Existence of solutions for parabolic variational inequalities

Abstract

In this paper, we are concerned with the study of parabolic variational inequality. Under appropriate assumptions on the main functions, we obtain the existence of weak solutions after the construction of the penalized Young measure by Galerkin’s method and the penalty method. The passage to the limit follows relying on the theory of Young measures.

1 Introduction

In this paper, we are concerned with the existence of weak solutions for parabolic systems. Let $\varOmega \subset \mathbb {R}^n$ ($n\ge 2$) be a bounded open domain, $p\in (2n/(n+2),\infty )$ and $0<T<\infty$ are given constants and denote $Q=\varOmega \times (0,T)$ with its boundary $\partial Q=\partial \varOmega \times (0,T)$. We deal with the following variational inequality

$$\begin{aligned} \begin{aligned} \int \limits _{Q_s}\frac{\partial u}{\partial t}(v-u)dxdt+\int \limits _{Q_s}a(x,t,Du):(Dv-Du)dxdt\ge \int \limits _{Q_s}f(v-u)dxdt, \end{aligned} \end{aligned}$$

(1)

for every $v\in L^p(0,T;W^{1,p}_0(\varOmega ;\mathbb {R}^m))$ and $Q_s=\varOmega \times (0,s)$ for all $s\in [0,T]$. Here $f\in L^{p'}(0,T;W^{-1,p'}(\varOmega ;\mathbb {R}^m))$, $p'=p/(p-1)$ and $a:Q\times \mathbb {M}^{m\times n}\rightarrow \mathbb {M}^{m\times n}$ is a function assumed to satisfy some conditions. Here $\mathbb {M}^{m\times n}$ stands for the set of $m\times n$ matrices equipped with the inner product $\xi :\eta =\sum _{i=1}^m\sum _{j=1}^n\xi _{ij}\eta _{ij}$. To deal with (1), we shall find a function $u(x,t)\in K$ satisfying the previous inequality, where

$$\begin{aligned} K=\Big \{w\in L^p(&0,T;W^{1,p}_0(\varOmega ;\mathbb {R}^m)\cap C(0,T;L^2(\varOmega ;\mathbb {R}^m)),\;\frac{\partial w}{\partial t}\in L^{p'}(0,T;W^{-1,p'}(\varOmega ;\mathbb {R}^m)):\\&\;0\le w(x,0)=u_0(x)\in L^2(\varOmega ;\mathbb {R}^m),\;w(x,t)\ge 0\quad \text {a.e.}\;(x,t)\in Q\Big \}. \end{aligned}$$

It should be noted, that the variational inequality (1) come from and is governed by the following quasilinear parabolic system

$$\begin{aligned} \frac{\partial u}{\partial t}-\text {div}\,a(x,t,Du)=f\quad \text {in}\;Q. \end{aligned}$$

(2)

There is a large number of papers to consider (2). By the theory of Young measures, the author in [17] has proved the existence of weak solutions, under mild monotonicity assumptions on the function a. This theory is used to serve the existence of weak solutions, since that problem can not be treated by the classical monotone operator method developed in [10, 11, 21, 22, 25]. And this is because a does not need to satisfy the strict monotonicity condition of Leray-Lions’s type. We refer the reader to [1,2,3,4,5,6,7] where the theory of Young measures has been applied for both elliptic and parabolic problems. The elliptic case of (1) was investigated in [8] where the authors have proved the existence of weak solutions employing the theory of Young measures and a theorem of Kinderlehrer and Stampacchia.

Variational inequalities as the development and extension of classic variational problems are a very useful tool to research partial differential equations, optimal control, and other fields. Many papers (see e.g. [15, 18, 21, 23, 24]) are interested in the solvability of the different kinds of parabolic variational inequalities, relying on the methods of time discretion, semigroup property of the corresponding differential quotient and a penalization method which transform a parabolic variational inequality into a parabolic equation with a penalty term. These works assumed the monotonicity or regularity condition of the obstacles. We find another method in [19], where only the continuity on obstacles was used. Works which are dealing with double-phase problems or multivalued problems can be found in [12, 26,27,28,29]. We point out these works are concerned with obstacle problems where the authors have used tools from the nonsmooth analysis.

Motivated by the works [17, 23, 24], we will study the existence of weak solutions to the problem (1) by using the penalty method (which transforms the inequality (1) into equality (3) below) and the theory of Young measures. To be more precise, we shall construct a Young measure $\nu _{(x,t)}^\epsilon$ generated by a penalized gradient sequence, with $\epsilon \in (0,1)$, which converges to the Young measure $\nu _{(x,t)}$ as $\epsilon$ tends to zero. To the best of our knowledge, this is the first paper treating the problem (1) by such methods.

This paper is organized as follows. Section 2 is devoted to recalling some necessary properties of Young measures. In Sect. 3, we prove the existence of weak solutions by Galerkin’s approximation and the theory of Young measures for (3), while Sect. 4 is concerned to show the existence of weak solutions for variational problem (1).

2 Young measures: necessary properties

Consider $C_0(\mathbb {R}^m)=\big \{\varphi \in C(\mathbb {R}^m):\;\lim _{|\lambda |\rightarrow \infty }\varphi (\lambda )=0\}$. Its dual is the well-known signed Radon measures ${\mathcal {M}}(\mathbb {R}^m)$ with finite mass. The duality of $\big ({\mathcal {M}}(\mathbb {R}^m),C_0(\mathbb {R}^m)\big )$ is given by the following integrand

$$\begin{aligned} \langle \nu ,\varphi \rangle =\int \limits _{\mathbb {R}^m}\varphi (\lambda )d\nu (\lambda ),\;\text {where}\;\nu :\varOmega \rightarrow {\mathcal {M}}(\mathbb {R}^m). \end{aligned}$$

Lemma 1

([14]) Let $(z_k)_k$ be a bounded sequence in $L^\infty (\varOmega ;\mathbb {R}^m)$. Then there exist a subsequence (still denoted $(z_k)$) and a Borel probability measure $\nu _x$ on $\mathbb {R}^m$ for a.e. $x\in \varOmega$, such that for almost each $\varphi \in C(\mathbb {R}^m)$ we have

$$\begin{aligned} \varphi (z_k)\rightharpoonup ^*\overline{\varphi }(x)=\langle \nu _x,\varphi \rangle \quad \text {weakly in}\;L^\infty (\varOmega ;\mathbb {R}^m) \end{aligned}$$

for a.e. $x\in \varOmega$.

Definition 1

The family $\nu =\{\nu _x\}_{x\in \varOmega }$ is called Young measures associated with (generated by) the subsequence $(z_k)_k$.

