Compact embedding for p(x, t)-Sobolev spaces and existence theory to parabolic equations with p(x, t)-growth

Abstract

In this paper, we establish the compact embedding of p(x, t)-Sobolev spaces into p(x, t)-Lebesgue spaces. Moreover, we prove some existence results for nonlinear parabolic problems of the form

$$\begin{aligned} \partial _tu-{\mathrm {div}\,}a(x,t,Du)=f-{\mathrm {div}\,}\left( |F|^{p(x,t)-2}F\right) \,\,\,\text {in}\,\Omega _T, \end{aligned}$$

where the vector-field $a(x,t,\cdot )$ satisfies certain p(x, t)-growth conditions.

1 Introduction

The first main aim of this paper is to establish the compact embedding of nonstandard p(x, t)-Sobolev spaces into nonstandard p(x, t)-Lebesgue spaces. This Aubin–Lions type Theorem is important among others for our existence result to general nonlinear parabolic equations with nonstandard p(x, t)-growth of the type

$$\begin{aligned} \partial _tu-{\mathrm {div}\,}a(x,t,Du)=f-{\mathrm {div}\,}\left( |F|^{p(x,t)-2}F\right) \,\,\,\text {in}~\Omega _T. \end{aligned}$$

(1.1)

Moreover, the results of this manuscript are also important to prove the existence of solutions to parabolic obstacle problems with p(x, t)-growth, see [20].

The motivation of this paper and the study of problems with nonstandard growth is on the one hand based on mathematical interest, on the other hand the consideration of problems in the sense of (1.1) are motivated by issues of life sciences. In general, parabolic problems are often motivated by physical aspects. In particular, evolutionary equations and systems can be used to model physical processes, e.g. heat conduction or diffusion processes. For example the basic equation of fluid mechanics is the Navier–Stokes equation. Some properties of solutions to the system of a modified Navier–Stokes equation describing electro-rheological fluids are studied in [5]. Such fluids are recently of high technological interest, because of their ability to change the mechanical properties under the influence of exterior electro-magnetic field, see [23, 27]. Many electro-rheological fluids are suspensions consisting of solid particles and a carrier oil. These suspensions change their material properties dramatically if they are exposed to an electric field, see [28].

Most of the known results concern the stationary models with p(x)-growth, see e.g. [1–4]. In the context of parabolic problems with p(x, t)-growth applications are e.g. the models for flows in porous media [11, 25] or nonlinear parabolic obstacle problems [19]. Moreover, in the last years parabolic problems with p(x, t)-growth arouse more and more interest in mathematics, also in regularity theory, cf. [12–15, 18–22, 31] and [32].

Finally, we want to highlight that in the case of certain parabolic equations with nonstandard growth condition first existence results are available, i.e. by Antontsev and Shmarev in [7–9], Alkutov and Zhikov in [6] and Diening et al. [17].

1.1 General assumptions

We consider a bounded domain $\Omega \subset \mathbb {R}^n$ of dimension $n\ge 2$ and we write $\Omega _T:=\Omega \times (0,T)$ for the space–time cylinder over $\Omega$ of height $T>0$. Here, $u_t$ resp. $\partial _tu$ denote the partial derivate with respect to the time variable t and Du denotes the one with respect to the space variable x.

The setting First of all, we should mention that we denote by $\partial _{\mathcal {P}}\Omega _T=(\bar{\Omega }\times \left\{ 0\right\} )\cup (\partial \Omega \times (0,T))$ the parabolic boundary of $\Omega _T$. Furthermore, we write $z=(x,t)$ for points in $\mathbb {R}^{n+1}$. We shall consider vector-fields $a:\Omega _T\times \mathbb {R}^n\rightarrow \mathbb {R}^n$ which are assumed to be Carathéodory functions—i.e. a(z, w) is measurable in the first argument for every $w\in \mathbb {R}^n$ and continuous in the second one for a.e. $z\in \Omega _T$—and satisfy the following nonstandard growth and monotonicity properties, for some growth exponent $p:\Omega _T\rightarrow (\frac{2n}{n+2},\infty )$ and structure constants $0<\nu \le 1\le L$ and $\mu \in [0,1]$:

$$\begin{aligned} |a(z,w)|&\le ~L(1+|w|)^{p(z)-1}, \end{aligned}$$

(1.2)

$$\begin{aligned} (a(z,w)-a(z,w_0))\cdot (w-w_0)&\ge \nu (\mu ^2+|w|^2+|w_0|^2)^{\frac{p{(z)}-2}{2}}|w-w_0|^2 \end{aligned}$$

(1.3)

for all $z\in \Omega _T$ and $w,~w_0\in \mathbb {R}^n$. Furthermore, the growth exponent $p:\Omega _T\rightarrow (\frac{2n}{n+2},\infty )$ satisfies the following conditions: There exist constants $\gamma _1,~\gamma _2<\infty$, such that

$$\begin{aligned} \frac{2n}{n+2}<\gamma _1\le p(z)\le \gamma _2\quad \text {and}\quad |p(z_1)-p(z_2)|\le \omega (d_\mathcal {P}(z_1,z_2)) \end{aligned}$$

(1.4)

hold for any choice of $z_1,~z_2\in \Omega _T$, where $\omega :[0,\infty )\rightarrow [0,1]$ denotes a modulus of continuity. More precisely, we shall assume that $\omega (\cdot )$ is a concave, non-decreasing function with $\lim _{\rho \downarrow 0}\omega (\rho )=0=\omega (0)$. Moreover, the parabolic distance is given by $d_\mathcal {P}(z_1,z_2):=\max \{|x_1-x_2|,\sqrt{|t_1-t_2|}\}$ for $z_1=(x_1,t_1),~z_2=(x_2,t_2)\in \mathbb {R}^{n+1}$. In addition, for the modulus of continuity $\omega (\cdot )$ we assume the following weak logarithmic continuity condition:

$$\begin{aligned} \limsup _{\rho \downarrow 0} \omega (\rho )\log \left( \frac{1}{\rho }\right) <+\infty . \end{aligned}$$

(1.5)

By virtue of (1.5) we may assume for a constant $L_1>0$ depending on $\omega (\cdot )$ that

$$\begin{aligned} \omega (\rho )\log \left( \frac{1}{\rho }\right) \le L_1,~~~\text {for all}~\rho \in (0,1]. \end{aligned}$$

(1.6)

Moreover, we denote by $p_1$ and $p_2$ the infimum resp. supremum of $p(\cdot )$ with respect to the domain we are going to deal with, e.g. $p_1:=\inf _{\Omega _T} p(\cdot ),~p_2:=\sup _{\Omega _T} p(\cdot )$. Finally, we point out that (1.3) implies, by using (1.2) and Young’s inequality, the coercivity property

$$\begin{aligned} a(z,w)\cdot w\ge \frac{\nu }{c(\gamma _1,\gamma _2)}|w|^{p(z)} -c(\gamma _1,\gamma _2,\nu ,L)\quad \forall ~z\in \Omega _T~\text {and}~w\in \mathbb {R}^n. \end{aligned}$$

(1.7)

1.2 The function spaces

The spaces $L^p(\Omega ), W^{1,p}(\Omega )$ and $W^{1,p}_0(\Omega )$ denote the usual Lebesgue and Sobolev spaces.

Parabolic Lebesgue–Orlicz spaces We start by the definition of the nonstandard p(z)-Lebesgue space. The space $L^{p(z)}(\Omega _T,\mathbb {R}^k)$ is defined as the set of those measurable functions $v:\Omega _T\rightarrow \mathbb {R}^k$ for $k\in \mathbb {N}$, which satisfy $|v|^{p(z)}\in L^1(\Omega _T,\mathbb {R}^k)$, i.e.

$$\begin{aligned} L^{p(z)}(\Omega _T,\mathbb {R}^k):=\left\{ v:\Omega _T \rightarrow \mathbb {R}^k~\text {is measurable in}~\Omega _T: \int _{\Omega _T}|v|^{p(z)}\mathrm {d}z<+\infty \right\} . \end{aligned}$$

The set $L^{p(z)}(\Omega _T,\mathbb {R}^k)$ equipped with the Luxemburg norm

$$\begin{aligned} \Vert v\Vert _{L^{p(z)}(\Omega _T)}:=\inf \left\{ \lambda >0: \int _{\Omega _T}\left| \frac{v}{\lambda }\right| ^{p(z)}\mathrm {d}z\le 1\right\} \end{aligned}$$

becomes a Banach space. This space is separable and reflexive, see [6, 17]. For elements of $L^{p(z)}(\Omega _T,\mathbb {R}^k)$ the generalized Hölder’s inequality holds in the following form: If $f\in L^{p(z)}(\Omega _T,\mathbb {R}^k), g\in L^{p'(z)}(\Omega _T,\mathbb {R}^k)$, where $p'(z)=\frac{p(z)}{p(z)-1}$, we have

$$\begin{aligned} \left| \int _{\Omega _T}fg\mathrm {d}z\right| \le \left( \frac{1}{\gamma _1}+\frac{\gamma _2-1}{\gamma _2}\right) \Vert f\Vert _{L^{p(z)}(\Omega _T)}\Vert g\Vert _{L^{p'(z)}(\Omega _T)}, \end{aligned}$$

(1.8)

see also [6]. Moreover, the norm $\Vert \cdot \Vert _{L^{p(z)}(\Omega _T)}$ can be estimated as follows:

$$\begin{aligned} -1+\Vert v\Vert _{L^{p(z)}(\Omega _T)}^{\gamma _1}&\le \int _{\Omega _T}|v|^{p(z)}\mathrm {d}z\le \Vert v\Vert _{L^{p(z)}(\Omega _T)}^{\gamma _2}+1. \end{aligned}$$

(1.9)

Finally, for the right-hand side of (1.1) we assume

$$\begin{aligned} F\in L^{p(z)}(\Omega _T,\mathbb {R}^n)~~~\text {and}~~~f\in L^{\gamma _1'}(0,T;W^{-1,\gamma _1'}(\Omega )). \end{aligned}$$

(1.10)

Notice that we will use also the abbreviation $p(\cdot )$ for the exponent p(z).

Parabolic Sobolev–Orlicz spaces First, we introduce nonstandard Sobolev spaces for fixed $t\in (0,T)$. From assumption (1.4), we know that $p(\cdot ,t)$ satisfy $|p(x_1,t)-p(x_2,t)|\le \omega (|x_1-x_2|)$ for any choice of $x_1,~x_2\in \Omega$ and for every $t\in (0,T)$. Next, we define for every fixed $t\in (0,T)$ the Banach space $W^{1,p(\cdot ,t)}(\Omega )$ as $W^{1,p(\cdot ,t)}(\Omega ):=\{u\in L^{p(\cdot ,t)}(\Omega ,\mathbb {R})\left. \right| ~Du\in L^{p(\cdot ,t)}(\Omega ,\mathbb {R}^n)\}$ equipped with the norm $\Vert u\Vert _{W^{1,p(\cdot ,t)}(\Omega )}:=\Vert u\Vert _{L^{p(\cdot ,t)}(\Omega )}+\Vert Du\Vert _{L^{p(\cdot ,t)}(\Omega )}.$ In addition, $W^{1,p(\cdot ,t)}_0(\Omega )\equiv$ the closure of $C^\infty _0(\Omega )~\text {in}~W^{1,p(\cdot ,t)}(\Omega )$ and denote by $W^{1,p(\cdot ,t)}(\Omega )'$ its dual. For every $t\in (0,T)$ the inclusion $W^{1,p(\cdot ,t)}_0(\Omega )\subset W^{1,\gamma _1}_0(\Omega )$ holds. Now, we consider more general nonstandard parabolic Sobolev spaces without fixed t. By $W_g^{p(\cdot )}(\Omega _T)$ we denote the Banach space

$$\begin{aligned} W_g^{p(\cdot )}(\Omega _T):=\begin{Bmatrix} u\in [g+L^1(0,T;W^{1,1}_0(\Omega ))]\cap L^{p(\cdot )}(\Omega _T)\left. \right| ~Du\in L^{p(\cdot )}(\Omega _T,\mathbb {R}^n)\end{Bmatrix} \end{aligned}$$

equipped by the norm $\Vert u\Vert _{W^{p(\cdot )}(\Omega _T)}:=\Vert u\Vert _{L^{p(\cdot )}(\Omega _T)}+\Vert Du\Vert _{L^{p(\cdot )}(\Omega _T)}.$ If $g=0$ we write $W_0^{p(\cdot )}(\Omega _T)$ instead of $W_g^{p(\cdot )}(\Omega _T)$. Here, it is worth to mention that the notion $(u-g)\in W_0^{p(\cdot )}(\Omega _T)$ respectively $u\in g+W_0^{p(\cdot )}(\Omega _T)$ indicate that u agrees with g on the lateral boundary of the cylinder $\Omega _T$, i.e. $u\in W_g^{p(\cdot )}(\Omega _T)$. Now, we are ready to give the definition of a weak solution to the nonstandard parabolic equation (1.1):

Definition 1.1

We identify a function $u\in L^1(\Omega _T)$ as a weak solution of the parabolic equation (1.1), if and only if $u\in C^0([0,T];L^2(\Omega ))\cap W^{p(\cdot )}(\Omega _T)$ and

$$\begin{aligned} \int _{\Omega _T}\left[ u\cdot \varphi _t-a(z,Du)\cdot D\varphi \right] \mathrm {d}z=-\int _{\Omega _T}\left[ f\cdot \varphi +|F|^{p(\cdot )-2}F\cdot D\varphi \right] \mathrm {d}z \end{aligned}$$

(1.11)

holds, whenever $\varphi \in C^\infty _0(\Omega _T)$.

