Existence of Solutions for the Evolution 𝑝(𝑥)-Laplacian Equation Not in Divergence Form

Abstract

The existence of weak solutions is studied to the initial Dirichlet problem of the equation 𝑢_𝑡=𝑢div(|∇𝑢|^{𝑝(𝑥)−2}∇𝑢), with inf 𝑝(𝑥)>2. We adopt the method of parabolic regularization. After establishing some necessary uniform estimates on the approximate solutions, we prove the existence of weak solutions.

1. Introduction

In this paper, we investigate the existence of solutions for the 𝑝(𝑥)-Laplacian equation𝜕𝑢𝜕𝑡=𝑢div||∇𝑢||^{𝑝(𝑥)−2}∇𝑢,(𝑥,𝑡)∈𝑄_𝑇.(1.1) The equation is supplemented the boundary condition𝑢(𝑥,𝑡)=0,𝑥∈𝜕Ω,(1.2) and the initial condition𝑢(𝑥,0)=𝑢₀(𝑥),𝑥∈Ω,(1.3) where 𝑄_𝑇=Ω×(0,𝑇), inf𝑝(𝑥)>2, Ω⊂𝑅^𝑁 is a bounded domain with smooth boundary 𝜕Ω and 0≤𝑢₀(𝑥)∈𝐶(Ω)∩𝑊^{01,𝑝(𝑥)}(Ω).

In the case when 𝑝 is a constant, there have been many results about the existence, localization and extendibility and of weak solutions. We refer the readers to the bibiography given in [1–5] and the references therein.

A new interesting kind of fluids of prominent technological interest has recently emerged, the so-called electrorheological fluids. This model includes parabolic equations which are nonlinear with respect to the gradient of the thought solution, and with variable exponents of nonlinearity. The typical case is the so-called evolution 𝑝-Laplace equation with exponent 𝑝 as a function of the external electromagnetic field (see [6–12] and the references therein). In [6], the authors studied the regularity for the parabolic systems related to a class of non-Newtonian fluids, and the equations involved are nondegenerated.

On the other hand, there are also many results to the corresponding elliptic 𝑝(𝑥)-Laplace equations [13–15].

In the present work, we will study the existence of the solutions to problem (1.1)–(1.3). As we know, when 𝑝 is a constant, the nondegenerate problems have classical solutions, and hence the weak solutions exist. But in the case of 𝑝(𝑥)-Laplace type, there are no results to the corresponding non-degenerate problems. Since (1.1) degenerates whenever 𝑢=0 and ∇𝑢=0, we need to regularize the problem in two aspects corresponding to two different degeneracy: the first is the initial and boundary value and the second is the equation. We will first consider the non-degenerate problems. Based on the uniform Schauder estimates and using the method of continuity, we obtain the existence of classical solutions for non-degenerate problems. After establishing some necessary uniform estimates on the approximate solutions, we prove the existence of weak solutions.

This paper is arranged as follows. We first state some auxiliary lemmas in Section 2, and then we study a general quasilinear equation in Section 3. Subsequently, we discuss the existence of weak solutions in Section 4.

2. Preliminaries

Denote that𝑝₊=esssup_Ω𝑝(𝑥),𝑝₋=essinf_Ω𝑝(𝑥).(2.1)

Throughout the paper, we assume that2<𝑝₋≤𝑝(𝑥)≤𝑝₊<∞,∀(𝑥,𝑡)∈Ω×[0,𝑇],(2.2) where 𝑝₋,𝑝₊ are given constants.

To study our problems, we need to introduce some new function spaces. Denote that𝐿^𝑝(𝑥)(Ω)=𝑢∶|𝑢𝑢|is_𝑝(𝑥)ameasurablereal-valuedfunction,_Ω||𝑢(𝑥)||^𝑝(𝑥)𝑑𝑥<∞,=inf𝜆∶𝜆>0,_Ω|||𝑢(𝑥)𝜆|||^𝑝(𝑥)𝐿𝑑𝑥≤1_𝑝(𝑥),𝐿𝑄_𝑇=𝑢∶|𝑢|^𝑝(𝑥)∈𝐿¹𝑄_𝑇_𝑝(𝑥),𝐶0,𝑇;𝑊^{01,𝑝(𝑥)}(Ω)=𝑢∶|𝑢|^𝑝(𝑥)∈𝐿¹𝑄_𝑇,||∇𝑢||^𝑝(𝑥)∈𝐿¹𝑄_𝑇,𝑢|_𝜕Ω=0_{2𝑘+𝛼,𝑘+𝛼/2},𝑄𝐶_𝑇=𝑢∶𝑢∈𝐶^2𝑘,𝑘𝑄_𝑇,𝐷^𝑟𝑡𝐷^𝑠𝑥𝑢∈𝐶^𝛼,𝛼/2,2𝑟+𝑠=2𝑘,0<𝛼≤1_2𝑘,𝑘,𝑊𝑄_𝑇=𝑢∶𝐷^𝑟𝑡𝐷^𝑠𝑥𝑢∈𝐶𝑄_𝑇,0≤2𝑟+𝑠≤2𝑘_1,𝑝(𝑥),|𝑢|_1,𝑝(𝑥)(Ω)=𝑢∶𝑢∈𝐿^𝑝(𝑥)(Ω),||∇𝑢||∈𝐿^𝑝(𝑥)(Ω),=|𝑢|_𝐿^𝑝(𝑥)_(Ω)+||∇𝑢||_{𝐿𝑝(𝑥)(Ω)},∀𝑢∈𝑊^1,𝑝(𝑥)(Ω).(2.3) We use 𝑊^{01,𝑝(𝑥)}(Ω) to denote the closure of 𝐶^∞0(Ω) in 𝑊^1,𝑝(𝑥).