In [9], it is shown that if for all $R>0$

$$\begin{aligned} \underset{L\rightarrow \infty }{\lim \sup }\big |\{x\in \varOmega \cap B_R(0):\,|z_k(x)|\ge L\}\big |=0, \end{aligned}$$

then for any measurable $\varOmega '\subset \varOmega$, we have

$$\begin{aligned} \varphi (x,z_k)\rightharpoonup \langle \nu _x,\varphi (x,.)\rangle =\int \limits _{\mathbb {R}^m}\varphi (x,\lambda )d\nu _x(\lambda )\quad \text {in}\;L^1(\varOmega '), \end{aligned}$$

for every Carathéodory function $\varphi :\varOmega \times \mathbb {R}^m\rightarrow \mathbb {R}$ such that $(\varphi (x,z_k(x)))_k$ is equiintegrable.

The following lemmas are useful throughout this paper.

Lemma 2

([16]) If $|\varOmega |<\infty$ and $\nu _x$ is the Young measure generated by the (whole) sequence $(z_k)$, then there holds

$$\begin{aligned} z_k\longrightarrow z\;\text {in measure}\;\Leftrightarrow \;\nu _x=\delta _{z(x)}\quad \text {for a.e.}\;x\in \varOmega . \end{aligned}$$

It should be noted that the above properties remain true when $z_k=Dw_k$, with $w_k:\varOmega \rightarrow \mathbb {R}^m$ and $\varOmega$ can be replaced by the cylinder Q.

Lemma 3

([13]) Let $\varphi :Q\times \mathbb {M}^{m\times n}\rightarrow \mathbb {R}$ be a Carathéodory function and $(w_k)$ be a sequence of measurable functions, where $w_k:Q\rightarrow \mathbb {R}^m$, such that $Dw_k$ generates the Young measure $\nu _{(x,t)}$. Then

$$\begin{aligned} \underset{k\rightarrow \infty }{\lim \inf }\int \limits _Q\varphi (x,t,Dw_k)dxdt\ge \int \limits _Q\int \limits _{\mathbb {M}^{m\times n}}\varphi (x,t,\lambda )d\nu _{(x,t)}(\lambda )dxdt \end{aligned}$$

provided that the negative part $\varphi ^-(x,t,Dw_k)$ is equiintegrable.

The following lemma describes limits points of gradient sequences utilizing the Young measures.

Lemma 4

([3]) The Young measure $\nu _{(x,t)}$ generated by $Dw_k$ in $L^p(0,T;L^p(\varOmega ))$ satisfy the following properties:

• (i) $\nu _{(x,t)}$ is a probability measure, i.e., $\Vert \nu _{(x,t)}\Vert _{{\mathcal {M}}(\mathbb {M}^{m\times n})}=1$ for a.e. $(x,t)\in Q$.

• (ii) The weak $L^1$-limit of $Dw_k$ is given by $\langle \nu _{(x,t)},id\rangle$.

• (iii) For a.e. $(x,t)\in Q$, $\langle \nu _{(x,t)},id\rangle =Dw(x,t)$.

Let $\nu _{(x,t)}^\epsilon$ be the Young measure generated by the penalized gradient sequence $(Dw_\epsilon )$.

Lemma 5

([9]) For every continuous function $\varphi$,

$$\begin{aligned} \langle \nu _{(x,t)}^\epsilon ,\varphi \rangle \longrightarrow \langle \nu _{(x,t)},\varphi \rangle \quad \text {as}\quad \epsilon \rightarrow 0\quad \text {for a.e.}\;(x,t)\in Q. \end{aligned}$$

3 Nonlinear parabolic systems with parameter $\epsilon$

Let $\varOmega$ be a bounded open domain of $\mathbb {R}^n$, $p\in (2n/(n+2),\infty )$ and $\epsilon \in (0,1)$ be fixed. In this section, we shall consider the existence result for the following parabolic system of Dirichlet’s type given in the form:

$$\begin{aligned} \left\{ \begin{array}{rl} \frac{\partial u}{\partial t}-\text {div}\,a(x,t,Du)-\frac{1}{\epsilon }|u^-|^{p-2}u^-&{}=f\quad \text {in}\;Q,\\ u&{}=0\quad \text {on}\;\partial Q,\\ u(x,0)&{}=u_0(x)\quad \text {in}\;\varOmega , \end{array} \right. \end{aligned}$$

(3)

where $u^-=\max \{-u,0\}$ and $a:Q\times \mathbb {M}^{m\times n}\rightarrow \mathbb {M}^{m\times n}$ satisfy the following hypothesis:

($H_0$):

a is a Carathéodory function, that is measurable in $(x,t)\in Q$ for fixed $\xi \in \mathbb {M}^{m\times n}$ and continuous in $\xi$ for fixed (x, t) in Q.

($H_1$):

There exist a function $l\in L^{p'}(Q)$ and a constant $\alpha _0>0$ such that

$$\begin{aligned} |a(x,t,\xi )|\le l(x,t)+|\xi |^{p-1} \end{aligned}$$

(4)

and

$$\begin{aligned} a(x,t,\xi ):\xi \ge \alpha _0|\xi |^p. \end{aligned}$$

(5)

($H_2$):

For all $\xi ,\xi '\in \mathbb {M}^{m\times n}$,

$$\begin{aligned} \big (a(x,t,\xi )-a(x,t,\xi ')\big ):(\xi -\xi ')\ge 0. \end{aligned}$$

Definition 2

A function $u_\epsilon \in L^p(0,T;W^{1,p}_0(\varOmega ;\mathbb {R}^m))$ with $\frac{\partial u_\epsilon }{\partial t}\in L^{p'}(0,T;W^{-1,p'}(\varOmega ;\mathbb {R}^m))$ is called a weak solution of (3), if for all $\varphi \in L^p(0,T;W^{1,p}_0(\varOmega ;\mathbb {R}^m))$, it holds

$$\begin{aligned} \int \limits _Q\frac{\partial u_\epsilon }{\partial t}\varphi dxdt+\int \limits _Qa(x,t,Du_\epsilon ):D\varphi dxdt-\frac{1}{\epsilon }\int \limits _Q|u^-_\epsilon |^{p-2}u^-_\epsilon \varphi dxdt=\int \limits _Qf(x,t)\varphi dxdt. \end{aligned}$$

Similar to that in [20], we take a sequence $\{w_j\}_{j\ge 1}\subset C^\infty _0(\varOmega ;\mathbb {R}^m)$, such that $C^\infty _0(\varOmega ;\mathbb {R}^m)\subset \overline{\bigcup _{k\ge 1}V_k}^{C^1(\overline{\varOmega })}$, where $\{w_j\}_{j\ge 1}$ is a standard orthogonal basis in $L^2(\varOmega ;\mathbb {R}^m)$ and $V_k=\text {span}\{w_1,...,w_k\}$. Firstly, remark that since $u_0\in L^2(\varOmega ;\mathbb {R}^m)$, there exists a sequence $\psi _k(x)\in V_k$ such that $\psi _k(x)\rightarrow u_0(x)$ in $L^2(\varOmega ;\mathbb {R}^m)$ as $k\rightarrow \infty$. Indeed, for $u_0\in L^2(\varOmega ;\mathbb {R}^m)$, there exists a sequence ${v_k}$ in $C^\infty _0(\varOmega ;\mathbb {R}^m)$ such that $v_k\rightarrow u_0$ in $L^2(\varOmega ;\mathbb {R}^m)$. Since $\{v_k\}\subset C^\infty _0(\varOmega ;\mathbb {R}^m)\subset \bigcup _{N\ge 1}\overline{V_N}^{C^1(\overline{\varOmega })}$, we can find a sequence $\{v_k^i\}\subset \bigcup _{N\ge 1}V_N$ such that $v_k^i\rightarrow v_k$ in $C^1(\overline{\varOmega };\mathbb {R}^m)$ as $i\rightarrow \infty$. For $\frac{1}{2^k}$, there exists $i_k\ge 1$ such that $\Vert v_k^{i_k}-v_k\Vert _{C^1(\overline{\varOmega })}\le \frac{1}{2^k}$. Therefore