Remark 1.2

In this paper we will consider certain initial value problems. Therefore, we should mention the meaning when referring to an initial condition of the type $u(\cdot ,0)=g(\cdot ,0)$ a.e. on $\Omega$. Here, we shall always mean

$$\begin{aligned} \frac{1}{h}\int _0^h\int _\Omega |u-g(\cdot ,0)|^2\mathrm {d}x\mathrm {d}t\rightarrow 0~~~ \text {as}~h\downarrow 0. \end{aligned}$$

(1.12)

In particular, when $u\in C^0([0,T];L^2(\Omega ))$, then (1.12) is obviously equivalent with saying $u(\cdot ,0)=g(\cdot ,0)$. $\square$

Our next aim is to introduce the dual space of $W_0^{p(\cdot )}(\Omega _T)$. Therefore, we denote by $W^{p(\cdot )}(\Omega _T)'$ the dual of the space $W_0^{p(\cdot )}(\Omega _T)$. Assume that $v\in W^{p(\cdot )}(\Omega _T)'$. Then, there exist functions $v_i\in L^{p'(\cdot )}(\Omega _T), i=0,1,\ldots ,n$, such that

$$\begin{aligned} \left\langle \left\langle v,w\right\rangle \right\rangle _{\Omega _T}=\int _{\Omega _T}\left( v_0w+\sum _{i=1}^nv_iD_iw\right) \mathrm {d}z\quad \forall ~w\in W_0^{p(\cdot )}(\Omega _T). \end{aligned}$$

(1.13)

Here and in the following, we will write $\left\langle \left\langle \cdot ,\cdot \right\rangle \right\rangle _{\Omega _T}$ for the dual pairing between $W^{p(\cdot )}(\Omega _T)'$ and $W^{p(\cdot )}_0(\Omega _T)$. Furthermore, if $v\in W^{p(\cdot )}(\Omega _T)'$, we define the norm

$$\begin{aligned} \Vert v\Vert _{W^{p(\cdot )}(\Omega _T)'}=\sup \{\left\langle \left\langle v,w\right\rangle \right\rangle _{\Omega _T}|w\in W_0^{p(\cdot )}(\Omega _T),~\Vert w\Vert _{W_0^{p(\cdot )}(\Omega _T)}\le 1\}. \end{aligned}$$

Notice, whenever (1.13) holds, we can write $v=v_0-\sum _{i=1}^nD_iv_i$, where $D_iv_i$ has to be interpreted as a distributional derivate. By

$$\begin{aligned} w\in W(\Omega _T):=\left\{ w\in W^{p(\cdot )}(\Omega _T)|w_t\in W^{p(\cdot )}(\Omega _T)'\right\} \end{aligned}$$

we mean that there exists $w_t\in W^{p(\cdot )}(\Omega _T)'$, such that

$$\begin{aligned} \left\langle \left\langle w_t,\varphi \right\rangle \right\rangle _{\Omega _T}=-\int _{\Omega _T}w\cdot \varphi _t\mathrm {d}z~~~\text {for all}~\varphi \in C^\infty _0(\Omega _T), \end{aligned}$$

see also [17]. The previous equality makes sense due to the inclusions

$$\begin{aligned} W^{p(\cdot )}(\Omega _T)\hookrightarrow L^2(\Omega _T)\cong (L^2(\Omega _T))'\hookrightarrow W^{p(\cdot )}(\Omega _T)' \end{aligned}$$

which allow us to identify w as an element of $W^{p(\cdot )}(\Omega _T)'$. Finally, from the definition of the norm $\Vert \cdot \Vert _{W^{p(\cdot )}(\Omega _T)'}$, we can conclude that the following holds: if $f\in W^{p(\cdot )}_0(\Omega _T)$ and $g\in W^{p(\cdot )}(\Omega _T)'$ we have

$$\begin{aligned} \left\langle \left\langle f,g\right\rangle \right\rangle _{\Omega _T}\le c(\gamma _1,\gamma _2)\Vert f\Vert _{W^{p(\cdot )}(\Omega _T)}\Vert g\Vert _{W^{p(\cdot )}(\Omega _T)'}, \end{aligned}$$

(1.14)

see [19]. Notice also that in the case $p(\cdot )=\text {const}.$, we deal with the standard Lebesgue and Sobolev spaces. This means for example, if $p(\cdot )=\gamma _1$, then we have $W^{\gamma _1}(\Omega _T)=L^{\gamma _1}(0,T;W^{1,\gamma _1}(\Omega ))$. Consequently, the dual space of $W^{\gamma _1}(\Omega _T)$ is given by $W^{\gamma _1}(\Omega _T)'=L^{\gamma _1'}(0,T;W^{-1,\gamma _1'}(\Omega ))$.

1.3 Statement of the result and plan of the paper

Here, we mention our main results and briefly describe the strategy of the proof to these results and the technical novelties of the paper. We start with some useful and important preliminary results, see Sect. 2, before we will prove the compact embedding of nonstandard p(z)-Sobolev spaces into nonstandard p(z)-Lebesgue spaces, see Sect. 3. This will be our first main result, i.e. the compactness theorem in the sense of Aubin–Lions and reads as follows.

Theorem 1.3

Let $\Omega \subset \mathbb {R}^n$ an open, bounded Lipschitz domain with $n\ge 2$ and $p(\cdot )>\frac{2n}{n+2}$ satisfying (1.4), (1.5). Furthermore, define $\hat{p}(\cdot ):=\max \left\{ 2,p(\cdot )\right\}$. Then, the inclusion $W(\Omega _T)\hookrightarrow L^{\hat{p}(\cdot )}(\Omega _T)$ is compact.

Theorem 1.3 is important for the strong convergence in p(z)-Lebesgue spaces and therefore, for our existence results. In Sect. 4, we prove the existence of solutions to the parabolic equation (1.1) under certain boundary and initial data conditions. First of all, we establish the existence of a weak solution to the Dirichlet problem of (1.1), i.e.

(1.15)

The approach to prove the existence of solutions to the Dirichlet problem is to construct a solution, which solve the problem (1.15). We start by constructing a sequence of the Galerkin’s approximations, where the limit of this sequence is equal to the solution in (1.15). Then, we show that this approximate solution converges to a general solution, where we used some energy bounds, which derive by the proof and finally, the compact embedding of Theorem 1.3 yields the desired convergence of the approximate solutions to general solutions. This yields

Theorem 1.4

Let $\Omega \subset \mathbb {R}^n$ be an open, bounded Lipschitz domain and $p:\Omega _T\rightarrow [\gamma _1,\gamma _2]$ satisfies (1.4), (1.5). Then, suppose that the vector-field $a:\Omega _T\times \mathbb {R}^n\rightarrow \mathbb {R}^n$ is a Carathéodory function and satisfies, for a given function $v\in C^0([0,T];L^2(\Omega ))\cap W^{p(\cdot )}_0(\Omega _T)$ with $\partial _tv\in L^{\gamma _1'}(0,T;W^{-1,\gamma _1'}(\Omega ))$, the growth condition

$$\begin{aligned} |a(z,w)|\le c(\gamma _2,L)\left( (1+|w|)^{p(\cdot )-1}+|v|^{p(\cdot )-1}\right) \end{aligned}$$

(1.16)

and the monotonicity property

$$\begin{aligned} \nu (\mu ^2+|w+v|^2+|w_0+v|^2)^\frac{p(\cdot )-2}{2}|w-w_0|^2\le (a(z,w)-a(z,w_0))\cdot (w-w_0) \end{aligned}$$

(1.17)

for all $z\in \Omega _T$ and $w,w_0\in \mathbb {R}^n$. Moreover, let (1.10) and $g(\cdot ,0)\in L^2(\Omega )$ hold. Furthermore, define

$$\begin{aligned} \mathcal {M}_0:=\int _{\Omega _T} 1+|F|^{p(\cdot )}+|v|^{p(\cdot )}dz+\Vert f\Vert _{L^{\gamma _1'}(0,T;W^{-1,\gamma _1'}(\Omega ))}^{\gamma _1'}+1\ge 1. \end{aligned}$$

(1.18)

Then, there exists an unique weak solution $u\in C^0([0,T];L^2(\Omega ))\cap W^{p(\cdot )}_0(\Omega _T)$ with $\partial _t u\in W^{p(\cdot )}(\Omega _T)'$ of (1.15) and this solution satisfies the following energy estimate

$$\begin{aligned} \sup _{0\le t\le T}\int _\Omega |u(\cdot ,t)|^2\mathrm {d}x+\int _{\Omega _T}|Du|^{p(\cdot )}\mathrm {d}z\le c\left( \Vert g(\cdot ,0)\Vert _{L^2(\Omega )}^2+\mathcal {M}_0\right) \end{aligned}$$

(1.19)

with $u(\cdot ,0)=g(\cdot ,0)$ and a constant $c=c(n,\gamma _1,\gamma _2,\mathrm {diam}(\Omega ))$.

Remark 1.5

We should mention the role of the function v of the preceding theorem. We will need this function in (1.16), (1.17) or in (4.5), (4.6) later for the proof of the next theorem. For the general existence result to the Dirichlet problem (1.15), we would choose v equal to zero. But for the proof of the existence of weak solution to the Cauchy–Dirichlet problem from below, we will re-write the Cauchy–Dirichlet problem into an Dirichlet problem and v will play the role of the boundary value. Then, this existence result derives immediately from Theorem 1.4 because of (1.16), (1.17). $\square$

Finally, the existence of solutions to initial value problem (1.15) can be extend to general boundary problems. Therefore, we consider the Cauchy–Dirichlet problem of the parabolic problem (1.1):

(1.20)

where g denotes the boundary data and satisfies

$$\begin{aligned} g\in C^0([0,T];L^2(\Omega ))\cap W^{p(\cdot )}(\Omega _T) \quad \text {and} \quad \partial _tg\in L^{\gamma _1'}(0,T;W^{-1,\gamma _1'}(\Omega )). \end{aligned}$$

(1.21)

We use the result of Theorem 1.4 to the Cauchy–Dirichlet problem (1.20) to get existence of solutions to (1.1) with general boundary data. Therefore, we have to change the problem (1.20) into a problem comparing to (1.15). Then, we can conclude the existence of solution to the modified problem. Hence, we get the existence result to the primal Cauchy–Dirichlet problem (1.20). This result is stated in the following Theorem.

Theorem 1.6

Let $\Omega \subset \mathbb {R}^n$ be an open, bounded Lipschitz domain and $p:\Omega _T\rightarrow [\gamma _1,\gamma _2]$ satisfies (1.4), (1.5). Then, suppose that the vector-field $a:\Omega _T\times \mathbb {R}^n\rightarrow \mathbb {R}^n$ is a Carathéodory function and satisfies the growth condition (1.2) and the monotonicity condition (1.3). Moreover, let (1.10) fulfilled. Furthermore, the boundary data g satisfy (1.21). Then, there exists an unique weak solution $u\in C^0([0,T];L^2(\Omega ))\cap W^{p(\cdot )}_g(\Omega _T)$ with $\partial _t u\in W^{p(\cdot )}(\Omega _T)'$ of the parabolic Cauchy–Dirichlet problem (1.20) and this solution satisfies the following energy estimate

$$\begin{aligned}&\sup _{0\le t\le T}\int _\Omega |u(\cdot ,t)|^2\mathrm {d}x+\int _{\Omega _T}|Du|^{p(\cdot )}\mathrm {d}z\nonumber \\&\quad \le c\left( \Vert g(\cdot ,0)\Vert _{L^2(\Omega )}^2+\Vert g\Vert _{L^\infty (0,T;L^2(\Omega ))}^2+\mathcal {M}_g\right) , \end{aligned}$$

(1.22)

where $c=c(n,\gamma _1,\gamma _2,\nu ,L,\mathrm {diam}(\Omega ))$ and $\mathcal {M}_g$ is defined as follows

$$\begin{aligned} \mathcal {M}_g:=\int _{\Omega _T} 1+|F|^{p(\cdot )}+|Dg|^{p(\cdot )}\mathrm {d}z+\Vert f\Vert _{W^{\gamma _1}(\Omega _T)'}^{\gamma _1'}+\Vert \partial _t g\Vert _{W^{\gamma _1}(\Omega _T)'}^{\gamma _1'}+1. \end{aligned}$$

Remark 1.7

Here, we would like to mention that in [20] we need the existence theorems 1.4 and 1.6. But we could also prove the existence of weak solutions to (1.15) and (1.20), if we assume that $a(\cdot )$ satisfies the growth condition (1.2), coercivity condition (1.7) and the monotonicity condition $(a(z,w)-a(z,w_0))\cdot (w-w_0)\ge 0$ for all $z\in \Omega _T$ and $w,~w_0\in \mathbb {R}^n$, see also [17]. $\square$

2 Preliminaries

2.1 Convolution and smoothing

Here, we introduce tools that will allow us to build smooth approximations to given functions. First, we will consider the so-called Friedrich’s mollifier. This mollifier can be used to smooth a function in space and time. Therefore, let $\delta \in C^\infty (\mathbb {R}^{n+1})$ be the Friedrich’s mollifying kernel

$$\begin{aligned} \delta (z):= {\left\{ \begin{array}{ll} \kappa \exp \left( -\frac{1}{1-|z|^2}\right) \,\,\,&{}\text {if}~|z|<1,\\ 0 ~&{}\text {if}~|z|\ge 1, \end{array}\right. } \quad \text {and}\quad \int _{\mathbb {R}^{n+1}}\delta (z)\mathrm {d}z=1, \end{aligned}$$

where $\kappa =\text {const.}~\text {and}~\delta (z)\ge 0$. Furthermore, we extend the given function $v\in W^{p(\cdot )}(\Omega _T)$ by zero to the whole $\mathbb {R}^{n+1}$ and define

$$\begin{aligned} v_h(z):=\int _{\mathbb {R}^{n+1}}v(s)\delta _h(z-s)ds ~~~\text {with}~\delta _h(s)=\frac{1}{h^{n+1}}\delta \left( \frac{s}{h}\right) ,~~~h>0. \end{aligned}$$

The next two propositions are stated in [10] and show some properties of Friedrich’s mollified functions.