Remark 2.1. In [16, 17], Zhikov showed 𝑊^{01,𝑝(𝑥)}(Ω)≠𝑣∶𝑣∈𝑊^1,𝑝(𝑥)(Ω),𝑣|_𝜕Ω=0.(2.4) Hence, the property of the space is different from the case when 𝑝 is a constant. This will bring us some difficulties in taking the limit of the weak solutions. Luckily, our approximating solutions are in 𝑊^{01,𝑝(𝑥)}, and hence the limit function is also in 𝑊^{01,𝑝(𝑥)} which avoids the above difficulties.

We now give the definition of the solutions to our problem.

Definition 2.2. A nonnegative function 𝑢∈𝐿^∞(𝑄_𝑇), |∇𝑢|∈𝐿^𝑝(𝑥)(𝑄_𝑇), and 𝑢_𝑡∈𝐿²(𝑄_𝑇) is said to be a weak solution of (1.1)–(1.3), if for all 𝜑∈𝐶^∞0(𝑄_𝑇) satisfies the following: _𝑄𝑇𝑢𝜑^𝑡−lim𝑢_𝑡→0||∇𝑢||^{𝑝(𝑥)−2}∇𝑢∇𝜑−||∇𝑢||^𝑝(𝑥)𝜑𝑑𝑥𝑑𝑡=0,_Ω||𝑢(𝑥,𝑡)−𝑢₀(𝑥)||𝑑𝑥=0.(2.5)

In the following, we state some of the properties of the function spaces introduced as above.

Proposition 2.3 (see [15, 18]). (i)The space (𝐿^𝑝(𝑥)(Ω),|⋅|_𝑝(𝑥)) is a separable, uniform convex Banach space, and its conjugate space is 𝐿^𝑞(𝑥)(Ω), where 1/𝑝(𝑥)+1/𝑞(𝑥)=1. For any 𝑢∈𝐿^𝑝(𝑥)(Ω) and 𝑣∈𝐿^𝑞(𝑥)(Ω), we have ||||_Ω𝑢𝑣𝑑𝑥||||≤1𝑝₋+1𝑞₋|𝑢|^𝑝(𝑥)|𝑣|_𝑞(𝑥).(2.6)(ii)If 𝑝₁,𝑝₂∈𝐶₊(Ω),𝑝₁(𝑥)≤𝑝₂(𝑥) for any 𝑥∈Ω, then 𝐿^𝑝2(𝑥)↪𝐿^𝑝1(𝑥) and the imbedding continuous.(iii)There is a constant 𝐶>0, such that |𝑢|_𝑝(𝑥)≤𝐶||∇𝑢||_𝑝(𝑥),∀𝑢∈𝑊^{01,𝑝(𝑥)}(Ω).(2.7) This implies that |∇𝑢|_𝑝(𝑥) and ‖𝑢‖_1,𝑝(𝑥) are equivalent norms of 𝑊^{01,𝑝(𝑥)}.(iv)We have ∫_Ω|𝑢|^𝑝(𝑥)𝑑𝑥≥|𝑢|^𝑝−_𝑝(𝑥)−1,forall𝑢∈𝐿^𝑝(𝑥)(Ω).

Proposition 2.4 (see [18]). If we denote 𝜌(𝑢)=_Ω|𝑢|^𝑝(𝑥)𝑑𝑥,∀𝑢∈𝐿^𝑝(𝑥)(Ω),(2.8) then(i)|𝑢|_𝑝(𝑥)<1(=1;>1)⇔𝜌(𝑢)<1(=1;>1), (ii)|𝑢|_𝑝(𝑥)>1⇒|𝑢|^𝑝−_𝑝(𝑥)≤𝜌(𝑢)≤|𝑢|^𝑝+_𝑝(𝑥), |𝑢|_𝑝(𝑥)<1⇒|𝑢|^𝑝−_𝑝(𝑥)≥𝜌(𝑢)≥|𝑢|^𝑝+_𝑝(𝑥), (iii)|𝑢|_𝑝(𝑥)→0⇔𝜌(𝑢)→0;|𝑢|_𝑝(𝑥)→∞⇔𝜌(𝑢)→∞.