$$\begin{aligned} \Vert v_k^{i_k}-u_0\Vert _{L^2(\varOmega )}\le C \Vert v_k^{i_k}-v_k\Vert _{C^1(\overline{\varOmega })}+\Vert v_k-u_0\Vert _{L^2(\varOmega )}. \end{aligned}$$

Hence $v_k^{i_k}\rightarrow u_0$ in $L^2(\varOmega ;\mathbb {R}^m)$ as $k\rightarrow \infty$. Let us denote $u_k=v_k^{i_k}$. Since $u_k\in \bigcup _{N\ge 1}V_N$, there exists $V_{N_k}$ such that $u_k\in V_{N_k}$, without loss of generality, we assume that $V_{N_1}\subset V_{N_2}$ as $N_1\le N_2$. We suppose that $N_1>1$ and define $\psi _k$ as follows: $\psi _k(x)=0$, $k=1,...,N_1-1$; $\psi _k=u_1$, $k=N_1,...,N_2-1$; $\psi _k=u_2$, $k=N_2,...,N_3-1$;..., then we obtain the desired sequence $\{\psi _k\}$ and $\psi _k\rightarrow u_0$ in $L^2(\varOmega ;\mathbb {R}^m)$ as $k\rightarrow \infty$.

Theorem 1

Let $f\in L^{p'}(0,T;W^{-1,p'}(\varOmega ;\mathbb {R}^m))$. Suppose that ($H_0$)-($H_2$) are satisfied. Then for every $\epsilon >0$ to be fixed, there exists a weak solution of Eq. (3).

Proof

(i) Galerkin approximation

For each $k\in \mathbb {N}$, $k\ge 1$, we define a vector-valued function $P_k(t,v):[0,\infty )\times \mathbb {R}^k\rightarrow \mathbb {R}^k$ as follows:

$$\begin{aligned} \big (P_k(t,v)\big )_i=\int \limits _\varOmega a\Bigg (x,t,\sum _{j=1}^k v_jDw_j\Bigg ):Dw_idx-\frac{1}{\epsilon }\int \limits _\varOmega \big |\sum _{j=1}^kv_jw_j\big |^{p-2}\Bigg (\sum _{j=1}^kv_jw_j\Bigg )w_idx, \end{aligned}$$

where $v=(v_1,...,v_k)$. Since a is continuous, the continuity of $P_k(t,v)$ follows.

Now, we shall construct the approximate solutions of problem (3) in the form

$$\begin{aligned} u_k(x,t)=\sum _{j=1}^k(\eta _k(t))_jw_j(x), \end{aligned}$$

where $(\eta _k(t))_k$ are unknown functions, which can be determined as solutions of the following system of ordinary differential equations

$$\begin{aligned} \left\{ \begin{array}{rl} \eta '(t)+P_k(t,\eta (t))&{}=F,\\ \eta (0)&{}=U_k(0), \end{array} \right. \end{aligned}$$

(6)

where $(F)_i=\int \limits _\varOmega fw_idx$, $(U_k(0))_i=\int \limits _\varOmega \psi _k(x)w_idx$, $\psi _k(x)\in V_k$, $\psi _k(x)\rightarrow u_0(x)$ in $L^2(\varOmega ;\mathbb {R}^m)$ as $k\rightarrow \infty$.

We multiply the Eq. (6) by $\eta (t)$, thus

$$\begin{aligned} \eta '\eta =P_k(t,\eta )\eta =F\eta . \end{aligned}$$

By the coercivity condition in ($H_1$), we have

$$\begin{aligned} \begin{aligned} P_k(t,\eta )\eta =&\int \limits _\varOmega a\big (x,t,\sum _{j=1}^k\eta _jDw_j\big ):\Bigg (\sum _{i=1}^k\eta _iDw_i\Bigg )dx\\&-\frac{1}{\epsilon }\int \limits _\varOmega \big |\Bigg (\sum _{j=1}^k\eta _jw_j\Bigg )^-\big |^{p-2} \Bigg (\sum _{j=1}^k\eta _jw_j\Bigg )^-\Bigg (\sum _{i=1}^k\eta _iw_i\Bigg )dx\ge 0. \end{aligned} \end{aligned}$$

(7)

From (7) and Young’s inequality, we arrive at

$$\begin{aligned} \frac{1}{2}\frac{\partial }{\partial t}|\eta (t)|^2\le |F\eta |\le \frac{1}{2}|F|^2+\frac{1}{2}|\eta |^2. \end{aligned}$$

Integrating the above inequality with respect to t from 0 to t, we obtain

$$\begin{aligned} |\eta (t)|\le c_k+\int \limits \limits _0^t|\eta (\tau )|^2d\tau , \end{aligned}$$

which implies, by Gronwall’s inequality, that $|\eta (t)|\le c_k(T)$. Consider

$$\begin{aligned} L_k=\max _{t\in [0,T]}|F-P_k(t,\eta )|\quad \text {and}\quad \tau _k=\min \big \{T,\frac{c_k(T)}{L_k}\big \}. \end{aligned}$$

Since $P_k(t,\eta )$ is continuous in t and $\eta$, the Peano Theorem implies that (6) has a $C^1$ solution locally in $[0,\tau _k]$. Let $\tau _k=t_1$ and $\eta (t_1)$ be an initial value, then we repeat the above process and get a $C^1$ solution on $[t_1,t_1+\tau _k]$. We can divide [0, T] into $[(i-1)\tau _k,i\tau _k]$, $i=1,...,L$, where $\frac{T}{L}\le \tau _k$, then there exist $C^1$ solution $\eta _k^i(t)$ in $[(i-1)\tau _k,i\tau _k]$, $i=1,...,L$. Consequently, we arrive at a solution $\eta _k(t)\in C^1([0,T])$ defined by

$$\begin{aligned} \eta _k(t)=\left\{ \begin{array}{rl} &{}\eta _k^1(t)\quad \text {if}\;t\in [0,\tau _k),\\ &{}\eta _k^2(t)\quad \text {if}\;t\in (\tau _k,2\tau _k],\\ &{}\vdots \\ &{}\eta _k^L(t)\quad \text {if}\;t\in ((L-1)\tau _k,L\tau _k]. \end{array} \right. \end{aligned}$$