Proposition 2.1

If $u\in W^{p(\cdot )}_0(\Omega _T)$ with exponent function p(z) satisfying (1.4), (1.5), then $\Vert u_h-u\Vert _{W^{p(\cdot )}(\Omega _T)}\longrightarrow 0$ as $h\downarrow 0$ and

$$\begin{aligned} \Vert u_h\Vert _{W^{p(\cdot )}(\Omega _T)}\le c\left( \Vert u\Vert _{W^{1,1}(\Omega _T)}+\Vert u\Vert _{W^{p(\cdot )}(\Omega _T)}\right) . \end{aligned}$$

Proposition 2.2

If $u\in W^{p(\cdot )}_0(\Omega _T)$ with exponent function p(z) satisfying (1.4), (1.5) and $u_t\in W^{p(\cdot )}(\Omega _T)'$, then $(u_h)_t\in W^{p(\cdot )}(\Omega _T)'$, and for every $\phi \in W^{p(\cdot )}_0(\Omega _T)$

$$\begin{aligned} \left\langle \left\langle (u_h)_t,\phi \right\rangle \right\rangle _{\Omega _T}\longrightarrow \left\langle \left\langle u_t,\phi \right\rangle \right\rangle _{\Omega _T}~~~\text {as}~h\downarrow 0. \end{aligned}$$

Since, $W^{1,\gamma _2}_0(\Omega )$ is separable, it is a span of a countable set of linearly independent functions $\left\{ \phi _k\right\} \subset W^{1,\gamma _2}_0(\Omega )$. Moreover, we have the dense embedding $W^{1,\gamma _2}_0(\Omega )\subset L^2(\Omega )$ for any $\gamma _2>\frac{2n}{n+2}$, see e.g. [29, 30]. Therefore, without loss of generality, we may assume that this system forms an orthonormal basis of $L^2(\Omega )$.

Moreover, since weak solutions u of parabolic equations possess only weak regularity properties with respect to the time variable t, i.e. they are not assumed to be weakly differentiable, in principle it is not possible to use the solution u itself (also disregarding boundary values) as a test-function in the weak formulation of the parabolic equation. In order to be able to test the equation properly, we smooth the solution u with respect to the time direction t using the so-called Steklov averages. Hence, we introduce the following: the Steklov averages of a function $f\in L^1(\Omega _T)$ are defined as

$$\begin{aligned} _h(x,t):= {\left\{ \begin{array}{ll} \frac{1}{h} \displaystyle \int _t^{t+h}f(x,s)\mathrm {d}s~~~&{}\text {for}~t\in (0,T-h),\\ 0 &{}\text {for}~t\in [T-h,T), \end{array}\right. } \end{aligned}$$

(2.1)

for $x\in \Omega$, for all $t\in (0,T)$ and $0<h<T$. Assuming that $u\in C^0([0,T];L^2(\Omega ))\cap W^{p(\cdot )}(\Omega _T)$ is a weak solution to the parabolic equation (1.1) the Steklov average $[u]_h$ satisfies the corresponding equation

$$\begin{aligned} \int _{\Omega \times \left\{ t\right\} }\partial _t([u]_h)\cdot \varphi +[a(\cdot ,Du)]_h\cdot D\varphi \mathrm {d}x~=\int _{\Omega \times \left\{ t\right\} }[f]_h\cdot \varphi +[F^{p(\cdot )-2}F]_h\cdot D\varphi \mathrm {d}x \end{aligned}$$

(2.2)

for a.e. $t\in (0,T)$ and all $\varphi \in C^\infty _0(\Omega )$.

2.2 Poincaré type estimate

Our next problem is, that we need a Poincaré inequality. It is only possible to use the elliptic Poincaré inequality slicewise for a.e. times t. For parabolic problems with nonstandard growth, it is not allowed to apply such an estimate, not even slicewise. There exists just a “Luxemburg-version”, see [7], i.e. $\Vert u\Vert _{L^{p(x)}(\Omega )}\le c\Vert Du\Vert _{L^{p(x)}(\Omega )}$ for all $u\in W^{1,p(x)}_0(\Omega )$, where $c>0$. But we need a “classical” Poincaré type inequality. The desired result is given by the following lemma, which is stated in [19].

Lemma 2.3

([19], Lemma 3.9) Let $\Omega \subset \mathbb {R}^n$ a bounded Lipschitz domain and $\gamma _2:=\sup _{\Omega _T}p(\cdot )$. Assume that $u\in C^0([0,T];L^2(\Omega ))\cap W^{p(\cdot )}_0(\Omega _T)$ and the exponent $p(\cdot )$ satisfies the conditions (1.4), (1.5). Then, there exists a constant $c=c(n,\gamma _1,\gamma _2,\mathrm {diam}(\Omega ),\omega (\cdot ))$, such that the two versions of the Poincaré type estimate are valid:

$$\begin{aligned} \int _{\Omega _T}|u|^{p(\cdot )}~\mathrm {d}z\le ~c\left( \Vert u\Vert _{L^\infty (0,T; L^2(\Omega ))}^\frac{4\gamma _2}{n+2}+1\right) \left( \int _{\Omega _T}|Du|^{p(\cdot )}+~1~\mathrm {d}z~\right) \end{aligned}$$

(2.3)

and

$$\begin{aligned} \Vert u\Vert _{L^{p(z)}(\Omega _T)}^{\gamma _1}\le ~c\left( \Vert u\Vert _{L^\infty (0,T;L^2(\Omega ))}^\frac{4\gamma _2}{n+2}+1\right) \left( \int _{\Omega _T}|Du|^{p(\cdot )}+~1~\mathrm {d}z~\right) . \end{aligned}$$

(2.4)

3 Proof of the compact embedding: compactness theorem

In this chapter, we will show some properties concerning distributional nonstandard Sobolev spaces. For usual Sobolev spaces $W^{1,p}(\Omega )$, these results are given in [29, Chapter III], [30, Chapter III]. Moreover, we would like to mention that in [17, Theorem 7.1] the authors proved a very similar assertion to Lemma 3.1.

Since $L^2(\Omega )$ is a Hilbert space which is identified with its dual $L^2(\Omega )\cong (L^2(\Omega ))'$ and in which $L^{p(\cdot ,t)}(\Omega )$ is dense and continuously embedded $\forall t\in [0,T]$, where $p(\cdot ,t)\ge 2$, see [16, Lemma 5.5], we have $L^{p(\cdot ,t)}(\Omega )\hookrightarrow L^2(\Omega )\hookrightarrow L^{p'(\cdot ,t)}(\Omega )$ for all $t\in [0,T]$. The fact that $L^2(\Omega )\cong (L^2(\Omega ))'$ can be demonstrated by the Riesz representation theorem. We denote the dual of $W^{1,p(\cdot ,t)}_0(\Omega )$ by $W^{1,p(\cdot ,t)}(\Omega )'$ and the natural pairing between $W^{1,p(\cdot ,t)}(\Omega )'$ and $W^{1,p(\cdot ,t)}_0(\Omega )$ by $\left\langle \cdot ,\cdot \right\rangle$. Moreover, we have the embeddings $W^{1,p(\cdot ,t)}_0(\Omega )\subset L^2(\Omega )$ and $(L^2(\Omega ))'\subset W^{1,p(\cdot ,t)}(\Omega )'$. Therefore, we can conclude that $W^{1,p(\cdot ,t)}_0(\Omega )\hookrightarrow L^2(\Omega )\cong (L^2(\Omega ))'\hookrightarrow W^{1,p(\cdot ,t)}(\Omega )',$ where the injections are compact. This also allows us to identify the duality product $\left\langle \cdot ,\cdot \right\rangle$ with the inner product between $L^2(\Omega )$ and $W^{1,p(\cdot ,t)}_0(\Omega )$, i.e.

$$\begin{aligned} f(v)=\left\langle f,v\right\rangle =\left\langle f,v\right\rangle _{L^2(\Omega )}=\int _{\Omega }f\cdot v\mathrm {d}x\end{aligned}$$

whenever $f\in L^2(\Omega )\subset W^{1,p(\cdot ,t)}(\Omega )'$ and $v\in W^{1,p(\cdot ,t)}_0(\Omega )$ and $t\in [0,T]$. Next, we consider the Banach space

$$\begin{aligned} W_0(\Omega _T):=\left\{ w\in W^{p(\cdot )}_0(\Omega _T)|w_t\in W^{p(\cdot )}(\Omega _T)'\right\} . \end{aligned}$$

Let $a<0<T<b$. We shall construct an extension of each $u\in W_0(\Omega _T)$ to $\tilde{u}\in W_0(\Omega _I)$, where $I:=(a,b)$. First extend u to (a, 0) and (T, b) (e.g. by symmetry). Let $\Theta \in C^\infty _0(a,b)$ with $\Theta =1$ on (0, T). We define $\tilde{u}=u\cdot \Theta$ and note that $\tilde{u}\in W_0(\Omega _I)$ and $\tilde{u}=u$ on (0, T). Therefore, $\Vert u\Vert _{W(\Omega _T)}\le \Vert \tilde{u}\Vert _{W(\Omega _I)}\le C(\Theta )\Vert u\Vert _{W(\Omega _T)},$ where $C(\Theta )$ depends only on $\Theta$ and $\tilde{u}=0$ in a neighborhood of a and b. Next, we regularize $\tilde{u}$ by the mollifier $u_h(t)=\int _{\mathbb {R}}\tilde{u}(s)\delta _h(t-s)\mathrm {d}s$, where $\delta _h(s)=\frac{1}{h}\delta \left( \frac{s}{h}\right) , \delta \in C^\infty _0(-1,1), \delta \ge 0$ and $\int _{\mathbb {R}}\delta (s)\mathrm {d}s=1$. It follows that $u_h\in C^\infty (\Omega _I)$ and $u_h(\cdot ,a)=0=u_h(\cdot ,b)$ for sufficiently small $h>0$. Moreover, we have $u_h\rightarrow \tilde{u}$ in $W_0(\Omega _I)$ with $\Vert u_h\Vert _{W(\Omega _I)}\le \Vert \tilde{u}\Vert _{W(\Omega _I)}$, cf. Propositions 2.1 and 2.2. By the preceding identification of spaces we have

$$\begin{aligned} \frac{1}{2}\frac{\mathrm {d}}{\mathrm {d}t}\Vert u_h(\cdot ,t)\Vert ^2_{L^2(\Omega )}=\frac{1}{2}\frac{\mathrm {d}}{\mathrm {d}t}\left\langle u_h(\cdot ,t),u_h(\cdot ,t)\right\rangle _{L^2(\Omega )}=\int _\Omega \partial _tu_h(\cdot ,t)\cdot u_h(\cdot ,t)\mathrm {d}x\end{aligned}$$

and this yields $\frac{1}{2}\Vert u_h(\cdot ,t)\Vert ^2_{L^2(\Omega )}\le \Vert \partial _t u_h(\cdot ,s)\Vert _{W^{p(\cdot ,t)}(\Omega \times (a,t))'} \Vert u_h(\cdot ,s)\Vert _{W^{p(\cdot ,t)}(\Omega \times (a,t))}\le \Vert u_h\Vert ^2_{W(\Omega _I)}$ for all $a\le t\le b$. Since $\left\{ u_h\right\}$ is a Cauchy sequence in $W_0(\Omega _I)$, such an estimate on differences $u_h-u_k$ from the sequence shows that it converges (uniformly) to $\tilde{u}$ in $C^0([0,T];L^2(\Omega ))$. Thus, we obtain the following:

Lemma 3.1

Let $n\ge 2$. Assume that $p:\Omega _T\rightarrow [\gamma _1,\gamma _2]$ satisfies (1.4), (1.5). Then $W(\Omega _T)$ is contained in $C^0([0,T];L^2(\Omega ))$. Moreover, if $u\in W_0(\Omega _T)$ then $t\mapsto \Vert u(\cdot ,t)\Vert ^2_{L^2(\Omega )}$ is absolutely continuous on [0, T],

$$\begin{aligned} \frac{\mathrm {d}}{\mathrm {d}t}\int _\Omega |u(\cdot ,t)|^2\mathrm {d}x=2\left\langle \partial _tu(\cdot ,t),u(\cdot ,t)\right\rangle ,~~~\text {for a.e.}~t\in [0,T], \end{aligned}$$

where $\left\langle \cdot ,\cdot \right\rangle$ denotes the duality pairing between $W^{1,p(\cdot ,t)}(\Omega )'$ and $W^{1,p(\cdot ,t)}_0(\Omega )$. Moreover, there is a constant c such that $\Vert u\Vert _{C^0([0,T];L^2(\Omega ))}\le c \Vert u\Vert _{W(\Omega _T)}$ for every $u\in W_0(\Omega _T)$.