Lemma 2.5 (see [4]). Let 𝜃≥0,𝐴(𝜂)=(𝐴₁(𝜂),…,𝐴_𝑁(𝜂))=(|𝜂|²+𝜃)^(𝑝−2)/2,𝜂=(𝜂₁,…,𝜂_𝑁)∈𝑅^𝑁. Then 𝐴(𝜂)−𝐴𝜂^′⋅𝜂−𝜂^≥𝐶||𝜂−𝜂′||^𝑝,∀𝜂,𝜂^∈𝑅^𝑁,(2.9) where 𝐶=𝐶(𝑝) is a positive constant depending on 𝑝.

3. A General Quasilinear Equation

Here, we will consider the general quasilinear equations𝑢_𝑡−𝑎_𝑖𝑗𝑥,𝑡,𝑢,𝑢_𝑥𝑢_{𝑥𝑖,𝑥𝑗}+𝑎𝑥,𝑡,𝑢,𝑢_𝑥=0,(3.1)𝑢|_Γ𝑇=𝜓||_Γ𝑇,(3.2) where Γ_𝑇=𝜕Ω×(0,𝑇]∪Ω×{0}.

Proposition 3.1 (see [19, Theorem  2.9 of Chapter I]). Let 𝑢(𝑥,𝑡) be a classical solution of (3.1) in 𝑄_𝑇. Suppose that the functions 𝑎_𝑖𝑗(𝑥,𝑡,𝑢,𝑝) and 𝑎(𝑥,𝑡,𝑢,𝑡) take finite value for any finite 𝑢,𝑝, and (𝑥,𝑡)∈𝑄_𝑇, and that for (𝑥,𝑡)∈𝑄_𝑇 and arbitrary 𝑢𝑢𝑎(𝑎𝑥_𝑖𝑗,𝑡(,𝑥𝑢,,𝑡0,)𝑢,≥0)−𝜉_𝑖𝜉𝑏_𝑗1≥0,𝑢²−𝑏₂,(3.3) where 𝑏₁ and 𝑏₂ are nonnegative constants. Then max_𝑄𝑇||𝑢(𝑥,𝑡)||≤𝑀,(3.4) where 𝑀 depends only on 𝑏₁,𝑏₂,𝑇, and max_Γ𝑇|𝑢|.

We suppose that for (𝑥,𝑡)∈𝑄_𝑇,max_𝑄𝑇|𝑢(𝑥,𝑡)|≤𝑀 and arbitrary 𝑞 the functions 𝑎_𝑖𝑗(𝑥,𝑡,𝑢,𝑞),𝑎(𝑥,𝑡,𝑢,𝑞) are continuous in 𝑥,𝑡,𝑢,𝑞, continuously differentiable with respect to 𝑥,𝑢, and 𝑞, and satisfy the inequalities𝜈1||||𝜕+𝑎_𝑖𝑗||𝑞||^{𝑚(𝑥,𝑡)−2}𝜉²≤𝑎_𝑖𝑗(𝑥,𝑡,𝑢,𝑞)𝜉_𝑖𝜉_𝑗≤𝜇₀1+||𝑞||^{𝑚(𝑥,𝑡)−2}𝜉²,𝜕𝑝_𝑘||||1+||𝑞||₃+|𝑎|+||||𝜕𝑎𝜕𝑞_𝑘||||1+||𝑞||≤𝜇₁||||𝜕𝑎_𝑖𝑗1+||𝑞||^{𝑚(𝑥,𝑡)},𝜕𝑥_𝑘||||1+||𝑞||₂+||||𝜕𝑎𝜕𝑥_𝑘||||≤𝜀+𝑃||𝑞||1+||𝑞||_{𝑚(𝑥,𝑡)+1}||||𝜕𝑎_𝑖𝑗,𝜕𝑢||||≤𝜀+𝑃||𝑞||1+||𝑞||^{𝑚(𝑥,𝑡)−2}−𝜕𝑎,𝜕𝑢≤𝜀+𝑃||𝑞||1+||𝑞||^{𝑚(𝑥,𝑡)},(3.5) where 𝑃(𝜌) is a nonnegative continuous function that tends to zero for 𝜌→∞ and 1<𝑚(𝑥,𝑡)∈𝐶¹(𝑄_𝑇) is an arbitrary function.

