned} \lim _{h \searrow 0}{\mathbf {I}}_h=\iint _{\Omega _T}\left| [u]_\varepsilon \right| ^{q-1}[u]_\varepsilon \partial _t[u]_\varepsilon \,dxdt. \end{aligned}$$

Performing similarly, we also have

$$\begin{aligned} \lim _{h \searrow 0}\mathbf {II}_h=\iint _{\Omega _T}\left| [u]_\varepsilon \right| ^{q-1}[u]_\varepsilon \partial _t[u]_\varepsilon \,dxdt. \end{aligned}$$

Merging these convergences in (B.5), we gain

$$\begin{aligned} 0\leqq \limsup _{h\searrow 0}\varvec{\Delta }^{(\varepsilon )}_1(h) \leqq \lim _{h \searrow 0}{\mathbf {I}}_h-\lim _{h \searrow 0}\mathbf {II}_h =0, \end{aligned}$$

proving the first assertion of (4.26). On the other hand, the integrand of $\varvec{\Delta }^{(\varepsilon )}_2(h)$ is estimated as

and, the first and second terms on the right-hand side converges as

in the limit $h \searrow 0$. Hence, this together with the fact that

$$\begin{aligned} \partial _t[u]_\varepsilon =\frac{u-[u]_\varepsilon }{\varepsilon }=\frac{u_0-u_0}{\varepsilon }=0 \quad \text {on}\,\,\,\Omega \times (-h,0) \end{aligned}$$

implies that

proving the second claim of (4.9). Therefore, the proof is complete. $\square$

Appendix C. Proof of Lemma 5.1

In this appendix, we record the proof of Lemma 5.1.

Proof of Lemma 5.1

We start with deducing that $v_h (t)$ is weakly continuous on $t \in [0, \infty )$ in $L^{\frac{q+1}{q}} (\Omega )$, uniformly on the parameter h. Let $t_2>t_1 \geqq 0$ be arbitrarily taken. Testing the approximating equation (3.5) with $\psi (x)\text{1 }\text{ l}_{(t_1,t_2)}(t)$, where $\psi =\psi (x) \in C^\infty _0(\Omega )$ and $\text{1 }\text{ l}_{(t_1,t_2)}(t)$ is the usual Lipschitz approximation of characteristic function of time interval $(t_1, t_2)$. Appealing to Hölder’s inequality and (3.6) in Lemma 3.2, we gain

$$\begin{aligned} \left| \int _{\Omega }\big (v_h(t_2)-v_h(t_1)\big ) \psi \,dx \right|&=\left| \int _{t_1}^{t_2}\int _\Omega |D{\bar{u}}_h|^{p-2}D{\bar{u}}_h \cdot D\psi \,dx\right| \\&\leqq \Vert D\psi \Vert _{L^p(\Omega )}\int _{t_1}^{t_2}\left\| D{\bar{u}}_h(t)\right\| ^{p-1}_{L^p(\Omega )}\,dt \\&{\mathop {\leqq }\limits ^{(3.6)}} \left( \frac{q}{q + 1}\right) ^{\frac{p -1}{p}}\Vert D\psi \Vert _{L^p(\Omega )}\Vert u_0\Vert _{L^{q+1}(\Omega )}^{\frac{(q+1) (p-1)}{p}}(t_2-t_1)^{\frac{1}{p}} \end{aligned}$$

for every $t_2>t_1 \geqq 0$. By the density of $C^\infty _0 (\Omega )$ in $L^{q+1} (\Omega )$, Hölder’s inequality and (3.6) in Lemma 3.2 again, the above display implies that $v_h (t)$ is weakly continuous on $t \in [0, \infty )$ in $L^{\frac{q+1}{q}} (\Omega )$, uniformly on approximation parameter h. For any $\varepsilon >0$, we define

$$\begin{aligned} \zeta _\varepsilon (t):={\left\{ \begin{array}{ll} \dfrac{t}{\varepsilon }\quad &{}\text {if}\quad 0 \leqq t \leqq \varepsilon , \\ 1 \quad &{}\text {if}\quad t>\varepsilon . \end{array}\right. } \end{aligned}$$