The proof of the compactness theorem will be based on the following interpolation lemma, which is established in [19]. Here, we consider $p>2n/(n+2)$ and we will utilize the fact that we have open, bounded Lipschitz domains $\Omega \subseteq \mathbb {R}^n$, the dense embeddings $W^{1,p}(\Omega )\subset L^2(\Omega )$ and $(L^2(\Omega ))'\subset W^{-1,p'}(\Omega )$. Moreover, the injection of $W^{1,p}(\Omega )$ into $L^{q}(\Omega )$ is compact, provided the exponents satisfy $1\le q<p^*$ if $\frac{2n}{n+2}< p <n$ and $q\ge 1$ if $p\ge n$, where $p^*:=np/(n-p)$ is the Sobolev exponent of p. Further, we want to infer from the Interpolation Lemma 3.2 a Gagliardo–Nirenberg inequality. This Gagliardo-Nirenberg inequality we use to conclude the desired compact embedding between $W(\Omega _T)$ and $L^{\hat{p}(\cdot )}(\Omega _T)$ with $\hat{p}(\cdot ):=\max \left\{ 2,p(\cdot )\right\}$, see the Aubin–Lions type Theorem 1.3.

Lemma 3.2

([19], Lemma 3.6) Let $p,r>1$ and $\Omega \subset \mathbb {R}^n$ be an open, bounded Lipschitz domain with $n\ge 2$. Moreover, suppose that the exponent q satisfies the following conditions $q\in [1,p^*)$ if $1< p<n$ or $q\in [1,\infty )$ if $n\le p<\infty$ with the Sobolev exponent $p^*:=\frac{np}{n-p}$, if $p<n$. Then, for each $\eta >0$, there exists some constant $C_\eta$ depending on $\eta ,p,q,r$ and $\Omega$, such that the following interpolation inequality holds:

$$\begin{aligned} \Vert v\Vert _{L^{q}(\Omega )}\le \eta \Vert v\Vert _{W^{1,p}(\Omega )}+C_\eta \Vert v\Vert _{W^{-1,r}(\Omega )}\quad \forall ~v\in W^{1,p}(\Omega )\cap W^{-1,r}(\Omega ). \end{aligned}$$

(3.1)

Our next aim is to conclude a needed version of (3.1). Therefore, we consider a bounded Lipschitz domain $A\subset \mathbb {R}^n$ and suppose that, $v\in W^{1,p}(A)\cap W^{-1,r}(A)$ for some $p>\frac{2n}{n+2}$ and $r>1$. Notice that, the Sobolev’s embedding implies $v\in L^2(A)$. Next, we consider q and s, such that $2<s<q<p^*$ with $p^*=\frac{np}{n-p}$ if $p<n$ and $p^*=\text {any number}>1$ if $p\ge n$. Now, we apply Hölder’s inequality with exponents $\frac{q-s}{q-2}$ and $\frac{s-2}{q-2}$ to $\Vert v\Vert _{L^s(\Omega )}^s$. This yields

$$\begin{aligned} \Vert v\Vert _{L^s(\Omega )}^s=\int _{\Omega }|v|^{2\frac{q-s}{q-2}}|v|^{s-2\frac{q-s}{q-2}}\mathrm {d}x\le \left( \int _\Omega |v|^2\mathrm {d}x\right) ^{\frac{q-s}{q-2}}\left( \int _\Omega |v|^q\mathrm {d}x\right) ^\frac{s-2}{q-2}. \end{aligned}$$

At this stage, we apply (3.1) to the last term on the right-hand side of the previous estimate. Hence, we have

$$\begin{aligned} \Vert v\Vert _{L^s(\Omega )}\le \left( \int _\Omega |v|^2\mathrm {d}x\right) ^{\frac{q-s}{q-2}\frac{1}{s}}\left( \eta \Vert v\Vert _{W^{1,p}(\Omega )}+C_\eta \Vert v\Vert _{W^{-1,r}(\Omega )}\right) ^{q\frac{s-2}{q-2}\frac{1}{s}}. \end{aligned}$$

(3.2)

Moreover, in the case $p\le 2$, we have again by Lemma 3.2

$$\begin{aligned} \Vert v\Vert _{L^2(\Omega )}= & {} \left( \int _A|v|^2\mathrm {d}x\right) ^\frac{2-p}{4} \left( \int _A|v|^2\mathrm {d}x\right) ^\frac{p}{4}\nonumber \\\le & {} \left( \int _A|v|^2\mathrm {d}x\right) ^\frac{2-p}{4}\left( \eta \Vert v\Vert _{W^{1,p}(\Omega )}+C_\eta \Vert v\Vert _{W^{-1,r}(\Omega )}\right) ^\frac{p}{2}. \end{aligned}$$

(3.3)

Now, we are in the situation to prove our compactness theorem 1.3 in the sense of Aubin and Lions.

Proof of Theorem 1.3

Let $\left\{ u_k\right\}$ be a bounded sequence in $W(\Omega _T)$. $W(\Omega _T)$ is reflexive. This is obvious, since $W^{p(\cdot )}(\Omega _T)$ and $W^{p(\cdot )}(\Omega _T)'$ are reflexive Banach spaces. Hence, there is a subsequence, which we denote again by $\left\{ u_k\right\}$, such that $u_k\rightharpoonup u$ weakly in $W(\Omega _T)$. Therefore, we have $u_k\rightharpoonup u$ weakly in $W^{p(\cdot )}(\Omega _T)$ and $\partial _tu_k\rightharpoonup \partial _tu$ weakly in $W^{p(\cdot )}(\Omega _T)'$. We shall show that, up to a subsequence, $u_k\rightarrow u$ strongly in $L^{\hat{p}(\cdot )}(\Omega _T)$. For this aim, we first have to conclude the strong convergence in $L^{\hat{p}(\cdot )}$ on subdomains of $\Omega _T$ with some restriction on the diameter of these subdomains and then, we observe the strong convergence in $L^{\hat{p}(\cdot )}$ on $\Omega _T$ by a covering argument. This is necessary, since in the nonstandard case with p(z)-growth, there do not exist such global arguments respectively estimates as in the standard case. For this reason, we have to bound the maximal oscillation of the exponent $p(\cdot )$ and use this bound to restrict the diameter of the subdomains of $\Omega _T$. Therefore, we fix $R_0\in (0,1]$, such that $\omega (R_0)\le \frac{1}{4n}$ and we consider an open, bounded Lipschitz domain $A\subset \Omega$ and $0\le t_1<t_2\le T$, such that $\mathrm {diam}(A)\le R_0$ and $t_2-t_1\le R_0^2$. Next, we set $p_1:=\inf _Q p(\cdot )$ and $p_2:=\sup _Q p(\cdot )$, where $Q:=A\times (t_1,t_2)$. Then, we define

$$\begin{aligned} q:= {\left\{ \begin{array}{ll} \frac{np_1}{n-\frac{1}{2}p_1}~~~&{}\text {if}~p_1<n,\\ 2p_2~~~&{}\text {if}~p_1\ge n. \end{array}\right. } \end{aligned}$$

(3.4)

In the following, we will show that in the case $p_1<n$, there holds

$$\begin{aligned} p_2<q<p_1^*. \end{aligned}$$

(3.5)

The second inequality is obvious from the definition of q, while the first one follows from (1.4) and the choice of $R_0$:

$$\begin{aligned} p_2-q&=p_2-\frac{np_1}{n-\frac{1}{2}p_1}=\frac{n(p_2-p_1)-\frac{1}{2}p_1p_2}{n-\frac{1}{2}p_1} \le \frac{n\omega (R_0)-\frac{1}{2}}{n-\frac{1}{2}p_1}\le \frac{\frac{1}{4}-\frac{1}{2}}{n-\frac{1}{2}p_1}<0. \end{aligned}$$

Next, we prove that in the case $p_2>2$, there holds

$$\begin{aligned} \frac{q(p_2-2)}{q-2}\le p_1. \end{aligned}$$

(3.6)

In the case $p_1\ge n$, this follows immediately from (1.4) and the fact that $\omega (R_0)\le 1$, since

$$\begin{aligned} q\frac{(p_2-2)}{q-2}-p_1 \le \frac{2\omega (R_0)p_2-4p_2+2p_1}{2p_2-2} \le \frac{2\omega (R_0)p_2-2p_2}{2p_2-2}\le ~0. \end{aligned}$$

In the case $p_1<n$, we first observe that $q>2$, which is a consequence of (1.4), the choice of $R_0$, the fact that $p_2>2$ and the following chain of inequalities:

$$\begin{aligned} 2-q&=~2-\frac{np_1}{n-\frac{1}{2}p_1}~=~\frac{2n-p_1-np_1}{n-\frac{1}{2}p_1}~\le ~\frac{2n-np_2+n\omega (R_0)-p_1}{n-\frac{1}{2}p_1}\\&\le ~\frac{n\omega (R_0)-p_1}{n-\frac{1}{2}p_1}~\le ~\frac{\frac{1}{4}-p_1}{n-\frac{1}{2}p_1}~<~0. \end{aligned}$$

Again from (1.4) and the choice of $R_0$, we obtain

$$\begin{aligned} \frac{q(p_2-2)}{q-2}-p_1~ \le ~\frac{np_1\omega (R_0)-p_1^2}{np_1-2n+p_1} \le ~\frac{\frac{p_1}{4}-p_1^2}{np_1-2n+p_1}~<~0, \end{aligned}$$

which proves (3.6) also in the case $p_1<n$. Therefore, we consider the exponent q from (3.4), which satisfies (3.5) and $2\le \hat{p}(\cdot )\le q$ on Q. Notice that we are allowed to assume that $2\le \hat{p}(\cdot )\le q$ on Q, since $p_2<q$ in the case $p_1<n$ by (3.5) and $2p_2=q$ in the case $p_1\ge n$ by (3.4). Finally, we assume without loss of generality that $u=0$.

First, we want to show that $u_k(\cdot ,t)\rightarrow 0$ in $W^{-1,p_2'}(A)$ for each $t\in [t_1,t_2]$. This will be done for $t=t_1$, with any other case being similar. For $t\in [t_1,t_2]$ we have that

$$\begin{aligned} u_k(\cdot ,t_1)=u_k(\cdot ,t)-\int _{t_1}^t\partial _tu_k(\cdot ,\tau )~\mathrm {d}\tau . \end{aligned}$$

Integration over $[t_1,s]$ with respect to t yields for any $s\in [t_1,t_2]$ the following

$$\begin{aligned} u_k(\cdot ,t_1)&=\frac{1}{s-t_1}\int _{t_1}^su_k(\cdot ,t)\mathrm {d}t-\frac{1}{s-t_1}\int _{t_1}^s\int _{t_1}^t\partial _tu_k(\cdot ,\tau )~d\tau \mathrm {d}t\\&=\frac{1}{s-t_1}\int _{t_1}^su_k(\cdot ,t)\mathrm {d}t-\frac{1}{s-t_1}\int _{t_1}^s(s-t)\partial _tu_k(\cdot ,t)\mathrm {d}t=a_k+b_k. \end{aligned}$$

Since $\partial _tu_k$ is bounded in $W^{p(\cdot )}(Q)'$ and $W^{p(\cdot )}(Q)'\subset L^1(t_1,t_2;W^{-1,p_2'}(A))$, for any $\varepsilon >0$, we can choose s close to $t_1$, such that

$$\begin{aligned} \Vert b_k\Vert _{W^{-1,p_2'}(A)}\le \int _{t_1}^s\Vert \partial _t u_k\Vert _{W^{-1,p_2'}(A)}\mathrm {d}t\le \frac{\varepsilon }{2}. \end{aligned}$$