Lemma 3.2. Let 𝑢(𝑥,𝑡) be a classical solution of (3.1) in 𝑄_𝑇. Suppose that the conditions of Proposition 3.1 hold and satisfy (3.5) with a sufficiently small 𝜀 determined by the numbers 𝑀, 𝜈, 𝜇, 𝜇₁, and 𝑃=max_𝜌≥0𝑃(𝜌).(3.6) Then max_𝑄𝑇||𝑢_𝑥(𝑥,𝑡)||≤𝑀₁.(3.7)

The proof of Lemma 3.2 is quite similar to the Theorem  4.1, chapter VI of [19]; one only has to replace 𝑚 with 𝑚(𝑥,𝑡) and remark that the constants in the proof are depending only on inf𝑚(𝑥,𝑡) and sup𝑚(𝑥,𝑡); we omit the details.

Theorem 3.3. Suppose that the following conditions are fulfilled.(a)For (𝑥,𝑡)∈𝑄_𝑇 and arbitrary 𝑢 either conditions (3.3) are fulfilled.(b)For (𝑥,𝑡)∈𝑄_𝑇, |𝑢|≤𝑀 (where 𝑀 is taken from estimate (3.4)) and arbitrary 𝑝, the functions 𝑎_𝑖𝑗(𝑥,𝑡,𝑢,𝑝) and 𝑎(𝑥,𝑡,𝑢,𝑝) are continuous and differentiable with respect to 𝑥, 𝑢, and 𝑝 and satisfy inequalities (3.5) with a sufficiently small 𝜀 determined by the numbers 𝑀, 𝜈, 𝜇, 𝜇₁, and 𝑃=max_𝜌≥0𝑃(𝜌).(3.8)(c)For (𝑥,𝑡)∈𝑄_𝑇|𝑢|≤𝑀 and |𝑝|≤𝑀₁ (where 𝑀₁ is taken from estimate (3.7)), the functions 𝑎_𝑖𝑗(𝑥,𝑡,𝑢,𝑝) and 𝑎(𝑥,𝑡,𝑢,𝑝) are continuously differentiable with respect to all of their arguments.(d)The boundary condition (3.2) is given by a function 𝜓(𝑥,𝑡) belonging to 𝐶^{2+𝛽,1+𝛽/2}(𝑄_𝑇) and satisfying on 𝑆₀={(𝑥,𝑡)∶𝑥∈𝜕Ω,𝑡=0} (3.1), that is, 𝜓_𝑡−𝑎_𝑖𝑗𝑥,0,𝜓(𝑥,0),𝜓_𝑥(𝑥,0)𝜓_{𝑥𝑖𝑥𝑗}+𝑎(𝑥,0,𝜓(𝑥,0))||_{𝑥∈𝜕Ω}=0(3.9) (in other words, the compatibility conditions of zero and first orders are assumed to be fulfilled).(e)We have 𝜕Ω∈𝐶^2+𝛽.
Then there exists a unique solution of problem (3.1) and (3.2) in the space 𝐶^{2+𝛽,1+𝛽/2}(𝑄_𝑇). This solution has derivatives 𝑢_𝑡𝑥𝑖 from 𝐿²(𝑄_𝑇).

Proof. We consider problem (3.1) and (3.2) along with a one-parameter family of problems of the same type 𝐿_𝜏𝑢=𝑢_𝑡−𝜏𝑎_𝑖𝑗𝑥,𝑡,𝑢,𝑢_𝑥+(1−𝜏)1+𝑢^2𝑥^{𝑚(𝑥)/2−1}𝛿^𝑗𝑖𝑢^𝑥_𝑖^𝑥_𝑗𝑢|_Γ𝑇+𝜏𝑎𝑥,𝑡,𝑢,𝑢_𝑥−(1−𝜏)𝜓_𝑡−1+𝜓^2𝑥^{𝑚(𝑥)/2−1}Δ𝜓=0,=𝜓||_Γ𝑇,0≤𝜏≤1,assuming𝜓∈𝐶^{2+𝛽,1+𝛽/2}𝑄_𝑇.(3.10) Define the Banach space 𝑋=𝑤∈𝐶^{1+𝛼,(1+𝛼)/2}𝑄_𝑇|||𝑤|^Γ_𝑇=0.(3.11) For any 𝑤∈𝑋, let 𝑣=𝑤+𝜓. Using Schauder theory, the linear problem 𝑣_𝑡−𝜏𝑎_𝑖𝑗𝑥,𝑡,𝑤,𝑤_𝑥+(1−𝜏)1+𝑤^2𝑥^{𝑚(𝑥)/2−1}𝛿^𝑗𝑖𝑣^𝑥_𝑖^𝑥_𝑗𝑢|_Γ𝑇+𝜏𝑎𝑥,𝑡,𝑤,𝑤_𝑥−(1−𝜏)𝜓_𝑡−1+𝜓^2𝑥^{𝑚(𝑥)/2−1}Δ𝜓=0,=𝜓||_Γ𝑇(3.12) admits a unique solution 𝑢∈𝐶^{2+𝛼,1+𝛼/2}(𝑄_𝑇). Let 𝑧=𝑢−𝜓, clearly 𝑧∈𝑋, and define the map 𝐺∶𝑋→𝑋 such that 𝑧=𝐺(𝑤). By [19], we know that 𝐺 is continuous and compact. By Proposition 3.1, Lemma 3.2, and the Leray-Schauder fixed point principle, the operator 𝐺 has a fixed point 𝑢.