Therefore, we get the approximate solutions $u_k(x,t)=\sum _{j=1}^k(\eta _k(t))_jw_j(x)$. From (6) it follows for $1\le i\le k$, that

$$\begin{aligned} \int \limits _\varOmega \frac{\partial u_k}{\partial t}w_idx+\int \limits _\varOmega a(x,t,Du_k):Dw_idx-\frac{1}{\epsilon }\int \limits _\varOmega |u_k^-|^{p-2}u_k^-w_idx=\int \limits _\varOmega fw_idx. \end{aligned}$$

(8)

Remark by (6), that $\eta _k(t)$ should be dependent on $\epsilon$, but for convenience we omit $\epsilon$, and for all $\varphi \in C^1(0,T;V_j)$, $j\le k$, there holds

$$\begin{aligned} \int \limits _Q\frac{\partial u_k}{\partial t}\varphi dxdt+\int \limits _Qa(x,t,Du_k):D\varphi dxdt-\frac{1}{\epsilon }\int \limits _Q|u_k^-|^{p-2}u_k^-\varphi dxdt=\int \limits _Q f\varphi dxdt. \end{aligned}$$

(9)

(ii) Passage to the limit

We multiply (8) by $(\eta _k(t))_i$ and sum up i from 1 to k, it holds by integrating with respect to t from 0 to $\tau$ ($\tau \in (0,T]$), that

$$\begin{aligned}&\int \limits _{Q_\tau }\frac{\partial u_k}{\partial t}u_kdxdt+\alpha _0\int \limits _{Q_s}|Du_k|^pdxdt-\frac{1}{\epsilon }\int \limits _{Q_s}|u_k^-|^{p-2}u_k^-u_kdxdt\\&\qquad \le \Vert f\Vert _{L^{p'}(0,T;W^{-1,p'}(\varOmega ;\mathbb {R}^m))}\Vert u_k\Vert _{L^p(0,\tau ;W^{1,p}_0(\varOmega ;\mathbb {R}^m))}, \end{aligned}$$

with $Q_\tau =\varOmega \times (0,\tau )$, where we have used the coercivity condition in ($H_1$) and Hölder’s inequality. We have $u_k(x,0)\rightarrow u_0$ in $L^2(\varOmega ;\mathbb {R}^m)$, this implies $\int _\varOmega u_k^2(x,0)dx\le c$, where c is a constant independent of $\epsilon$ and k. Moreover $\Vert f\Vert _{L^{p'}(0,T;W^{-1,p'}(\varOmega ;\mathbb {R}^m))}\le c$. Therefore

$$\begin{aligned}{} & {} \frac{1}{2}\int \limits _\varOmega |u_k(x,\tau )|^2dx+\alpha _0\int \limits _{Q_\tau }| Du_k|^pdx+\frac{1}{\epsilon }\int \limits _{Q_\tau }|u_k^-|^pdxdt\nonumber \\{} & {} \quad \le c(\Vert u_k\Vert _{L^p(0,\tau ;W^{1,p}_0(\varOmega ;\mathbb {R}^m))}+1). \end{aligned}$$

(10)

From this inequality, we deduce, that

$$\begin{aligned} \begin{aligned} \Vert u_k\Vert _{L^p(0,T;W^{1,p}_0(\varOmega ;\mathbb {R}^m))}\le c,\quad \int \limits _\varOmega |u_k(x,T)|^2dx\le c\quad \text {and}\quad \frac{1}{\epsilon }\int \limits _{Q}|u_k^-|^pdxdt\le c. \end{aligned} \end{aligned}$$

(11)

By (11) and the growth condition in ($H_1$), we have

$$\begin{aligned} \int \limits _Q|a(x,t,Du_k)|^{p'}dxdt\le c\left( \int \limits _Q|Du_k|^pdxdt+1\right) \le c, \end{aligned}$$

(12)

where c is a constant independent of $\epsilon$ and k. From (11) and (12), there exist a subsequence of $(u_k)_k$ (still denoted by $(u_k)$), $\chi \in L^{p'}(Q;\mathbb {M}^{m\times n})$ and $g\in L^{p'}(Q;\mathbb {R}^m)$ such that

$$\begin{aligned} \left\{ \begin{array}{rl} &{}u_k\rightharpoonup u_\epsilon \quad \text {in}\; L^p(0,T;W^{1,p}_0(\varOmega ;\mathbb {R}^m)),\\ &{}u_k\rightharpoonup ^* u_\epsilon \quad \text {in}\;L^\infty (0,T;L^2(\varOmega ;\mathbb {R}^m)),\\ &{}a(x,t,Du_k)\rightharpoonup \chi \quad \text {in}\;L^{p'}(Q;\mathbb {M}^{m\times n}),\\ &{}\frac{1}{\epsilon }|u_k^-|^{p-2}u_k^-\rightharpoonup g\quad \text {in}\; L^{p'}(Q;\mathbb {R}^m). \end{array} \right. \end{aligned}$$

By the compact embedding $W^{1,p}_0(\varOmega ;\mathbb {R}^m)\hookrightarrow L^p(\varOmega ;\mathbb {R}^m)$, one has $u_k\rightarrow u_\epsilon$ in $L^p(Q;\mathbb {R}^m)$ and almost everywhere in Q (for a subsequence). Thus, as $k\rightarrow \infty$, we have

$$\begin{aligned} \left\{ \begin{array}{rl} &{}u_k^-\longrightarrow u_\epsilon ^-\quad \text {a.e.}\;(x,t)\in Q,\\ &{}\frac{1}{\epsilon }|u_k^-|^{p-2}u_k^-\longrightarrow \frac{1}{\epsilon }|u_\epsilon ^-|^{p-2}u_\epsilon ^-\quad \text {a.e.}\;(x,t)\in Q. \end{array} \right. \end{aligned}$$

(13)

From (11) and (13), it follows that $g=\frac{1}{\epsilon }|u_\epsilon ^-|^{p-2}u_\epsilon ^-$. Let $\varphi \in L^p(0,T;W^{1,p}_0(\varOmega ;\mathbb {R}^m))$, then there exists a sequence $\varphi _k\in C^1(0,T;V_k)$ such that $\varphi _k\rightarrow \varphi$ in $L^p(0,T;W^{1,p}_0(\varOmega ;\mathbb {R}^m))$. By virtue of (9) and Hölder’s inequality, we get

$$\begin{aligned}&\Big |\int \limits _Q\frac{\partial u_k}{\partial t}\varphi _kdxdt\Big |\\&\quad =\Big |\int \limits _Qf(x,t)\varphi _kdxdt-\int \limits _Qa(x,t,Du_k):D\varphi _kdxdt +\frac{1}{\epsilon }\int \limits _Q|u_k^-|^{p-2}u_k^-\varphi _kdxdt\Big |\\&\quad \le c\Vert \varphi _k\Vert _{L^p(0,T;W^{1,p}_0(\varOmega ;\mathbb {R}^m))}, \end{aligned}$$

where we have used (11) and (12), and c is a constant independent of k and $\epsilon$. Consequently, $\Vert \frac{\partial u_k}{\partial t}\Vert _{L^{p'}(0,T;W^{-1,p'}(\varOmega ;\mathbb {R}^m))}\le c$. It immediately follows the existence of a subsequence of $(u_k)$ (still denoted as $(u_k)$) such that

$$\begin{aligned} \frac{\partial u_k}{\partial t}\rightharpoonup \alpha \quad \text {in}\; L^{p'}(0,T;W^{-1,p'}(\varOmega ;\mathbb {R}^m)). \end{aligned}$$