Next, we conclude from $u_k\rightharpoonup 0$ weakly in $W^{p(\cdot )}(Q)$ that $a_k\rightharpoonup 0$ weakly in $W^{1,p_1}(A)$, as $k\rightarrow \infty$. Therefore, $a_k\rightarrow 0$ strongly in $L^\mathfrak {q}(A)$ for all $\mathfrak {q}<p_1^*$ in the case $p<n$ and $\mathfrak {q}<\infty$ in the case $n\le p$, as $k\rightarrow \infty$ by compactness. Since $2<p_1^*=\frac{np_1}{n-p_1}$ for $p_1<n$ by (3.5), we can conclude from the fact that $a_k\rightarrow 0$ strongly in $L^\mathfrak {q}(A)$, also $a_k\rightarrow 0$ strongly in $L^{2}(A)$, as $k\rightarrow \infty$. Next, we can infer from the continuous embedding $L^{2}(A)\hookrightarrow W^{-1,p'_2}(A)$ and the strong convergence of $a_k$ in $L^{2}(A)$, that $a_k\rightarrow 0$ in $W^{-1,p_2'}(A)$, i.e. we have for sufficiently large k that $\Vert a_k\Vert _{W^{-1,p_2'}(A)}\le \frac{\varepsilon }{2}$. Finally, we have also shown that $u_k(\cdot ,t_1)\rightarrow 0$ in $W^{-1,p_2'}(A)$, as $k\rightarrow \infty$. Second, observe that the continuity of

$$\begin{aligned} W(Q)\hookrightarrow C([t_1,t_2];W^{-1,p_2'}(A)) \end{aligned}$$

shows that $\{\Vert u_k\Vert _{C([t_1,t_2];W^{-1,p_2'}(A))}\}$ is bounded, i.e. $\Vert u_k\Vert _{C([t_1,t_2];W^{-1,p_2'}(A))}\le c\Vert u_k\Vert _{W(Q)}.$ Since $u_k$ is bounded, we can obtain by the Dominated Convergence Theorem, see [24, Theorem 5, p. 648] respectively [29, Theorem 1.4, Chapter III], that

$$\begin{aligned} u_k\rightarrow 0~\text {in}~L^{p_1}(t_1,t_2;W^{-1,p_2'}(A)),~\text {as}~k\rightarrow \infty . \end{aligned}$$

Our next aim is to prove, that $u_k\rightarrow 0$ in $L^{\hat{p}(\cdot )}(Q)$, as $k\rightarrow \infty$. This will be a consequence of the fact, that $u_k\rightarrow 0$ in $L^{\max \left\{ p_2,2\right\} }(Q)$. We start with the case $p_2>2$. Here, we recall the definition of q from (3.4). Due to (3.5), we are allowed to apply (3.2) with $(p_1,p_2,p_2')$ instead of (p, s, r). This yields, for $\eta >0$ that

$$\begin{aligned} \int _Q|u_k|^{p_2}\mathrm {d}z&\le \int _{t_1}^{t_2}\left( \int _A|u_k(\cdot ,t)|^2\mathrm {d}x~\right) ^\frac{q-p_2}{q-2}\\&\quad \times \, \left( \eta \Vert u_k(\cdot ,t)\Vert _{W^{1,p_1}(A)}+C_\eta \Vert u_k(\cdot ,t)\Vert _{W^{-1,p_2'}(A)}\right) ^{q\frac{(p_2-2)}{q-2}}\mathrm {d}t\\&\le 2^{q-1} \int _{t_1}^{t_2}\left( \eta ^{q\frac{(p_2-2)}{q-2}}\Vert u_k(\cdot ,t)\Vert _{W^{1,p_1}(A)}^{q\frac{(p_2-2)}{q-2}}\right. \\&\left. \quad +\,C_\eta ^{q\frac{(p_2-2)}{q-2}}\Vert u_k(\cdot ,t)\Vert _{W^{-1,p_2'}(A)}^{q\frac{(p_2-2)}{q-2}}\right) \mathrm {d}t\times \sup _{t_1< t < t_2}\Vert u_k(\cdot ,t)\Vert _{L^2(A)}^{2\frac{q-p_2}{q-2}}. \end{aligned}$$

Now, we recall (3.6), which allows us to apply Hölder’s inequality and thus obtain, that

$$\begin{aligned} \int _{t_1}^{t_2}\Vert u_k(\cdot ,t)\Vert _{W^{1,p_1}(A)}^{q\frac{ (p_2-2)}{q-2}}\mathrm {d}t~&\le ~c\left( \int _{t_1}^{t_2}\Vert u_k(\cdot ,t)\Vert _{W^{1,p_1}(A)}^{p_1}\mathrm {d}t~\right) ^\frac{q(p_2-2)}{p_1(q-2)}\\&\le ~c\left( \int _Q|u_k|^{p(\cdot )}+~|Du_k|^{p(\cdot )} ~+~1~\mathrm {d}z~\right) ^\frac{q(p_2-2)}{p_1(q-2)}\\&\le ~c\left( \Vert u_k\Vert _{W^{p(\cdot )}(Q)}^{p_2\frac{q(p_2-2)}{p_1(q-2)}}+~1~\right) \le ~c\left( \Vert u_k\Vert _{W^{p(\cdot )}(Q)}^{p_2}+~1~\right) \end{aligned}$$

with a constant $c=c(p_1,p_2,q)$, where we also used (1.9), (3.6) and that $t_2-t_1\le R_0^2\le 1$ and $|A|\le \alpha _n R_0^n\le c$. Plugging this into the previous estimate, we gain

$$\begin{aligned} \int _Q|u_k|^{p_2}\mathrm {d}z&\le \sup _{t_1< t < t_2}\Vert u_k(\cdot ,t)\Vert _{L^2(A)}^{2\frac{q-p_2}{q-2}}\left[ c(p_1,p_2,q)\eta ^{q\frac{p_2-2}{q-2}}\left( \Vert u_k\Vert _{W^{p(\cdot )}(Q)}^{p_2}+~1~\right) \right. \\&\quad +\left. c(q)C_\eta ^{q\frac{(p_2-2)}{q-2}}\int _{t_1}^{t_2}\Vert u_k(\cdot ,t)\Vert _{W^{-1,p_2'}(A)}^{q\frac{(p_2-2)}{q-2}}\mathrm {d}t\right] . \end{aligned}$$

Further, since $u_k$ is bounded in $W^{p(\cdot )}(\Omega _T)$ and by Lemma 3.1, we can infer that

$$\begin{aligned} \int _{Q}|u_k|^{p_2}~\mathrm {d}z&\le ~c_1\cdot \eta ^{q\frac{p_2-2}{q-2}}~+~c_2\int _{t_1}^{t_2}\Vert u_k(\cdot ,t)\Vert _{W^{-1,p_2'}(A)}^{q\frac{(p_2-2)}{q-2}}\mathrm {d}t, \end{aligned}$$

(3.7)

where $c_1=c_1(n,\gamma _1,\gamma _2,\sup _{k\in \mathbb {N}}\Vert u_k\Vert _{L^\infty (t_1,t_2;L^2(A))},\sup _{k\in \mathbb {N}}\Vert u_k\Vert _{W^{p(\cdot )}(Q)})$ and $c_2=c_2(\eta ,n,\gamma _1,\gamma _2,\sup _{k\in \mathbb {N}}\Vert u_k\Vert _{L^\infty (t_1,t_2;L^2(A))})$. Since, the dependencies on $p_1,~p_2$ and q is continuous, it can be replaced by a dependencies on $\gamma _1$ and $\gamma _2$. Next, we consider the case $p_2\le 2$. Here, we use (3.3) applied with (p, r) replaced by $(p_1,p_2')$ to infer that

$$\begin{aligned} \int _Q|u_k|^2\mathrm {d}z&\le \int _{t_1}^{t_2}\left( \int _A|u_k(\cdot ,t)|^2\mathrm {d}x\right) ^{1-\frac{p_1}{2}}\\&\quad \times \, \left( \eta \Vert u_k(\cdot ,t)\Vert _{W^{1,p_1}(A)}+C_\eta \Vert u_k(\cdot ,t)\Vert _{W^{-1,p_2'}(A)}\right) ^{p_1}\mathrm {d}t\\&\le ~2^{p_1-1}\eta ^{p_1}\sup _{t_1< t< t_2}\Vert u_k(\cdot ,t)\Vert _{L^2(A)}^{1-\frac{p_1}{2}}\left( \Vert u_k\Vert _{W^{p(\cdot )}(Q)}^{p_1}+1\right) \\&\quad + \, c C_\eta ^{p_1}\sup _{t_1< t < t_2}\Vert u_k(\cdot ,t)\Vert _{L^2(A)}^{1-\frac{p_1}{2}}\Vert u_k\Vert _{L^{p_1}(t_1,t_2;W^{-1,p_2'}(A))}^{p_1} \end{aligned}$$

with a constant $c=c(p_1,p_2)$, where we utilized $\Vert u_k\Vert _{W^{1,p_1}(Q)}\le c\Vert u_k\Vert _{W^{p(\cdot )}(Q)}$ by the compact embedding $W^{p(\cdot )}(Q)\hookrightarrow W^{1,p_1}(Q)$. Here, we have also used that $t_2-t_1\le R_0^2\le 1$ and $|A|\le \alpha _n R_0^n\le c$. Thus, since $u_k$ is bounded in $W^{p(\cdot )}(\Omega _T)$ and by Lemma 3.1, we can infer that

$$\begin{aligned} \int _{Q}|u_k|^2~\mathrm {d}z\le ~c_1\cdot \eta ^{p_1}~+~c_2\Vert u_k\Vert _{L^{p_1}(t_1,t_2;W^{-1,p_2'}(A))}^{p_1}, \end{aligned}$$

(3.8)

where $c_1=c_1(n,\gamma _1,\gamma _2,\sup _{k\in \mathbb {N}}\Vert u_k\Vert _{L^\infty (t_1,t_2;L^2(A))},\sup _{k\in \mathbb {N}}\Vert u_k\Vert _{W^{p(\cdot )}(Q)})$ and the constant $c_2$ depends on $\eta ,n,\gamma _1, \gamma _2,\sup _{k\in \mathbb {N}}\Vert u_k\Vert _{L^\infty (t_1,t_2;L^2(A))})$. Since, $u_k\rightarrow 0$ in $L^{p_1}(t_1,t_2;W^{-1,p_2'}(A))$ and the last term on the right-hand side of (3.7) and (3.8) converges to zero, so

$$\begin{aligned} \limsup _{k\rightarrow \infty }\int _{Q}|u_k|^{\max \left\{ 2,p_2\right\} }~\mathrm {d}z\le ~c_1\max \left\{ \eta ^{q\frac{p_2-2}{q-2}},\eta ^{p_1}\right\} , \end{aligned}$$

where we also used the fact $L^{p_1}(t_1,t_2;W^{-1p_2'}(A))\hookrightarrow L^{q\frac{p_2-2}{q-2}}(t_1,t_2;W^{-1p_2'}(A))$ by (3.6). Since, $\eta >0$ is chosen arbitrary, this upper limit is 0. This shows $u_k\rightarrow 0$ strongly in $L^{\max \left\{ 2,p_2\right\} }(Q)$, as $k\rightarrow \infty$ for every open, bounded Lipschitz domain $Q=A\times (t_1,t_2)$, provided the condition $\mathrm {diam}(A)\le R_0=R_0(n,\omega (\cdot ))$ and $t_2-t_1\le R_0^2$ holds. Moreover, the compact embedding of $L^{p_2}$ into $L^{p(\cdot )}$ implies that $u_k\rightarrow 0$ strongly in $L^{\hat{p}(\cdot )}(Q)$, as $k\rightarrow \infty$ for every open, bounded Lipschitz domain $Q=A\times (t_1,t_2)$, provided the condition $\mathrm {diam}(A)\le R_0=R_0(n,\omega (\cdot ))$ and $t_2-t_1\le R_0^2$ holds. Since, we have shown the desired strong convergence in $L^{\hat{p}(\cdot )}$ on subdomains Q of $\Omega _T$ with the restriction on the diameter of these Q, our next goal is to deduce the strong convergence in $L^{\hat{p}(\cdot )}$ on $\Omega _T$. This will be done by a covering argument. Therefore, we choose a family of dyadic cuboids $\left\{ C_i\right\} _{i=1}^\infty$ such that $\bigcup _{i=1}^\infty C_i=\mathbb {R}^{n+1}$, where $C_i$ denotes the cuboid

$$\begin{aligned} C_{\frac{R_0}{2},\left( \frac{R_0}{2}\right) ^2}(z_i):=\left\{ x\in \mathbb {R}^n,s\in \mathbb {R}\left| \right. |x_i-x|< \frac{1}{2}R_0,~|t_i-s|< \frac{1}{4}R_0^2,~1\le i\le n\right\} \end{aligned}$$

with center in $z_i=(x_i,t_i)$, side length $\frac{1}{2}R_0$ and height $\left( \frac{1}{2}R_0\right) ^2$. These cuboids, we use to partition the $\mathbb {R}^{n+1}$ into dyadic cuboids $C_i$, where $1\le i\le \infty$. Since, $Q\subseteq \Omega _T$ was arbitrary, we can consider $Q_i=\tilde{C}_i\cap \Omega _T\subseteq \Omega _T,~i=1,\ldots ,M$, such that $\Omega _T=\bigcup _{i=1}^{M}Q_i$, where $\tilde{C}_i=C_{R_0,(R_0)^2}(z_i)$. Since, every cuboid in $\mathbb {R}^n$ has a Lipschitz boundary, $C_i$ respectively $\tilde{C}_i$ are open, bounded domains and the intersection of two open, bounded domains yields again an open, bounded domain, it follows that $Q_i$ are open, bounded Lipschitz domains. Therefore, we can conclude that $u_k\rightarrow u$ strongly in $L^{\hat{p}(\cdot )}(Q_i)$, as $k\rightarrow \infty$, for all $i\in \left\{ 1,\ldots ,M\right\}$. Moreover, the covering of $\Omega _T$ yields that $u_k\rightarrow u$ strongly in $L^{\hat{p}(\cdot )}(\Omega _T)$ and thus, $W(\Omega _T)\hookrightarrow L^{\hat{p}(\cdot )}(\Omega _T)$ compact as desired. This completes the proof. $\square$