4. Existence

In this section, we are going to prove the existence of solutions of the problem (1.1)–(1.3).

Theorem 4.1. Assume that 𝑝(𝑥)∈𝐶¹(Ω),inf𝑝(𝑥)>2, and 0≤𝑢₀(𝑥)∈𝐶(Ω)∩𝑊^{01,𝑝(𝑥)}(Ω). Then the problem (1.1)–(1.3) admits a weak solution 𝑢.

Consider the following problem:𝜕𝑢_𝜀,𝜂𝜕𝑡=𝑢^𝜀,𝜂div||∇𝑢_𝜀,𝜂||²+𝜂^{(𝑝(𝑥)−2)/2}∇𝑢_𝜀,𝜂,(4.1)𝑢_𝜀,𝜂||_𝑆𝑇=𝜀,𝑢_𝜀,𝜂||_𝑡=0=𝑢₀+𝜀,(4.2) where 𝑆_𝑇=𝜕Ω×(0,𝑇], 𝜀∈(0,1), and 𝜂∈(0,𝜀). Roughly speaking, here we use to regularize the initial-boundary value and use 𝜂 to regularize the equation. Thus, we have to carry out two limit processes, that is, first let 𝜂→0 (along a certain subsequence) and then let 𝜀→0.

We first change (4.1) into the form𝑢_𝑡−𝑎_𝑖𝑗𝑥,𝑡,𝑢,𝑢_𝑥𝑢_{𝑥𝑖,𝑥𝑗}+𝑎𝑥,𝑡,𝑢,𝑢_𝑥=0,(4.3) where𝑎_𝑖𝑗𝑥,𝑡,𝑢𝑎,𝑢_𝑥𝑥,𝑡=,𝑢𝑢,𝛿𝑢_𝑖𝑗𝑥||∇𝑢||²+𝜂^{(𝑝(𝑥)−2)/2}+𝑢(𝑝(𝑥)−2)||∇𝑢||²+𝜂^{(𝑝(𝑥)−4)/2}𝜕_𝑖𝑢𝜕_𝑗𝑢,=𝑢12||∇𝑢||²+𝜂^{(𝑝(𝑥)−2)/2}𝜕_𝑖𝑝(𝑥)𝜕_𝑖𝑢ln||∇𝑢||²+𝜂.(4.4) It is easily seen that (4.4) satisfies (3.3) and (3.5), where 𝑝(𝑥) instead of 𝑚(𝑥,𝑡). By Theorem 3.3, we know that (4.1)-(4.2) has a classical solution 𝑢_𝜀,𝜂.

Proposition 4.2. We have 𝜀≤𝑢_𝜀,𝜂≤||𝑢₀||_∞+𝜀,𝑢_𝜀1,𝜂≤𝑢_𝜀2,𝜂,for𝜀₁≤𝜀₂.(4.5)