We apply (3.5) with the testing function $\zeta _\varepsilon (t)\varphi$, where $\varphi$ is a smooth function with support in $\Omega \times [0,T)$, to gain

Since by definition $v_h(0)=|u_0(x)|^{q-1}u_0(x)$ and the weak continuous of $v_h(t)$ in $L^{\frac{q+1}{q}}(\Omega )$, we pass to the limit $\varepsilon \searrow 0$ in the above display to have

$$\begin{aligned} \iint _{\Omega _T} \Big [-v_h\partial _t\varphi +|D{\bar{u}}_h|^{p-2}D{\bar{u}}_h\cdot D\varphi \Big ]\,dxdt=\int _\Omega |u_0|^{q-1}u_0\varphi (0)\,dx, \end{aligned}$$

therefore, sending $h \searrow 0$ finally concludes that

$$\begin{aligned} \iint _{\Omega _T}\Big [-|u|^{q-1}u \,\partial _t\varphi +|Du|^{p-2}Du\cdot D\varphi \Big ]\,dxdt=\int _\Omega |u_0|^{q-1}u_0\varphi (0)\,dx, \end{aligned}$$

where we used (4.23) in Lemma 4.7 and Lemma 4.8 again. $\square$

Appendix D. Proof of further initial regularity (5.6)

In this final appendix, we prove the further initial regularity (5.6). For this, we recall a fundamental convergence result by Clarkson [11] and Hanner [15], which states that a weak convergent sequence in $L^{p},\,1<p<\infty$ fulfilling the norm convergence is strong convergent in $L^p$.

Lemma D.1

With $p \in (1,\infty )$ let $(f_{i})_{i \in {\mathbb {N}}}$ be a sequence in $L^{p}(\Omega )$. If $f_{i} \rightarrow f$ weakly in $L^{p}(\Omega )$ and $\Vert f_{i}\Vert _{L^p(\Omega )} \rightarrow \Vert f\Vert _{L^p(\Omega )}$, then

$$\begin{aligned} f_{i} \rightarrow f \quad \text {strongly in}\,\,\, L^{p}(\Omega )\quad \text {as}\,\,\,i \rightarrow \infty . \end{aligned}$$

Proof

The proof is simple but fundamental. For this reason, we only give a sketch of it. Without loss of generality, we may assume that $\Vert f_i\Vert _{L^p(\Omega )}=\Vert f\Vert _{L^p(\Omega )}=1$. Since by the assumption,

$$\begin{aligned} \frac{f_i+f}{2} \rightarrow f \quad \text {weakly}\,\,\text {in} \quad L^p(\Omega ) \end{aligned}$$

holds true, the lower semi-continuity along the weak convergence yields that

$$\begin{aligned} 1=\Vert f\Vert _{L^p(\Omega )} \leqq \liminf _{i \rightarrow \infty } \left\| \frac{f_{i}+f}{2} \right\| _{L^p(\Omega )} \leqq \limsup _{i \rightarrow \infty } \frac{\Vert f_{i}\Vert _{L^p(\Omega )}+\Vert f\Vert _{L^p(\Omega )}}{2}=1, \end{aligned}$$

that is,

$$\begin{aligned} \lim _{i \rightarrow \infty } \left\| \frac{f_{i}+f}{2} \right\| _{L^p(\Omega )}=1. \end{aligned}$$

This together with the uniform convexity¹ of the Lebesgue space $L^{p}$ (refer to [11, 15]) shows that

$$\begin{aligned} \lim _{i \rightarrow \infty }\Vert f_{i}-f\Vert _{L^p(\Omega )}=0, \end{aligned}$$

which concludes the proof. $\square$

We are now in position to prove (5.6).