Let $\psi \in C^\infty _0(Q;\mathbb {R}^m)$, by letting $k\rightarrow \infty$ in $\int \limits _Q\frac{\partial u_k}{\partial t}\psi dxdt=-\int \limits _Q u_k\frac{\partial \psi }{\partial t}dxdt$, it results

$$\begin{aligned} \int \limits _Q\alpha \psi dxdt=-\int \limits _Qu_\epsilon \frac{\partial \psi }{\partial t}dxdt. \end{aligned}$$

This implies $\alpha =\frac{\partial u_\epsilon }{\partial t}$. On the other hand, since $\int _\varOmega |u_k(x,T)|^2dx\le c$, there is a subsequence of $(u_k(x,T))$ (still labelled by $(u_k(x,T))$) and a function $u^*$ in $L^2(\varOmega ;\mathbb {R}^m)$ such that $u_k(x,T)\rightharpoonup u^*$ in $L^2(\varOmega ;\mathbb {R}^m)$. To identify $u^*$ with u(x, T), we use the fact that

$$\begin{aligned} \int \limits _Q\frac{\partial u_k}{\partial t}w_idxdt=\int \limits _\varOmega u_k(x,T)w_idx-\int \limits _\varOmega u_k(x,0)w_idx. \end{aligned}$$

Passing k to infinity, it results by integration by parts, that $\int _\varOmega u^*w_idx=\int _\varOmega u_\epsilon (x,T)w_idx$. We conclude, by the completeness of $\{w_i\}_i$, that $u^*=u_\epsilon (x,T)$, i.e.,

$$\begin{aligned} \int \limits _\varOmega u_\epsilon ^2(x,T)dx\le \underset{k\rightarrow \infty }{\lim \inf }\int \limits _\varOmega u_k^2(x,T)dx. \end{aligned}$$

(14)

Note that, since $(u_k)$ is bounded in $L^p(0,T;W^{1,p}_0(\varOmega ;\mathbb {R}^m))$, it follows by Lemma 1 the existence of a Young measure $\nu _{(x,t)}^\epsilon$ generated by $(Du_k)$ in $L^p(Q;\mathbb {M}^{m\times n})$ and satisfying Lemma 4. Remark that the generated Young measure is labeled by $\epsilon$, as well as $Du_k$, depending on it.

To identify $\chi$ with $a(x,t,Du_\epsilon )$, we will need the following inequality:

$$\begin{aligned} \int \limits _Q\int \limits _{\mathbb {M}^{m\times n}}\big (a(x,t,\lambda )-a(x,t,Du_\epsilon )\big ):(\lambda -Du_\epsilon )d\nu _{(x,t)}^\epsilon (\lambda )dxdt\le 0. \end{aligned}$$

(15)

To see this, consider the sequence

$$\begin{aligned} Y_k&=\big (a(x,t,Du_k)-a(x,t,Du_\epsilon )\big ):(Du_k-Du_\epsilon )\\&=Y_{k,1}-Y_{k,2}, \end{aligned}$$

where $Y_{k,1}=(a(x,t,Du_k):(Du_k-Du_\epsilon )$ and $Y_{k,2}=a(x,t,Du_\epsilon ):(Du_k-Du_\epsilon )$. As in (12), it follows that $a(x,t,Du_\epsilon )\in L^{p'}(Q;\mathbb {M}^{m\times n})$. On the one hand, because of the weak limit defined in Lemma 4, we obtain

$$\begin{aligned} \begin{aligned} \underset{k\rightarrow \infty }{\lim \inf }\int \limits _Q Y_{k,2}dxdt&=\int \limits _Q\int \limits _{\mathbb {M}^{m\times n}}a(x,t,Du_\epsilon ):(\lambda -Du_\epsilon )dxdt\\&=\int \limits _Qa(x,t,Du_\epsilon )\Big (\underset{:=Du_\epsilon (x,t)}{\underbrace{\int \limits _{\mathbb {M}^{m\times n}}\lambda d\nu _{(x,t)}^\epsilon (\lambda )}}-Du_\epsilon \Big )dxdt=0. \end{aligned} \end{aligned}$$

(16)

On the other hand, since $(a(x,t,Du_k):(Du_k-Du_\epsilon ))$ is equiintegrable (by (11), (12) and Hölder’s inequality), Lemma 3 implies

$$\begin{aligned} \begin{aligned} \underset{k\rightarrow \infty }{\lim \inf }\ {}&\int \limits _Qa(x,t,Du_k):(Du_k-Du_\epsilon )dxdt\\&\ge \int \limits _Q\int \limits _{\mathbb {M}^{m\times n}}a(x,t,\lambda ):(\lambda -Du_\epsilon )d\nu _{(x,t)}^\epsilon (\lambda )dxdt. \end{aligned} \end{aligned}$$

(17)

The next step is to show, that the left-hand side of the above inequality is $\le 0$. We have

$$\begin{aligned} \int \limits _Q\frac{\partial u_k}{\partial t}u_kdxdt+\int \limits _Qa(x,t,Du_k):Du_kdxdt -\frac{1}{\epsilon }\int \limits _Q|u_k^-|^{p-2}u_k^-u_kdxdt=\int \limits _Q fu_kdxdt. \end{aligned}$$

By using this equation, it results

$$\begin{aligned} \begin{aligned}&Y:=\underset{k\rightarrow \infty }{\lim \inf }\int \limits _Qa(x,t,Du_k):(Du_k-Du_\epsilon )dxdt\\&\quad =\underset{k\rightarrow \infty }{\lim \inf }\left( \int \limits _Qfu_kdxdt-\int \limits _Q\frac{\partial u_k}{\partial t}u_kdxdt+\frac{1}{\epsilon }\int \limits _Q|u_k^-|^{p-2}u_k^-u_kdxdt \right. \\&\quad \left. -\int \limits _Qa(x,t,Du_k):Du_\epsilon dxdt\right) . \end{aligned} \end{aligned}$$

(18)

Passing to the limit as $k\rightarrow \infty$ in (9), we then have the following energy equality:

$$\begin{aligned} \int \limits _Q\frac{\partial u_\epsilon }{\partial t}\varphi dxdt+\int \limits _Q\chi :D\varphi dxdt-\frac{1}{\epsilon }\int \limits _Q|u_\epsilon ^-|^{p-2}u_\epsilon ^-\varphi dxdt=\int \limits _Qf\varphi dxdt. \end{aligned}$$