Proof. By the maximum principle, we know that 𝜀≤𝑢_𝜀,𝜂≤|𝑢₀|_∞+𝜀.
A simple calculation shows that 𝜕𝑢_𝜀1,𝜂𝜕𝑡−𝑎^𝑖𝑗𝑥,𝑡,𝑢_𝜀1,𝜂,∇𝑢_𝜀1,𝜂𝜕²𝑢_𝜀1,𝜂𝜕𝑥_𝑖𝜕𝑥_𝑗𝜕𝑢_𝜀2,𝜂+𝑎𝑥,𝑡,𝑢_𝜀1,𝜂,∇𝑢_𝜀1,𝜂=0,𝜕𝑡−𝑎^𝑖𝑗𝑥,𝑡,𝑢_𝜀2,𝜂,∇𝑢_𝜀2,𝜂𝜕²𝑢_𝜀2,𝜂𝜕𝑥_𝑖𝜕𝑥_𝑗+𝑎𝑥,𝑡,𝑢_𝜀2,𝜂,∇𝑢_𝜀2,𝜂=0,(4.6) where 𝑎_𝑖𝑗(𝑥,𝑡,𝑢,𝑢_𝑥) and 𝑎(𝑥,𝑡,𝑢,𝑢_𝑥) are defined as (4.4).
It is easy to prove that 𝜉_𝑖𝑎_𝑖𝑗𝑥,𝑡,𝑢_{𝜀𝑖,𝜂},∇𝑢_{𝜀𝑖,𝜂}𝜉_𝑗≥𝜀_𝑖𝜂^{(𝑝(𝑥)−2)/2}||𝜉||²,∀𝜉∈𝑅^𝑁.(4.7) Hence, we have 𝜕𝑢_𝜀1,𝜂𝜕𝑡−𝜕𝑢𝜕𝑡^𝜀2,𝜂=𝑎^𝑖𝑗𝑥,𝑡,𝑢_𝜀1,𝜂,∇𝑢_𝜀1,𝜂𝜕²𝑢_𝜀1,𝜂−𝑢_𝜀2,𝜂𝜕𝑥_𝑖𝜕𝑥_𝑗+𝑎_𝑖𝑗𝑥,𝑡,𝑢_𝜀1,𝜂,∇𝑢_𝜀1,𝜂−𝑎_𝑖𝑗𝑥,𝑡,𝑢_𝜀2,𝜂,∇𝑢_𝜀2,𝜂𝜕²𝑢_𝜀2,𝜂𝜕𝑥_𝑖𝜕𝑥_𝑗+𝑎𝑥,𝑡,𝑢_𝜀1,𝜂,∇𝑢_𝜀1,𝜂−𝑎𝑥,𝑡,𝑢_𝜀2,𝜂,∇𝑢_𝜀2,𝜂.(4.8) Let 𝑤=𝑢_𝜀1,𝜂−𝑢_𝜀2,𝜂,𝑎_𝑖𝑗(𝑥,𝑡)=𝑎_𝑖𝑗(𝑥,𝑡,𝑢_𝜀1,𝜂,∇𝑢_𝜀1,𝜂). Using the mean value theorem, we have 𝑎_𝑖𝑗𝑥,𝑡,𝑢_𝜀1,𝜂,∇𝑢_𝜀1,𝜂−𝑎_𝑖𝑗𝑥,𝑡,𝑢_𝜀2,𝜂,∇𝑢_𝜀2,𝜂𝜕²𝑢_𝜀2,𝜂𝜕𝑥_𝑖𝜕𝑥_𝑗=𝑎_𝑖𝑗𝑥,𝑡,𝑢_𝜀1,𝜂,∇𝑢_𝜀1,𝜂−𝑎_𝑖𝑗𝑥,𝑡,𝑢_𝜀1,𝜂,∇𝑢_𝜀2,𝜂𝜕²𝑢_𝜀2,𝜂𝜕𝑥_𝑖𝜕𝑥_𝑗+𝑎_𝑖𝑗𝑥,𝑡,𝑢_𝜀1,𝜂,∇𝑢_𝜀2,𝜂−𝑎_𝑖𝑗𝑥,𝑡,𝑢_𝜀2,𝜂,∇𝑢_𝜀2,𝜂𝜕²𝑢_𝜀2,𝜂𝜕𝑥_𝑖𝜕𝑥_𝑗=𝑑_𝑘(𝑥,𝑡)𝜕𝑤𝜕𝑥_𝑘+𝑒(𝑥,𝑡)𝑤.(4.9) Similarly, we get 𝑎𝑥,𝑡,𝑢_𝜀1,𝜂,∇𝑢_𝜀1,𝜂−𝑎𝑥,𝑡,𝑢_𝜀2,𝜂,∇𝑢_𝜀2,𝜂=𝑓_𝑘(𝑥,𝑡)𝜕𝑤𝜕𝑥_𝑘+𝑔(𝑥,𝑡)𝑤,(4.10) where 𝑑_𝑘(𝑥,𝑡),𝑒(𝑥,𝑡),𝑓_𝑘(𝑥,𝑡), and 𝑔(𝑥,𝑡) are bounded functions. Hence, we see that 𝜕𝑤𝜕𝑡=𝑎^𝑖𝑗(𝑥,𝑡)𝜕²𝑤𝜕𝑥_𝑖𝜕𝑥_𝑗+𝑏_𝑘(𝑥,𝑡)𝜕𝑤𝜕𝑥_𝑘+𝑐(𝑥,𝑡)𝑤.(4.11) Since 𝑤≤0 on Γ_𝑇, by comparison principle of linear parabolic equation, we have 𝑤≤0.