4 Proof of Theorems 1.4 and 1.6

We start with the

Proof of Theorem 1.4

First of all, we construct a sequence of the Galerkin’s approximations, where the limit of this sequence is equal to the solution in (1.15). Therefore, $\left\{ \phi _i(x )\right\} _{i=1}^{\infty }\subset W^{1,\gamma _2}_0(\Omega )$ is an orthonormal basis in $L^2(\Omega )$. Now, we fix a positive integer m and define the approximate solution to (1.15) as follows

$$\begin{aligned} u^{(m)}(z):=\sum _{i=1}^mc_i^{(m)}(t)\phi _i(x), \end{aligned}$$

where the coefficients $c_i^{(m)}(t)$ are defined via the identity

$$\begin{aligned} \int _\Omega \left( u^{(m)}_t\phi _i(x)~+~\left( a(x,t,Du^{(m)})~-~|F|^{p(\cdot ,t)-2}F\right) D\phi _i(x)~-~f\phi _i(x)\right) \mathrm {d}x=0, \end{aligned}$$

(4.1)

for $i=0,\ldots ,m$ and $t\in (0,T)$ with the initial condition

$$\begin{aligned} c^{(m)}_i(0)=\int _\Omega g(\cdot ,0)\phi _i\mathrm {d}x,~~~i=1,\dots ,m. \end{aligned}$$

Then, the equation (4.1) together with the initial boundary condition generates a system of m ordinary differential equations:

$$\begin{aligned} {\left\{ \begin{array}{ll} \left( c^{(m)}_i\right) '(t)=F_i\left( t,c^{(m)}_1(t),\dots ,c^{(m)}_m(t)\right) ,\\ c^{(m)}_i(0)=\displaystyle \int _\Omega g(\cdot ,0)\phi _i\mathrm {d}x,~~~i=1,\dots ,m, \end{array}\right. } \end{aligned}$$

(4.2)

where we abbreviated

$$\begin{aligned} F_i(t,\cdot ):= -\int _\Omega (a(\cdot ,t,Du^{(m)})-|F(\cdot ,t)|^{p(\cdot ,t)-2}F(\cdot ,t))D\phi _i(x)-f(\cdot ,t)\phi _i(x)\mathrm {d}x, \end{aligned}$$

since $\left\{ \phi _i(x)\right\}$ is orthonormal in $L^2(\Omega )$. From this starting point, we will conclude the existence result to the Dirichlet problem (1.15). By [26, Theorem 1.44, p. 25]—see also [26, p. 240 ff.]—we know that, there is for every finite system (4.2) a solution $c^{(m)}_i(t), i=1,\dots ,m$ on the interval $(0,T_m)$ for some $T_m>0$. First, we multiply the equation (4.1) by the coefficients $c^{(m)}_i(t), i=1,\dots ,m$. Then, we need a priori estimates that permit us to extend the solution to the whole domain $(0,T_m)$. Therefore, we integrate the equation over $(0,\tau )$ for an arbitrarily $\tau \in (0,T_m)$. Next, we sum the resulting equation over $i=1,\ldots ,m$. Therefore, it follows

$$\begin{aligned} \int _{\Omega _\tau }\partial _tu^{(m)}\cdot u^{(m)}+ \left( a(z,Du^{(m)})-|F|^{p(\cdot )-2}F\right) \cdot Du^{(m)}-fu^{(m)}\mathrm {d}z=0 \end{aligned}$$

(4.3)

for a.e. $\tau \in (0,T_m)$. Next, we convert the first term on the left-hand side of (4.3) as follows

$$\begin{aligned} \int _{\Omega _\tau }\partial _tu^{(m)}\cdot u^{(m)}\mathrm {d}z~&=\frac{1}{2}\int _{\Omega _\tau }\partial _t[u^{(m)}]^2\mathrm {d}z\\&=\frac{1}{2}\int _{\Omega }|u^{(m)}(\cdot ,\tau )|^2\mathrm {d}x-~\frac{1}{2}\int _{\Omega }|u^{(m)}(\cdot ,0)|^2\mathrm {d}x\\&\ge \frac{1}{2}\int _{\Omega }|u^{(m)}(\cdot ,\tau )|^2\mathrm {d}x-\frac{1}{2}\int _{\Omega }|g(\cdot ,0)|^2\mathrm {d}x\end{aligned}$$

for a.e. $\tau \in (0,T_m)$, since $g(\cdot ,0)\in L^2(\Omega ), \left\{ \phi _i\right\} _{i=1}^\infty \subset L^2(\Omega )$ and

$$\begin{aligned} \int _{\Omega }|u^{(m)}(\cdot ,0)|^2\mathrm {d}x&=\int _\Omega \big |\sum _{i=1}^mc_i^{(m)}(0)\phi _i(x)\big |^2\mathrm {d}x\\&=\int _\Omega \big |\sum _{i=1}^m\int _\Omega g(\cdot ,0)\phi _i(x)\mathrm {d}x\phi _i(x)\big |^2\mathrm {d}x\\&\le \int _\Omega \big |\sum _{i=1}^\infty \int _\Omega g(\cdot ,0)\phi _i(x)dx\phi _i(x)\big |^2\mathrm {d}x=\int _\Omega |g(\cdot ,0)|^2\mathrm {d}x, \end{aligned}$$

cf. [19]. Therefore, it holds

$$\begin{aligned} \int _{\Omega _\tau }\partial _tu^{(m)}\cdot u^{(m)}\mathrm {d}z\ge \frac{1}{2}\int _{\Omega }|u^{(m)}(\cdot ,\tau )|^2\mathrm {d}x-\frac{1}{2}\Vert g(\cdot ,0)\Vert _{L^2(\Omega )}^2 \end{aligned}$$

for a.e. $\tau \in (0,T_m)$. Now, we have from (4.3) that

$$\begin{aligned}&\frac{1}{2}\int _{\Omega }|u^{(m)}(\cdot ,\tau )|^2\mathrm {d}x+\int _{\Omega _\tau }a(z,Du^{(m)})\cdot Du^{(m)}\mathrm {d}z\nonumber \\&\quad \le \frac{1}{2}\Vert g(\cdot ,0)\Vert _{L^2(\Omega )}^2~ +\int _{\Omega _\tau }|F|^{p(\cdot )-2}F\cdot Du^{(m)}+fu^{(m)}\mathrm {d}z \end{aligned}$$

(4.4)

for a.e. $\tau \in (0,T_m)$. Using the coercivity condition (1.7) on left-hand side of (4.4) and estimating the right-hand side of (4.4) by the absolute value and (1.14), then we get the following estimate

$$\begin{aligned}&\frac{1}{2}\int _{\Omega }|u^{(m)}(\cdot ,\tau )|^2\mathrm {d}x+\frac{\nu }{c(\gamma _1,\gamma _2)}\int _{\Omega _\tau }|Du^{(m)}|^{p(\cdot )} \mathrm {d}z-c\int _{\Omega _\tau }(1+|v|^{p(\cdot )})\mathrm {d}z\nonumber \\&\quad \le \frac{1}{2}\Vert g(\cdot ,0)\Vert _{L^2(\Omega )}^2~+\int _{\Omega _\tau }|F|^{p(\cdot )-1} |Du^{(m)}|\mathrm {d}z+c\Vert f\Vert _{W^{\gamma _1'}(\Omega _\tau )}\Vert u^{(m)}\Vert _{W^{\gamma _1}(\Omega _\tau )}, \end{aligned}$$

(4.5)

where $c=c(\gamma _1,\gamma _2,\nu ,L)$. Now, we adopt Young’s inequality to the second term on the right-hand side of (4.5) with the exponents $p(\cdot )$ and $\frac{p(\cdot )}{p(\cdot )-1}$, where we use the factors $\varepsilon ^{-\frac{1}{p(\cdot )}}\cdot \varepsilon ^\frac{1}{p(\cdot )}=1$ and $\varepsilon ^{-\frac{1}{\gamma _1}}\cdot \varepsilon ^\frac{1}{\gamma _1}=1, \varepsilon \in (0,1)$ and get the following estimate

$$\begin{aligned}&\frac{1}{2}\int _{\Omega }|u^{(m)}(\cdot ,\tau )|^2\mathrm {d}x+\frac{\nu }{c(\gamma _1,\gamma _2)}\int _{\Omega _\tau }|Du^{(m)}|^{p(\cdot )}\mathrm {d}z-c\int _{\Omega _\tau }1+|v|^{p(\cdot )}\mathrm {d}z\nonumber \\&\quad \le \frac{1}{2}\Vert g(\cdot ,0)\Vert _{L^2(\Omega )}^2+\int _{\Omega _\tau }\frac{p(\cdot )-1}{p(\cdot )}\varepsilon ^\frac{1}{1-p(\cdot )}|F|^{p(\cdot )}+\frac{1}{p(\cdot )}\varepsilon |Du^{(m)}|^{p(\cdot )}\mathrm {d}z\nonumber \\&\qquad +\,\varepsilon ^{-\frac{1}{\gamma _1}}\Vert f\Vert _{L^{\gamma _1'}(0,\tau ;W^{-1,\gamma _1'}(\Omega ))}\varepsilon ^\frac{1}{\gamma _1}\Vert u^{(m)}\Vert _{L^{\gamma _1}(0,\tau ;W^{1,\gamma _1}(\Omega ))} \nonumber \\&\quad \le \frac{1}{2}\Vert g(\cdot ,0)\Vert _{L^2(\Omega )}^2+\frac{\gamma _2-1}{\gamma _1}\varepsilon ^\frac{1}{1-\gamma _1}\left( \int _{\Omega _\tau } |F|^{p(\cdot )}\mathrm {d}z+\Vert f\Vert _{L^{\gamma _1'}(0,\tau ;W^{-1,\gamma _1'}(\Omega ))}^{\gamma _1'}\right) \nonumber \\&\qquad +\frac{1}{\gamma _1}\varepsilon \left( \int _{\Omega _\tau }|Du^{(m)}|^{p(\cdot )}\mathrm {d}z+\int _{\Omega _\tau }|u^{(m)}|^{\gamma _1}+|Du^{(m)}|^{\gamma _1}\mathrm {d}z\right) \end{aligned}$$

(4.6)

for a.e. $\tau \in (0,T_m)$ with a constant $c=c(\gamma _1,\gamma _2,\nu ,L)$, where we used Young’s estimate with exponents $\frac{1}{\gamma _1}+\frac{\gamma _1-1}{\gamma _1}=1$ for the last estimate. Next, we apply the standard Poincaré inequality slicewise to get the following estimate

$$\begin{aligned}&\frac{1}{2}\int _{\Omega }|u^{(m)}(\cdot ,\tau )|^2\mathrm {d}x+\frac{\nu }{c(\gamma _1,\gamma _2)}\int _{\Omega _\tau }|Du^{(m)}|^{p(\cdot )}\mathrm {d}z-c\int _{\Omega _\tau }1+|v|^{p(\cdot )}\mathrm {d}z\\&\quad \le \frac{1}{2}\Vert g(\cdot ,0)\Vert _{L^2(\Omega )}^2+\frac{\gamma _2-1}{\gamma _1}\varepsilon ^\frac{1}{1-\gamma _1}\left( \int _{\Omega _\tau } |F|^{p(\cdot )}\mathrm {d}z+\Vert f\Vert _{L^{\gamma _1'}(0,\tau ;W^{-1,\gamma _1'}(\Omega ))}^{\gamma _1'}\right) \\&\qquad +\left( \frac{1}{\gamma _1}+c_*\right) \varepsilon \left( \int _{\Omega _\tau }|Du^{(m)}|^{p(\cdot )}+1\mathrm {d}z\right) \end{aligned}$$

with a constant $c=c(\gamma _1,\gamma _2,\nu ,L)$ and the constant $c_*$, which depends on $n,\gamma _1,\gamma _2$ and $\mathrm {diam}(\Omega )$, where we used

$$\begin{aligned} \int _{\Omega _\tau }~|Du^{(m)}|^{\gamma _1}\mathrm {d}z\le c(\gamma _2)\int _{\Omega _\tau }~|Du^{(m)}|^{p(\cdot )}+~1\mathrm {d}z. \end{aligned}$$

This inequality can be converted as follows

$$\begin{aligned}&\int _{\Omega }|u^{(m)}(\cdot ,\tau )|^2\mathrm {d}x+2\left( \frac{\nu }{c}-\left( \frac{1}{\gamma _1}+c_*\right) \varepsilon \right) \int _{\Omega _\tau }|Du^{(m)}|^{p(\cdot )}\mathrm {d}z-c\int _{\Omega _\tau }1+|v|^{p(\cdot )}\mathrm {d}z\\&\quad \le \Vert g(\cdot ,0)\Vert _{L^2(\Omega )}^2+2\frac{\gamma _2-1}{\gamma _1}\varepsilon ^\frac{1}{1-\gamma _1}\left( \int _{\Omega _\tau } |F|^{p(\cdot )}\mathrm {d}z+\Vert f\Vert _{L^{\gamma _1'}(0,\tau ;W^{-1,\gamma _1'}(\Omega ))}^{\gamma _1'}\right) +c_*\varepsilon , \end{aligned}$$

where $c=c(\gamma _1,\gamma _2,\nu ,L)$ and $c_*=c(n,\gamma _1,\gamma _2,\mathrm {diam}(\Omega ))$. Next, we choose $\varepsilon$, such that

$$\begin{aligned} 2\left( \frac{\nu }{c(\gamma _1,\gamma _2)}-\left( \frac{1}{\gamma _1}+c_*\right) \varepsilon \right) \ge \frac{\nu }{c(\gamma _1,\gamma _2)}. \end{aligned}$$