Passing to the limit as $k\rightarrow \infty$ on the right hand-side of (18), we get

$$\begin{aligned} Y\le&\int \limits _Qfu_\epsilon dxdt-\frac{1}{2}\int \limits _\varOmega u_\epsilon ^2(x,T)dx+\frac{1}{2}\int \limits _\varOmega u_\epsilon ^2(x,0)dx\\&\qquad -\frac{1}{\epsilon }\int \limits _Q|u_\epsilon ^-|^pdxdt-\int \limits _Q\chi :Du_\epsilon dxdt. \end{aligned}$$

By taking $\varphi =u_\epsilon$ in the energy equality and plugging it in the right-hand side of the above inequality, we arrive at $Y\le 0$ as desired. From this and (16), the Eq. (15) follows. In virtue of the monotonicity of the function a, we conclude the following localization of the support of $\nu _{(x,t)}^\epsilon$:

$$\begin{aligned} \big (a(x,t,\lambda )-a(x,t,Du_\epsilon )\big ):(\lambda -Du_\epsilon )=0\quad \text {on supp}\,\nu _{(x,t)}^\epsilon . \end{aligned}$$

(19)

Now, we identify $\chi$ with $a(x,t,Du_\epsilon )$ as follows:

From the monotonicity assumption, we can write for all $\tau \in \mathbb {R}$ and $\xi \in \mathbb {M}^{m\times n}$

$$\begin{aligned} \begin{aligned} 0&\le \big (a(x,t,\lambda )-a(x,t,Du_\epsilon +\tau \xi )\big ):(\lambda -Du_\epsilon -\tau \xi )\\&=a(x,t,\lambda ):(\lambda -Du_\epsilon ) -a(x,t,\lambda ):\tau \xi -a(x,t,Du_\epsilon +\tau \xi ):(\lambda -Du_\epsilon -\tau \xi ), \end{aligned} \end{aligned}$$

(20)

which implies by (19)

$$\begin{aligned} -a(x,t,\lambda ):\tau \xi \ge -a(x,t,Du_\epsilon ):(\lambda -Du_\epsilon )+a(x,t,Du_\epsilon +\tau \xi ):(\lambda -Du_\epsilon -\tau \xi ). \end{aligned}$$

Note that

$$\begin{aligned}&a(x,t,Du_\epsilon +\tau \xi ):(\lambda -Du_\epsilon -\tau \xi )\\&\quad =a(x,t,Du_\epsilon +\tau \xi ):(\lambda -Du_\epsilon )-a(x,t,Du_\epsilon +\tau \xi ):\tau \xi \\&\quad =a(x,t,Du_\epsilon ):(\lambda -Du_\epsilon )\\&\qquad +\tau \Big (\big (\nabla a(x,t,Du_\epsilon )\xi \big ) :(\lambda -Du_\epsilon )-a(x,t,Du_\epsilon ):\xi \Big )+o(\tau ), \end{aligned}$$

where $\nabla$ is the derivative of a with respect to its third variable. Therefore

$$\begin{aligned} -a(x,t,\lambda ):\tau \xi \ge \tau \Big (\big (\nabla a(x,t,Du_\epsilon )\xi \big ):(\lambda -Du_\epsilon )-a(x,t,Du_\epsilon ):\xi \Big )+o(\tau ). \end{aligned}$$

Since $\tau$ is arbitrary in $\mathbb {R}$, we get

$$\begin{aligned} a(x,t,\lambda ):\xi = a(x,t,Du_\epsilon ):\xi + \big (\nabla a(x,t,Du_\epsilon )\xi \big ):(Du_\epsilon -\lambda ) \end{aligned}$$

(21)

holds on the support of $\nu _{(x,t)}^\epsilon$. The equiintegrability of $a(x,t,Du_k)$ implies that its weak $L^1$-limit $\overline{a}_\epsilon$ is given by

$$\begin{aligned} \begin{aligned}&\overline{a}_\epsilon (x,t):=\int \limits _{\mathbb {M}^{m\times n}}a(x,t,\lambda )d\nu _{(x,t)}^\epsilon (\lambda )\\&=\int \limits _{\text {supp}\,\nu _{(x,t)}^\epsilon }\Big (a(x,t,Du_\epsilon )+\big (\nabla a(x,t,Du_\epsilon )\big ):(Du_\epsilon -\lambda )\Big )d\nu _{(x,t)}^\epsilon (\lambda )\quad (\text {by}\;(21))\\&=a(x,t,Du_\epsilon ), \end{aligned} \end{aligned}$$

(22)

where we used

$$\begin{aligned} \Vert \nu _{(x,t)}^\epsilon \Vert _{{\mathcal {M}}(\mathbb {M}^{m\times n})}=1\quad \text {and}\quad \big (\nabla a(x,t,Du_\epsilon )\big )\int \limits _{\text {supp}\,\nu _{(x,t)}^\epsilon }(Du_\epsilon -\lambda )d\nu _{(x,t)}^\epsilon (\lambda )=0. \end{aligned}$$

Consequently

$$\begin{aligned} a(x,t,Du_k)\rightharpoonup \chi =a(x,t,Du_\epsilon )\quad \text {in}\;L^{p'}(Q;\mathbb {M}^{m\times n}). \end{aligned}$$

In view of (9), for all $\varphi \in C^1(0,T;V_j)$ with $j\le k$, letting $k\rightarrow \infty$ it holds

$$\begin{aligned} \int \limits _Q\frac{\partial u_\epsilon }{\partial t}\varphi dxdt+\int \limits _Qa(x,t,Du_\epsilon ):D\varphi dxdt-\frac{1}{\epsilon }\int \limits _Q|u_\epsilon ^-|^{p-2}u_\epsilon ^-\varphi dxdt=\int \limits _Qf\varphi dxdt,\nonumber \\ \end{aligned}$$

(23)

and since $C^1(0,T;\bigcup _{j\ge 1}V_j)$ is dense in $L^p(0,T;W^{1,p}_0(\varOmega ;\mathbb {R}^m))$, the Eq. (23) holds for all $\varphi \in L^p(0,T;W^{1,p}_0(\varOmega ;\mathbb {R}^m))$. $\square$

4 Variational inequality

We shall prove the main result of this paper. Denote

$$\begin{aligned}&K=\Big \{w\in L^p(0,T;W^{1,p}_0(\varOmega ;\mathbb {R}^m)\cap C(0,T;L^2(\varOmega ;\mathbb {R}^m)),\;\frac{\partial w}{\partial t}\in L^{p'}(0,T;W^{-1,p'}(\varOmega ;\mathbb {R}^m)):\\&\;0\le w(x,0)=u_0(x)\in L^2(\varOmega ;\mathbb {R}^m),\;w(x,t)\ge 0\quad \text {a.e.}\;(x,t)\in Q\Big \}. \end{aligned}$$

The main theorem can be stated as follows:

Theorem 2

Let $f\in L^{p'}(0,T;W^{-1,p'}(\varOmega ;\mathbb {R}^m))$ and suppose that ($H_0$)-($H_2$) are satisfied. Then there exists a function $u(x,t)\in K$ such that for all $v\in L^p(0,T;W^{1,p}_0(\varOmega ;\mathbb {R}^m))$ with $v(x,t)\ge 0$ for a.e. $(x,t)\in Q$, there holds

$$\begin{aligned} \int \limits _{Q_s}\frac{\partial u}{\partial t}(v-u)dxdt+\int \limits _{Q_s}a(x,t,Du):(Dv-Du)dxdt\ge \int \limits _{Q_s}f(v-u)dxdt, \end{aligned}$$

for almost every $s\in [0,T]$.