Lemma 4.3. For all 𝛼∈[0,1) and 𝜂∈(0,𝜀), there hold (1)_𝑄𝑇||∇𝑢^𝜀,𝜂||²+𝜂^𝑝(𝑥)/2𝑢^{𝛼𝜀,𝜂}(2)_𝑄𝑇𝑑𝑥𝑑𝑡≤𝐶,𝜕𝑢𝜕𝑡^𝜀,𝜂²𝑑𝑥𝑑𝑡≤𝐶.(4.12)

Proof. Multiplying (4.1) by 𝑢^{−𝛼𝜀,𝜂}, integrating both sides of the equality over 𝑄_𝑇 and integrating by parts, we derive _𝑄𝑇𝜕𝑢_𝜀,𝜂𝜕𝑡𝑢^{−𝛼𝜀,𝜂}𝑑𝑥𝑑𝑡=11−𝛼_Ω𝑢^{1−𝛼𝜀,𝜂}=(𝑥,𝑇)−𝑢^{1−𝛼𝜀,𝜂}(𝑥,0)𝑑𝑥_𝑄𝑇𝑢^{1−𝛼𝜀,𝜂}div||∇𝑢_𝜀,𝜂||²+𝜂^{(𝑝(𝑥)−2)/2}∇𝑢_𝜀,𝜂=𝑑𝑥𝑑𝑡_𝑇0_𝜕Ω𝑢^{1−𝛼𝜀,𝜂}||∇𝑢^𝜀,𝜂||²+𝜂^{(𝑝(𝑥)−2)/2}𝜕𝑢_𝜀,𝜂−(1−𝛼)_𝑄𝑇𝜕𝜈𝑑𝜎𝑑𝑡||∇𝑢^𝜀,𝜂||²+𝜂^{(𝑝(𝑥)−2)/2}||∇𝑢_𝜀,𝜂||²𝑢^{−𝛼𝜀,𝜂}𝑑𝑥𝑑𝑡,(4.13) where 𝜈 denotes the outward normal to 𝜕Ω×(0,𝑇). Since from (4.5), 𝑢_𝜀,𝜂≥𝜀, we have 𝜕𝑢_𝜀,𝜂/𝜕𝜈≤0 on 𝜕Ω×(0,𝑇). Hence _𝑄𝑇||∇𝑢^𝜀,𝜂||²+𝜂^{(𝑝(𝑥)−2)/2}||∇𝑢_𝜀,𝜂||²𝑢^{−𝛼𝜀,𝜂}𝑑𝑥𝑑𝑡≤1(1−𝛼)²_Ω𝑢^{1−𝛼𝜀,𝜂}(𝑥,0)𝑑𝑥≤𝐶,(4.14) where 𝐶=𝐶(𝛼,Ω,|𝑢₀|_∞).
Using 𝑢_𝜀,𝜂≥𝜀>𝜂 and Young’s inequality, we have 𝜂≤_𝑄𝑇||∇𝑢^𝜀,𝜂||²+𝜂^{(𝑝(𝑥)−2)/2}𝑢^{−𝛼𝜀,𝜂}𝑑𝑥𝑑𝑡_𝑄𝑇||∇𝑢^𝜀,𝜂||²+𝜂^{(𝑝(𝑥)−2)/2}𝑢^{(𝑝(𝑥)−2)𝛼/𝑝(𝑥)𝜀,𝜂}𝜂^{2𝛼/𝑝(𝑥)}𝑢^{2𝛼/𝑝(𝑥)𝜀,𝜂}≤1𝑑𝑥𝑑𝑡2_𝑄𝑇||∇𝑢^𝜀,𝜂||²+𝜂^𝑝(𝑥)/2𝑢^{−𝛼𝜀,𝜂}𝑑𝑥𝑑𝑡+𝐶𝑝₋,𝑝₊,𝑇,Ω.(4.15)
Combining (4.14) with (4.15) yields _𝑄𝑇||∇𝑢^𝜀,𝜂||²+𝜂^𝑝(𝑥)/2𝑢^{−𝛼𝜀,𝜂}𝑑𝑥𝑑𝑡=_𝑄𝑇||∇𝑢^𝜀,𝜂||²+𝜂^{(𝑝(𝑥)−2)/2}𝑢^{−𝛼𝜀,𝜂}||∇𝑢^𝜀,𝜂≤𝜂_𝑄𝑇||²+𝜂𝑑𝑥𝑑𝑡||∇𝑢^𝜀,𝜂≤1||²+𝜂^{(𝑝(𝑥)−2)/2}𝑢^{−𝛼𝜀,𝜂}𝑑𝑥𝑑𝑡+𝐶2_𝑄𝑇||∇𝑢^𝜀,𝜂||²+𝜂^𝑝(𝑥)/2𝑢^{−𝛼𝜀,𝜂}𝑑𝑥𝑑𝑡+𝐶(𝑝,𝑇,Ω).(4.16) Hence, _𝑄𝑇||∇𝑢^𝜀,𝜂||²+𝜂^𝑝(𝑥)/2𝑢^{−𝛼𝜀,𝜂}𝑑𝑥𝑑𝑡≤𝐶.(4.17) Multiplying (4.1) by (𝜕𝑢_𝜀,𝜂/𝜕𝑡)𝑢^{−1𝜀,𝜂}, integrating both sides of the equality over 𝑄_𝑇 and integrating by parts and noticing that (𝑢_𝜀,𝜂)_𝑡=0 on 𝜕Ω×(0,𝑇), we derive _𝑄𝑇𝜕𝑢𝜕𝑡^𝜀,𝜂²=𝑢^{−1𝜀,𝜂}𝑑𝑥𝑑𝑡_𝑄𝑇div||∇𝑢_𝜀,𝜂||²+𝜂^{(𝑝(𝑥)−2)/2}∇𝑢_𝜀,𝜂=𝜕𝑢𝜕𝑡^𝜀,𝜂𝑑𝑥𝑑𝑡_𝑄𝑇div||∇𝑢_𝜀,𝜂||²+𝜂^{(𝑝(𝑥)−2)/2}∇𝑢_𝜀,𝜂𝜕𝑢_𝜀,𝜂−𝜕𝑡𝑑𝑥𝑑𝑡_𝑄𝑇||∇𝑢^𝜀,𝜂||²+𝜂^{(𝑝(𝑥)−2)/2}∇𝑢_𝜀,𝜂=−_𝑄𝑇∇𝜕𝑢𝜕𝑡^𝜀,𝜂𝑑𝑥𝑑𝑡||∇𝑢^𝜀,𝜂||²+𝜂^{(𝑝(𝑥)−2)/2}∇𝑢_𝜀,𝜂=−_Ω∇𝜕𝑢𝜕𝑡^𝜀,𝜂𝑑𝑥𝑑𝑡1𝑝(𝑥)||∇𝑢^𝜀,𝜂+(𝑥,𝑇)||²+𝜂^𝑝(𝑥)/2𝑑𝑥_Ω1𝑝(𝑥)||∇𝑢^𝜀,𝜂≤(𝑥,0)||²+𝜂^𝑝(𝑥)/2𝑑𝑥_Ω1𝑝(𝑥)||∇𝑢⁰||²+𝜂^𝑝(𝑥)/2𝑑𝑥.(4.18)