Hence, we have:

$$\begin{aligned}&\int _{\Omega }|u^{(m)}(\cdot ,\tau )|^2\mathrm {d}x+\int _{\Omega _\tau }|Du^{(m)}|^{p(\cdot )}\mathrm {d}z\\&\quad \le c\Vert g(\cdot ,0)\Vert _{L^2(\Omega )}^2 +c\left( \int _{\Omega _\tau } |F|^{p(\cdot )}+|v|^{p(\cdot )}+1\mathrm {d}z\Vert f\Vert _{L^{\gamma _1'}-W^{-1,\gamma _1'}}^{\gamma _1'}+1\right) , \end{aligned}$$

where $c=c(n,\gamma _1,\gamma _2,\nu ,L,\mathrm {diam}(\Omega ))$. This estimate holds for a.e. $\tau \in (0,T_m)$. Therefore, we have that $u^{(m)}$ is uniformly bounded in $L^\infty (0,T_m;L^2(\Omega ))$ and $Du^{(m)}$ is uniformly bounded in $L^{p(\cdot )}(\Omega _{T_m})$. Now, we can estimate the right-hand side from above by the bound $\mathcal {M}_0$ introduced in (1.18). This yields

$$\begin{aligned} \sup _{0\le t\le T_m}\int _{\Omega }|u^{(m)}(\cdot ,\tau )|^2\mathrm {d}x+\int _{\Omega _{T_m}}|Du^{(m)}|^{p(\cdot )}\mathrm {d}z\le c \Vert g(\cdot ,0)\Vert _{L^2(\Omega )}^2+c\cdot \mathcal {M}_0, \end{aligned}$$

(4.7)

where $c=c(n,\gamma _1,\gamma _2,\nu ,L,\mathrm {diam}(\Omega ))$. Next, we apply the Poincaré type inequality (2.4) to get an uniform $L^{p(\cdot )}$-bound for $u^{(m)}$ in the following way:

$$\begin{aligned} \Vert u^{(m)}\Vert _{L^{p(\cdot )}(\Omega _{T_m})}^{\gamma _1} \le c\left( \Vert u^{(m)}\Vert _{L^\infty (0,T_m;L^2(\Omega ))}^\frac{4\gamma _2}{n+2}+1\right) \left( \int _{\Omega _{T_m}}|Du^{(m)}|^{p(\cdot )}+1\mathrm {d}z\right) \end{aligned}$$

where $c=c(n,\gamma _1,\gamma _2,\mathrm {diam}(\Omega ),\omega (\cdot ))$. Using (4.7) then we have the following estimate

$$\begin{aligned} \Vert u^{(m)}\Vert _{L^{p(\cdot )}(\Omega _{T_m})}\le ~c\left[ \left( \Vert u^{(m)}\Vert _{L^\infty (0, T_m;L^2(\Omega ))}^{\frac{4\gamma _2}{n+2}}+1\right) \left( \Vert g(\cdot ,0)\Vert ^2_{L^2(\Omega )}+\mathcal {M}_0\right) \right] ^\frac{1}{\gamma _1} \end{aligned}$$

where $c=c(n,\gamma _1,\gamma _2,\nu ,L,\mathrm {diam}(\Omega ),\omega (\cdot ))$. Applying again (4.7), we get the following $L^{p(\cdot )}$-bound for $u^{(m)}$:

$$\begin{aligned} \Vert u^{(m)}\Vert _{L^{p(\cdot )}(\Omega _{T_m})}\le c\left( \Vert g(\cdot ,0)\Vert ^2_{L^2(\Omega )}+\mathcal {M}_0\right) ^{\left( \frac{2\gamma _2}{n+2}+1\right) \frac{1}{\gamma _1}} \end{aligned}$$

with a constant $c=c(n,\gamma _1,\gamma _2,\nu ,L,\mathrm {diam}(\Omega ),\omega (\cdot ))$, where we finally used Young’s inequality. Therefore, we have shown that $u^{(m)}$ is uniformly bounded in $W^{p(\cdot )}(\Omega _{T_m})$ and $L^\infty (0,T_m;L^2(\Omega ))$ independently of m. Thus, the solution of system (4.2) can be continued to the maximal interval (0, T).

Next, we want to derive an uniform bound for $\partial _t u^{(m)}$ in $W^{p(\cdot )}(\Omega _T)'$. Therefore, we define a subspace of the set of admissible test functions

$$\begin{aligned} \mathcal {W}_m(\Omega _T):=\left\{ \eta :\eta =\sum _{i=1}^md_i\phi _i,~d_i\in C^1([0,T])\right\} \subset W^{p(\cdot )}_0(\Omega _T). \end{aligned}$$

Then, we choose a test function

$$\begin{aligned} \varphi (z)=\sum _{i=1}^m d_i(t)\phi _i(x)\in \mathcal {W}_m(\Omega _T)~~~\text {with}~d_i(0)=d_i(T)=0. \end{aligned}$$

Note that $\partial _t\varphi$ exists, since the coefficients $d_i(t)$ lie in $C^1([0,T])$. Moreover, we know that $C^1([0,T],W^{1,\gamma _2}_0(\Omega _T))\subset W^{p(\cdot )}_0(\Omega _T)$ and therefore, we have also $\varphi \in W^{p(\cdot )}_0(\Omega _T)$. Thus, we can conclude by the definition of $u^{(m)}$ and (4.1) that

$$\begin{aligned} -\int _{\Omega _T}u^{(m)}\varphi _t\mathrm {d}z&= \int _{\Omega _T}u^{(m)}_t\varphi \mathrm {d}z=-\int _{\Omega _T}[a(z,Du^{(m)})\\&\quad - \, |F|^{p(\cdot )-2}F]\cdot D\varphi \mathrm {d}z+\left\langle \left\langle f,\varphi \right\rangle \right\rangle _{\Omega _T}. \end{aligned}$$

Then, we derive from the growth condition (1.16), the generalized Hölder’s inequality (1.8), (1.14) and the fact that $L^{\gamma _1'}(0,T;W^{-1,\gamma _1'}(\Omega ))\subset W^{p(\cdot )}(\Omega _T)'$ implies $f\in W^{p(\cdot )}(\Omega _T)'$, the following

$$\begin{aligned} \left| \int _{\Omega _T}u^{(m)}\varphi _t\mathrm {d}z\right|&\le \int _{\Omega _T}\left( |a(z,Du^{(m)})|+|F|^{p(\cdot )-1}\right) \cdot |D\varphi |\mathrm {d}z+\left\langle \left\langle f,\varphi \right\rangle \right\rangle _{\Omega _T}\\&\le \int _{\Omega _T} \left( |a(z,Du^{(m)})|+|F|^{p(\cdot )-1} \right) \cdot \left( |D\varphi |+|\varphi |\right) \mathrm {d}z\\&\quad + \, c(\gamma _1,\gamma _2)\Vert f\Vert _{W^{p(\cdot )}(\Omega _T)'}\Vert \varphi \Vert _{W^{p(\cdot )}(\Omega _T)}\\&\le c~\left[ \Vert (1+|Du^{(m)}|^{p(\cdot )-1}\right. \\&\left. \quad + \, |v|^{p(\cdot )-1}+|F|^{p(\cdot )-1})\Vert _{L^{p'(\cdot )}(\Omega _T)} +\Vert f\Vert _{W^{p(\cdot )}(\Omega _T)'}\right] \\&\quad \times \, \left[ \Vert \varphi \Vert _{W^{p(\cdot )}(\Omega _T)}\right] , \end{aligned}$$

where $c=c(\gamma _1,\gamma _2,L)$. Next, we consider the term

$$\begin{aligned} \Vert (1+|Du^{(m)}|^{p(\cdot )-1}+|v|^{p(\cdot )-1}+|F|^{p(\cdot )-1})\Vert _{L^{p'(\cdot )}(\Omega _T)} \end{aligned}$$

and use (1.9) to get the following bound

$$\begin{aligned}&\Vert (1+|Du^{(m)}|^{p(\cdot )-1}+|v|^{p(\cdot )-1}+|F|^{p(\cdot )-1})\Vert _{L^{p'(\cdot )}(\Omega _T)}\\&\quad \le \left( \int _{\Omega _T}|(1+|Du^{(m)}|^{p(\cdot )-1}+|v|^{p(\cdot )-1}+|F|^{p(\cdot )-1})|^{p'(\cdot )}\mathrm {d}z+1\right) ^\frac{1}{\gamma _1} \\&\quad \le \left( \int _{\Omega _T}(1+|Du^{(m)}|^{p(\cdot )}+|v|^{p(\cdot )}+|F|^{p(\cdot )})\mathrm {d}z+1\right) ^\frac{1}{\gamma _1}\\&\quad \le c(\gamma _1,\gamma _2)(\mathcal {M}_0)^\frac{1}{\gamma _1}\le c(\gamma _1,\gamma _2,\mathcal {M}_0). \end{aligned}$$

Summarized, we have for every $\varphi \in \mathcal {W}_m(\Omega _T)\subset W^{p(\cdot )}_0(\Omega _T)$ and an arbitrary m that

$$\begin{aligned} \left| \int _{\Omega _T}u^{(m)}\varphi _t\mathrm {d}z\right| \le c\Vert \varphi \Vert _{W^{p(\cdot )}(\Omega _T)} \end{aligned}$$

with a constant $c=c(\gamma _1,\gamma _2,L,\Vert f\Vert _{W^{p(\cdot )}(\Omega _T)'},\mathcal {M}_0)$, where c is independent of m. This shows that $u_t^{(m)}\in W^{p(\cdot )}(\Omega _T)'$ with the estimate

$$\begin{aligned} \Vert u_t^{(m)}\Vert _{W^{p(\cdot )}(\Omega _T)'}\le c(\gamma _1,\gamma _2,L,\Vert f\Vert _{W^{p(\cdot )}(\Omega _T)'},\mathcal {M}_0). \end{aligned}$$

Therefore, we have an uniform bound of $u_t^{(m)}$ in $W^{p(\cdot )}(\Omega _T)'$ and it follows that

$$\begin{aligned} {\left\{ \begin{array}{ll} u^{(m)}\in W^{p(\cdot )}_0(\Omega _T)\subseteq L^{\gamma _1}(0,T;W^{1,\gamma _1}_0(\Omega ))\\ u^{(m)}_t\in W^{p(\cdot )}(\Omega _T)'\subseteq L^{\gamma _2'}(0,T;W^{-1,\gamma _2'}(\Omega )) \end{array}\right. } \end{aligned}$$

are bounded. This imply the following weak convergences for the sequence $\left\{ u^{(m)}\right\}$ (up to a subsequence):

$$\begin{aligned} {\left\{ \begin{array}{ll} u^{(m)}\rightharpoonup ^*u~\text {weakly* in}~L^\infty (0,T;L^2(\Omega )),\\ Du^{(m)}\rightharpoonup Du~\text {weakly in}~L^{p(\cdot )}(\Omega _T,\mathbb {R}^n),\\ u^{(m)}_t\rightharpoonup u_t~\text {weakly in}~W^{p(\cdot )}(\Omega _T)'. \end{array}\right. } \end{aligned}$$

Moreover, by Theorem 1.3 we can conclude that the sequence $\left\{ u^{(m)}\right\}$ (up to a subsequence) converges strongly in $L^{\hat{p}(\cdot )}(\Omega _T)$ with $\hat{p}(\cdot ):=\max \left\{ 2,p(\cdot )\right\}$ to some function $u\in W(\Omega _T)$. Thus, we get the desired convergences

$$\begin{aligned} {\left\{ \begin{array}{ll} u^{(m)}\rightarrow u~\text {strongly in}~L^{\hat{p}(\cdot )}(\Omega _T)\\ u^{(m)}\rightarrow u~\text {a.e. in}~\Omega _T \end{array}\right. } \end{aligned}$$

for the sequence $\left\{ u^{(m)}\right\}$ (up to a subsequence). Further, the growth assumption of $a(z,\cdot )$ and the energy estimate (4.7) imply that the sequence $\left\{ a(z,Du^{(m)})\right\} _{m\in \mathbb {N}}$ is bounded in $L^{p'(\cdot )}(\Omega _T,\mathbb {R}^n)$. Consequently, after passing to a subsequence once more, we can find a limit map $A_0\in L^{p'(\cdot )}(\Omega _T,\mathbb {R}^n)$ with

$$\begin{aligned} a(z,Du^{(m)})\rightarrow A_0~~~\text {as}~m\rightarrow \infty . \end{aligned}$$

(4.8)