Proof

(i) A prior estimates

Let us take $\varphi =u_\epsilon .\chi _{(0,t)}$ as a test function in Definition 2 (where $\chi _{(0,t)}$ is the characteristic function of (0, t)), $t\in (0,T]$, thus

$$\begin{aligned} \int \limits _{Q_t}\frac{\partial u_\epsilon }{\partial t}u_\epsilon dxdt+\int \limits _{Q_t}a(x,t,Du_\epsilon ):Du_\epsilon dxdt&-\frac{1}{\epsilon }\int \limits _{Q_t}|u_\epsilon ^-|^{p-2}u_\epsilon ^-u_\epsilon dxdt\\&=\int \limits _{Q_t}f(x,t)u_\epsilon dxdt, \end{aligned}$$

where $Q_t=\varOmega \times (0,t)$. Integrate the first term, it follows by the coercivity condition in ($H_1$) and Hölder’s inequality, that

$$\begin{aligned}{} & {} \frac{1}{2}\int \limits _\varOmega |u_\epsilon (x,t)|^2dx+\alpha _0\int \limits _{Q_t}|Du_\epsilon |^pdxdt +\frac{1}{\epsilon }\int \limits _{Q_t}|u_\epsilon ^-|^pdxdt\nonumber \\{} & {} \qquad \le c(1+\Vert u_\epsilon \Vert _{L^p(0,T;W^{1,p}_0(\varOmega ;\mathbb {R}^m))}), \end{aligned}$$

(24)

where c is a constant independent of $\epsilon$ and t. Consequently

$$\begin{aligned} \begin{aligned} (u_\epsilon )_\epsilon \quad \text {is bounded in}&\;L^\infty (0,T;L^2(\varOmega ;\mathbb {R}^m))\cap L^p(0,T;W^{1,p}_0(\varOmega ;\mathbb {R}^m))\\ \text {and}\quad&\frac{1}{\epsilon }\int \limits _Q|u_\epsilon ^-|^pdxdt\le c. \end{aligned} \end{aligned}$$

(25)

Similar to that in (12), one has

$$\begin{aligned} \Vert a(x,t,Du_\epsilon )\Vert _{L^{p'}(Q)}\le c\quad \text {and}\quad \big \Vert \frac{1}{\epsilon }|u_\epsilon ^-|^{p-2}u_\epsilon ^-\big \Vert _{L^{p'}(Q)}\le c. \end{aligned}$$

(26)

Using (25) and (26), we deduce from Definition 2 that for all $\varphi \in L^p(0,T;W^{1,p}_0(\varOmega ;\mathbb {R}^m))$

$$\begin{aligned} \big \Vert \frac{\partial u_\epsilon }{\partial t}\big \Vert _{L^{p'}(0,T;W^{-1,p'}(\varOmega ;\mathbb {R}^m))}=\underset{\Vert \varphi \Vert _{L^p(0,T;W^{1,p}_0(\varOmega ;\mathbb {R}^m))}\le 1}{\sup } \Big |\int \limits _Q\frac{\partial u_\epsilon }{\partial t}\varphi dxdt\Big |\le c. \end{aligned}$$

(27)

(ii) Passage to the limit

As in the previous section, from (25 to 27) there exists a subsequence of $(u_\epsilon )$ (still labeled by $(u_\epsilon )$), such that

$$\begin{aligned} \left\{ \begin{array}{rl} &{}u_\epsilon \rightharpoonup u\quad \text {in}\;L^p(0,T;W^{1,p}_0(\varOmega ;\mathbb {R}^m)),\\ &{}u_\epsilon \rightharpoonup ^*u\quad \text {in}\;L^\infty (0,T;L^2(\varOmega ;\mathbb {R}^m)),\\ &{}a(x,t,Du_\epsilon )\rightharpoonup \sigma \quad \text {in}\;L^{p'}(Q;\mathbb {M}^{m\times n}),\\ &{}u_\epsilon ^-\longrightarrow 0\quad \text {in}\; L^p(Q;\mathbb {R}^m),\\ &{}\frac{\partial u_\epsilon }{\partial t}\rightharpoonup \alpha \quad \text {in}\; L^{p'}(0,T;W^{-1,p'}(\varOmega ;\mathbb {R}^m)). \end{array} \right. \end{aligned}$$

(28)

Let $\varphi \in C^\infty _0(Q;\mathbb {R}^m)$, we have $\int _Q\frac{\partial u_\epsilon }{\partial t}\varphi dxdt=-\int _Qu_\epsilon \frac{\partial \varphi }{\partial t}dxdt$. Passing to the limit and using (28), there holds $\int _Q\alpha \varphi dxdt=-\int _Qu\frac{\partial \varphi }{\partial t}dxdt$, and therefore $\alpha =\frac{\partial u}{\partial t}$. By virtue of (28), there exists a subsequence, still denoted as $(u_\epsilon )$, such that $u_\epsilon \rightarrow u$ in $L^p(Q;\mathbb {R}^m)$ and almost everywhere, thus $u_\epsilon ^-\rightarrow u^-$ a.e. $(x,t)\in Q$. Moreover, from (13) we have $u^-=0$ for a.e. $(x,t)\in Q$, that is to say $u(x,t)\ge 0$ for a.e. $(x,t)\in Q$. Since $u_\epsilon \in L^\infty (0,T;L^2(\varOmega ;\mathbb {R}^m))$, for all $s\in [0,T]$, we have $u_\epsilon (x,s)\rightharpoonup u^*$ in $L^2(\varOmega ;\mathbb {R}^m)$. Let $\varphi \in C^\infty _0(\varOmega ;\mathbb {R}^m)$ and $\eta (t)\in C([0,s])$. By passing to the limit in

$$\begin{aligned}&\int \limits _{Q_s}\frac{\partial u_\epsilon }{\partial t}\eta (t)\varphi (x)dxdt\\&\quad =\int \limits _\varOmega u_\epsilon (x,s)\eta (s)\varphi (x)dx-\int \limits _\varOmega u_0(x)\eta (0)\varphi (x)dx-\int \limits _{Q_s}u_\epsilon \frac{\partial \eta }{\partial t}\varphi dxdt, \end{aligned}$$

it follows by the integration by parts, that

$$\begin{aligned} \int \limits _\varOmega \Big (\big (u^*-u(x,s)\big )\eta (s)\varphi (x)-\big (u(x,0)-u_0(x)\big )\eta (0)\varphi (x)\Big )dx=0. \end{aligned}$$

If we choose $\eta (s)=1$ and $\eta (0)=0$, or $\eta (s)=0$ and $\eta (0)=1$, we then get $u^*=u(x,s)$ and $u(x,0)=u_0(x)$ (by the density of $C^\infty _0(\varOmega ;\mathbb {R}^m)$ in $L^2(\varOmega ;\mathbb {R}^m)$).