Proof

First we take $\varphi \in L^{p^\prime }(\Omega )$ arbitrarily, where $p^\prime :=\frac{p-1}{p}$. By density, we approximate $\varphi \in L^{p^\prime }(\Omega )$ by $(\varphi _k)_{k \in {\mathbb {N}}} \subset C^\infty _0(\Omega )$. By the initial condition (D3) we infer that, for every $\alpha =1,\ldots , n$,

$$\begin{aligned} \int _\Omega \Big (D_\alpha u(t)-D_\alpha u_0\Big )\cdot \varphi _k\,dx=-\int _\Omega \Big (u(t)-u_0\Big )D_\alpha \varphi _k\,dx \rightarrow 0 \end{aligned}$$

(D.1)

as $t \searrow 0$, where we abbreviate $D_\alpha :=\partial _{x_\alpha }$ for short. In view of Lemma 4.8 and (3.9) in Lemma 3.3, we gain, as $h \searrow 0$, the energy inequality (1.4) in Theorem 1.1 together with Hölder’s inequality renders that, for every $\varphi \in L^{p^\prime }(\Omega )$ and $\alpha =1,\ldots , n$,

$$\begin{aligned}&\left| \int _\Omega \Big (D_\alpha u(t)-D_\alpha u_0\Big )\cdot \varphi \,dx\right| \\&\quad =\left| \int _\Omega \Big (D_\alpha u(t)-D_\alpha u_0\Big )\cdot \varphi _k\,dx\right| +\left| \int _\Omega \Big (D_\alpha u(t)-D_\alpha u_0\Big )\cdot (\varphi -\varphi _k)\,dx\right| \\&\quad \leqq \left| \int _\Omega \Big (D_\alpha u(t)-D_\alpha u_0\Big )\cdot \varphi _k\,dx\right| +\Big (\Vert D_\alpha u(t)\Vert _{L^p(\Omega )}+\Vert D_\alpha u_0\Vert _{L^p(\Omega )}\Big )\Vert \varphi -\varphi _k\Vert _{L^{p^\prime }(\Omega )} \\&\quad {\mathop {\leqq }\limits ^{(1.4)}}\left| \int _\Omega \Big (D_\alpha u(t)-D_\alpha u_0\Big )\cdot \varphi _k\,dx\right| +2\Vert Du_0\Vert _{L^p(\Omega )}\Vert \varphi -\varphi _k\Vert _{L^{p^\prime }(\Omega )}, \end{aligned}$$

therefore, sending $t \searrow 0$ and subsequently, passing to the limit $k \rightarrow \infty$, the weak convergence (D.1) implies that

$$\begin{aligned} Du(t) \rightarrow Du_0\quad \text {weakly in}\,\,\,L^p(\Omega )\quad \text {as}\,\,\,t \searrow 0 \end{aligned}$$

(D.2)

and, the lower semi-continuity of the norm with respect to this weak convergence further yields

$$\begin{aligned} \Vert Du_0\Vert _{L^p (\Omega )} \leqq \liminf _{t \searrow 0} \Vert Du(t) \Vert _{L^p (\Omega )}. \end{aligned}$$

Estimate (1.4) in Theorem 1.1 again leads to

$$\begin{aligned} \limsup _{t\searrow 0}\Vert Du(t)\Vert _{L^p(\Omega )} \leqq \Vert Du_0\Vert _{L^p(\Omega )}. \end{aligned}$$

The preceding convergences conclude that

$$\begin{aligned} \lim _{t\searrow 0}\Vert Du(t)\Vert _{L^p(\Omega )} = \Vert Du_0\Vert _{L^p(\Omega )}. \end{aligned}$$

(D.3)

Eventually, by (D.2) and (D.3), we apply Lemma D.1 to infer that

$$\begin{aligned} \lim _{t\searrow 0}\Vert Du(t)-Du_0\Vert _{L^p(\Omega )}=0, \end{aligned}$$

finishing the proof. $\square$