Our next aim is to show that $A_0=a(z,Du)$ for almost every $z\in \Omega _T$. First of all, we should mention that each of $u^{(m)}$ satisfies the identity (4.1) with a test function $\varphi \in \mathcal {W}_m(\Omega _T)$. This follows by the method of construction, see [7]. Then, we fix an arbitrary $m\in \mathbb {N}$. Thus, we have for every $s\le m$

$$\begin{aligned} -\int _{\Omega _T}u^{(m)}_t\varphi +[a(z,Du^{(m)})+|F|^{p(\cdot )-2}F]D\varphi -f\varphi \mathrm {d}z=0 \end{aligned}$$

for all test functions $\varphi \in \mathcal {W}_s(\Omega _T)$. Passing to the limit $m\rightarrow \infty$, we can conclude that for all test functions $\varphi \in \mathcal {W}_s(\Omega _T)$

$$\begin{aligned} -\int _{\Omega _T}u_t\varphi +[A_0+|F|^{p(\cdot )-2}F]D\varphi -f\varphi \mathrm {d}z=0 \end{aligned}$$

(4.9)

with an arbitrary $s\in \mathbb {N}$, by the convergence from above. Therefore, it follows that the identity (4.9) holds for every $\varphi \in W^{p(\cdot )}_0(\Omega _T)$. According to monotonicity assumption (1.3), we know that for every $w\in \mathcal {W}_s(\Omega _T), s\le m$

$$\begin{aligned} \int _{\Omega _T}[a(z,Du^{(m)})-a(z,Dw)]D(u^{(m)}-w)\mathrm {d}z\ge 0. \end{aligned}$$

(4.10)

Moreover, it follows from (4.1), the conclusion from above and a test function $\varphi =u^{(m)}-w$ with $w\in \mathcal {W}_s(\Omega _T)$ that

$$\begin{aligned} -\int _{\Omega _T}u^{(m)}_t\varphi +[a(z,Du^{(m)})+|F|^{p(\cdot )-2}F]D\varphi -f\varphi \mathrm {d}z=0. \end{aligned}$$

(4.11)

Adding (4.10) and (4.11), we have

$$\begin{aligned}&-\int _{\Omega _T}u^{(m)}_t\varphi +[a(z,Du^{(m)})+|F|^{p(\cdot )-2}F]D\varphi -f\varphi \mathrm {d}z\\&\quad +\int _{\Omega _T}[a(z,Du^{(m)})-a(z,Dw)]D\varphi \mathrm {d}z\ge 0 \end{aligned}$$

with a test function $\varphi =u^{(m)}-w$. This yields

$$\begin{aligned} -\int _{\Omega _T}u^{(m)}_t\varphi +[a(z,Dw)+|F|^{p(\cdot )-2}F]D\varphi -f\varphi \mathrm {d}z\ge 0. \end{aligned}$$

Then, we test (4.9) with $\varphi =u^{(m)}-w$ and subtract the resulting equation from the last estimate. Passing to the limit $m\rightarrow \infty$, we arrive at

$$\begin{aligned} -\int _{\Omega _T}[A_0-a(z,Dw)]D(u-w)\mathrm {d}z\ge 0 \end{aligned}$$

for all $w\in \mathcal {W}_s(\Omega _T)$. Since, $\mathcal {W}_s(\Omega _T)\subset W^{p(\cdot )}_0(\Omega _T)$ is dense, we are allowed to choose $w\in W^{p(\cdot )}_0(\Omega _T)$. Hence, we choose $w=u\pm \varepsilon \zeta$ with an arbitrary $\zeta \in W^{p(\cdot )}_0(\Omega _T)$. This yields

$$\begin{aligned} -\varepsilon \int _{\Omega _T}[A_0-a(z,D(u\pm \varepsilon \zeta ))]D\zeta \mathrm {d}z\ge 0. \end{aligned}$$

Finally, passing to the limit $\varepsilon \downarrow 0$, we can conclude that

$$\begin{aligned} \int _{\Omega _T}[A_0-a(z,Du)]D\zeta \mathrm {d}z=0 \end{aligned}$$

for all $\zeta \in W^{p(\cdot )}_0(\Omega _T)$. This shows that

$$\begin{aligned} A_0=a(z,Du)~~~\text {for almost every}~z\in \Omega _T. \end{aligned}$$

Moreover we have to show, that $u(\cdot ,0)=g(\cdot ,0)$. First of all, we should mention that we get from (4.9) and integration by parts the following

$$\begin{aligned} \int _{\Omega _T}u\varphi _t-[a(z,Du)+|F|^{p(\cdot )-2}F]D\varphi +f\varphi \mathrm {d}z=\int _\Omega (u\cdot \varphi )(\cdot ,0)\mathrm {d}x\end{aligned}$$

for all $\varphi \in W^{p(\cdot )}_0(\Omega _T)$ with $\varphi (\cdot ,T)=0$. Moreover, we can conclude (similar to (4.11)) that

$$\begin{aligned} \int _{\Omega _T}u^{(m)}\varphi _t-[a(z,Du^{(m)})+|F|^{p(\cdot )-2}F]D\varphi +f\varphi \mathrm {d}z=\int _\Omega (u^{(m)}\cdot \varphi )(\cdot ,0)\mathrm {d}x\end{aligned}$$

for all $\varphi \in W^{p(\cdot )}_0(\Omega _T)$ with $\varphi (\cdot ,T)=0$. Passing to the limit $m\rightarrow \infty$ and using the convergences from above we get

$$\begin{aligned} \int _{\Omega _T}u\varphi _t-[a(z,Du)+|F|^{p(\cdot )-2}F]D\varphi +f\varphi \mathrm {d}z=\int _\Omega g(\cdot ,0)\cdot \varphi (\cdot ,0)\mathrm {d}x, \end{aligned}$$

where $u^{(m)}(\cdot ,0)\rightarrow g(\cdot ,0)$ as $m\rightarrow \infty$, since

$$\begin{aligned} u^{(m)}(\cdot ,0)&=\sum _{i=1}^mc_i^{(m)}(0)\phi _i(x)\\&=\sum _{i=1}^m\int _\Omega g(\cdot ,0)\phi _i(x)\mathrm {d}x\phi _i(x) \rightarrow \sum _{i=1}^\infty \int _\Omega g(\cdot ,0)\phi _i(x)\mathrm {d}x\phi _i(x)=g(\cdot ,0) \end{aligned}$$

as $m\rightarrow \infty$. Furthermore, $\varphi (\cdot ,0)$ is arbitrary. Therefore, we can conclude that $u(\cdot ,0)=g(\cdot ,0)$.

Finally, we show the uniqueness of the weak solution. Therefore, we assume that there exist two weak solution u and $u_*\in C^0([0,T];L^2(\Omega ))\cap W^{p(\cdot )}_0(\Omega _T)$ with $\partial _t u,\partial _t u_*\in W^{p(\cdot )}(\Omega _T)'$ of the Dirichlet Problem (1.15). Thus, we have the following weak formulations

$$\begin{aligned} \int _{\Omega _T}\left[ u\cdot \varphi _t-a(z,Du)\cdot D\varphi \right] \mathrm {d}z=-\int _{\Omega _T}\left[ f\cdot \varphi +|F|^{p(\cdot )-2}F\cdot D\varphi \right] \mathrm {d}z\end{aligned}$$

and

$$\begin{aligned} \int _{\Omega _T}\left[ u_*\cdot \varphi _t-a(z,Du_*)\cdot D\varphi \right] \mathrm {d}z=-\int _{\Omega _T}\left[ f\cdot \varphi +|F|^{p(\cdot )-2}F\cdot D\varphi \right] \mathrm {d}z\end{aligned}$$

with the admissible test function $\varphi =u-u_*\in W^{p(\cdot )}_0(\Omega _T)$, since $W^{p(\cdot )}_0(\Omega _T)'$ is the dual of $W^{p(\cdot )}_0(\Omega _T)$. Hence, we can conclude that

$$\begin{aligned} \int _{\Omega _T}\left[ (u-u_*)\cdot (u-u_*)_t-(a(z,Du)-a(z,Du_*))\cdot D(u-u_*)\right] \mathrm {d}z=0. \end{aligned}$$

Using the monotonicity condition (1.3), we derive at

$$\begin{aligned} 0\ge \int _{\Omega _T}(u-u_*)\cdot (u-u_*)_t\mathrm {d}z=\frac{1}{2}\int _{\Omega _T}\partial _t (u-u_*)^2\mathrm {d}z. \end{aligned}$$

Finally, we have that $0\ge \frac{1}{2}\Vert u(t)-u_*(t)\Vert _{L^2(\Omega )}^2\ge 0$ for every $t\in (0,T]$, since $u(\cdot ,0)=u_*(\cdot ,0)=g(\cdot ,0)$. This shows the conclusion of the Theorem. $\square$

The proof of Theorem 1.6 is very short and the conlusion of Theorem 1.6 derives immediately from Theorem 1.4 as follows.

Proof of Theorem 1.6

First, we define a modified vector-field $\tilde{a}(z,w):=a(z,w+Dg(z))$ for all $z\in \Omega _T$ and $w\in \mathbb {R}^n$. Moreover, we let $v\in L^\infty (0,T;L^2(\Omega ))\cap W^{p(\cdot )}_0(\Omega _T)$ be a solution to the following initial value problem

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t v-{\mathrm {div}\,}~\tilde{a}(z,Dv)&{}=~~f-{\mathrm {div}\,}~(|F|^{p(\cdot )-2}F)-\partial _tg~~~\text {in}~\Omega _T\\ v&{}=~~0~~~\text {on}~\partial \Omega \times (0,T)\\ v&{}=~~g(\cdot ,0)-g~~~\text {on}~\Omega \times \left\{ 0\right\} . \end{array}\right. } \end{aligned}$$

(4.12)

The existence of the solution is guaranteed by Lemma 1.4, since we have $f-\partial _tg \in L^{\gamma _1'}(0,T;W^{-1,\gamma _1'}(\Omega ))$. It is easy to show, that $u=v+g$ is the desired solution to the boundary value problem (1.20). From the energy estimate (1.19) with u replaced by v, we get the following energy estimate

$$\begin{aligned} \sup _{0\le t\le T}\int _\Omega |v(\cdot ,t)|^2\mathrm {d}x+\int _{\Omega _T}|Dv|^{p(\cdot )}\mathrm {d}z\le ~c\Vert v(\cdot ,0)\Vert _{L^2(\Omega )} +c\cdot \mathcal {M}_0, \end{aligned}$$

where $\mathcal {M}_0$ is introduced in (1.18) with v replaced by g and f replaced by $f-\partial _tg$. Using the fact that $v=u-g$, we get the energy estimate (1.22). Therefore, we can conclude that

$$\begin{aligned}&\sup _{0\le t\le T}\int _\Omega |u(\cdot ,t)|^2\mathrm {d}x-2\sup _{0\le t\le T}\int _\Omega |g(\cdot ,t)|^2\mathrm {d}x+\int _{\Omega _T}|Du|^{p(\cdot )}-c(\gamma _2)|Dg|^{p(\cdot )}\mathrm {d}z\\&\quad \le c(\gamma _2)\left[ \sup _{0\le t\le T}\int _\Omega |v(\cdot ,t)|^2\mathrm {d}x+\int _{\Omega _T}|Dv|^{p(\cdot )}\mathrm {d}z\right] \\&\quad \le c\Vert g(\cdot ,0)\Vert _{L^2(\Omega )}^2 +c\cdot \mathcal {M}_0, \end{aligned}$$

since $|Du|^{p(\cdot )}\le 2^{\gamma _2-1}\left[ |D(u-g)|^{p(\cdot )}+|Dg|^{p(\cdot )}\right]$ and therefore, $|Du|^{p(\cdot )}-2^{\gamma _2-1}|Dg|^{p(\cdot )}\le 2^{\gamma _2-1}|D(u-g)|^{p(\cdot )}$ and $|u|^2-2|g|^2\le 2|u-g|^2.$ This yields the energy estimate (1.22). Finally, we show the uniqueness of the weak solution. Therefore, we assume that there exist two weak solution u and $u_*\in C^0([0,T];L^2(\Omega ))\cap W^{p(\cdot )}_g(\Omega _T)$ with $\partial _t u,\partial _t u_*\in W^{p(\cdot )}(\Omega _T)'$ of the Cauchy–Dirichlet Problem (1.20). Thus, we consider again the difference of the weak formulations with the admissible test function $\varphi =u-u_*\in W^{p(\cdot )}_0(\Omega _T)$, since $W^{p(\cdot )}_0(\Omega _T)'$ is the dual of $W^{p(\cdot )}_0(\Omega _T)$. Hence, we can conclude that

$$\begin{aligned} \int _{\Omega _T}\left[ (u-u_*)\cdot (u-u_*)_t-(a(z,Du)-a(z,Du_*))\cdot D(u-u_*)\right] \mathrm {d}z=0. \end{aligned}$$

Using the monotonicity condition (1.3), we derive at

$$\begin{aligned} 0\ge \int _{\Omega _T}(u-u_*)\cdot (u-u_*)_t\mathrm {d}z=\frac{1}{2}\int _{\Omega _T}\partial _t (u-u_*)^2\mathrm {d}z. \end{aligned}$$

Finally, we have that $0\ge \frac{1}{2}\Vert u(t)-u_*(t)\Vert _{L^2(\Omega )}^2\ge 0$ for every $t\in (0,T]$, since $u(\cdot ,0)=u_*(\cdot ,0)=g(\cdot ,0)$. This completes the proof. $\square$