By (25), there exists a Young measure $\nu _{(x,t)}$ generated by $Du_\epsilon$ in $L^p(Q;\mathbb {M}^{m\times n})$ and verify the properties of Lemma 4. The next step has for goal to identify $\sigma$ with a(x, t, Du). To do this, we consider the sequence

$$\begin{aligned} I_\epsilon =\big (a(x,t,Du_\epsilon )-a(x,t,Du)):(Du_\epsilon -Du) \end{aligned}$$

According to the weak limit in Lemma 4, we have

$$\begin{aligned}&\lim _{\epsilon \rightarrow 0}\int \limits _Qa(x,t,Du):(Du_\epsilon -Du)dxdt\\&\quad =\int \limits _Qa(x,t,Du):\Big (\underset{:=Du(x,t)}{\underbrace{\int \limits _{\mathbb {M}^{m\times n}}\lambda d\nu _{(x,t)}(\lambda )}}-Du\Big )dxdt=0. \end{aligned}$$

This and Lemma 3 implies

$$\begin{aligned} \underset{\epsilon \rightarrow 0}{\lim \inf }\int \limits _Q I_\epsilon dxdt\ge \int \limits _Q\int \limits _{\mathbb {M}^{m\times n}}a(x,t,\lambda ):(\lambda -Du)d\nu _{(x,t)}(\lambda )dxdt. \end{aligned}$$

Similar to the previous section, there holds

$$\begin{aligned} \big (a(x,t,\lambda )-a(x,t,Du)\big ):(\lambda -Du)=0\quad \text {on supp}\,\nu _{(x,t)}. \end{aligned}$$

(29)

By the same procedure from (20) to (22) and equiintegrability of $(a(x,t,Du_\epsilon ))$, it follows that the weak $L^1$-limit of $a(x,t,Du_\epsilon )$ is a(x, t, Du). Therefore $\sigma =a(x,t,Du)$.$\square$

Remark 1

Note that, since $a(x,t,Du_\epsilon )=\int \limits _{\mathbb {M}^{m\times n}}a(x,t,\lambda )d\nu _{(x,t)}^\epsilon (\lambda )$, thus one can directly pass to the limit using Lemma 5 and (21) as follows:

$$\begin{aligned} a(x,t,Du_\epsilon )&=\int \limits _{\text {supp}\,\nu _{(x,t)}^\epsilon }a(x,t,\lambda )d\nu _{(x,t)}^\epsilon (\lambda )\\&\rightharpoonup \int \limits _{\text {supp}\,\nu _{(x,t)}}a(x,t,\lambda )d\nu _{(x,t)}(\lambda )\\&=\int \limits _{\text {supp}\,\nu _{(x,t)}}\Big (a(x,t,Du)+\big (\nabla a(x,t,Du)\big ):(Du-\lambda )\Big )d\nu _{(x,t)}(\lambda )\\&=a(x,t,Du). \end{aligned}$$

Proof

(iii) Existence of weak solutions

Let $v\in L^p(0,T;W^{1,p}_0(\varOmega ;\mathbb {R}^m))$, $v\ge 0$. By taking $\varphi =v-u_\epsilon$ as a test function in Definition 2, we get

$$\begin{aligned}&\int \limits _{Q_s}\frac{\partial u_\epsilon }{\partial t}vdxdt+\int \limits _{Q_s}a(x,t,Du_\epsilon ):(Dv-Du_\epsilon )dxdt-\int \limits _{Q_s}f(v-u_\epsilon )dxdt\\&\quad =\int \limits _{Q_s}\frac{\partial u_\epsilon }{\partial t}u_\epsilon dxdt+\frac{1}{\epsilon }\int \limits _{Q_s}|u_\epsilon ^-|^{p-2}u_\epsilon ^-(v-u_\epsilon )dxdt\\&\quad \ge \frac{1}{2}\int \limits _\varOmega |u_\epsilon (x,s)|^2dx-\frac{1}{2}\int \limits _\varOmega |u_\epsilon (x,0)|^2dx, \end{aligned}$$

i.e.,

$$\begin{aligned} \begin{aligned}&\int \limits _{Q_s}\frac{\partial u_\epsilon }{\partial t}vdxdt+\int \limits _{Q_s}a(x,t,Du_\epsilon ):Dvdxdt-\int \limits _{Q_s}f(v-u_\epsilon )dxdt\\&\quad \ge \frac{1}{2}\int \limits _\varOmega |u_\epsilon (x,s)|^2dx -\frac{1}{2}\int \limits _\varOmega |u_\epsilon (x,0)|^2dx+\int \limits _{Q_s}a(x,t,Du_\epsilon ):Du_\epsilon dxdt. \end{aligned} \end{aligned}$$

(30)

Since $\frac{\partial u_\epsilon }{\partial t}\rightharpoonup \frac{\partial u}{\partial t}$ in $L^{p'}(0,T;W^{-1,p'}(\varOmega ;\mathbb {R}^m))$, $a(x,t,Du_\epsilon )\rightharpoonup \sigma =a(x,t,Du)$ in $L^{p'}(Q;\mathbb {M}^{m\times n})$ and $Du_\epsilon \rightharpoonup \langle \nu _{(x,t)},id\rangle =Du(x,t)$ in $L^p(Q;\mathbb {M}^{m\times n})$, we conclude as $\epsilon \rightarrow 0$, that

$$\begin{aligned} \int \limits _{Q_s}\frac{\partial u}{\partial t}(v-u)dxdt+\int \limits _{Q_s}a(x,t,Du):(Dv-Du)dxdt\ge \int \limits _{Q_s}f(v-u)dxdt, \end{aligned}$$

for almost every $s\in [0,T]$. Remark that, since $u\in L^p(0,T;W^{1,p}_0(\varOmega ;\mathbb {R}^m))$, $\frac{\partial u}{\partial t}\in L^{p'}(0,T;W^{-1,p'}(\varOmega ;\mathbb {R}^m))$ and $\big \{u\in L^p(0,T;W^{1,p}_0(\varOmega ;\mathbb {R}^m)):\;\frac{\partial u}{\partial t}\in L^{p'}(0,T;W^{-1,p'}(\varOmega ;\mathbb {R}^m))\big \}$ is continuously embedded in $C(0,T;L^2(\varOmega ;\mathbb {R}^m))$, thus $u\in C(0,T;L^2(\varOmega ;\mathbb {R}^m))$ and the proof is complete. $\square$