Equation (4.5), Lemma 4.3, and Proposition 2.3 imply that, for any 𝜀∈(0,1), there exists a subsequence of 𝑢_𝜀,𝜂, denoted by 𝑢_{𝜀,𝜂𝑘}, and a function 𝑢_𝜀∈𝐿^∞(𝑄_𝑇)  ∇𝑢∈𝐿^𝑝(𝑥)(𝑄_𝑇), such that, as 𝜂=𝜂_𝑘→0,𝑢_𝜀,𝜂⟶𝑢_𝜀,a.e.in𝑄_𝑇,(4.19)∇𝑢_𝜀,𝜂⟶∇𝑢_𝜀,weaklyin𝐿^𝑝(𝑥)𝑄_𝑇,(4.20)𝜕𝑢_𝜀,𝜂𝜕𝑡⟶𝜕𝜕𝑢𝑡^𝜀,weaklyin𝐿²𝑄_𝑇.(4.21)

Lemma 4.4. As 𝜂=𝜂_𝑘→0, we have ((12))_{𝑄𝑇𝑄𝑇}||∇𝑢_𝜀,𝜂−∇𝑢_𝜀||^𝑝(𝑥)𝑑𝑥𝑑𝑡⟶0,|||||∇𝑢_𝜀,𝜂||^𝑝(𝑥)−||∇𝑢_𝜀||^𝑝(𝑥)(3)_𝑄𝑇|||𝑑𝑥𝑑𝑡⟶0,||||||∇𝑢_𝜀,𝜂||²+𝜂^{(𝑝(𝑥)−2)/2}||∇𝑢_𝜀,𝜂||²−||∇𝑢_𝜀||^𝑝(𝑥)(4)_𝑄𝑇||||𝑑𝑥𝑑𝑡⟶0,||||𝑢_𝜀,𝜂||∇𝑢^𝜀,𝜂||²+𝜂^{(𝑝(𝑥)−2)/2}∇𝑢_𝜀,𝜂−𝑢_𝜀||∇𝑢_𝜀||^{𝑝(𝑥)−2}∇𝑢_𝜀||||𝑑𝑥𝑑𝑡⟶0.(4.22)

Proof. Observe that (𝑢_𝜀,𝜂−𝑢_𝜀)/𝑢_𝜀,𝜂∈𝐿^𝑝(𝑥)(0,𝑇;𝑊^{01,𝑝(𝑥)}(Ω)). Multiplying (4.1) by (𝑢_𝜀,𝜂−𝑢_𝜀)/𝑢_𝜀,𝜂, integrating both sides of the equality over 𝑄_𝑇 and integrating by parts, we derive _𝑄𝑇𝜕𝑢𝜕𝑡^𝜀,𝜂𝑢^𝜀,𝜂−𝑢_𝜀𝑢_𝜀,𝜂+||∇𝑢_𝜀,𝜂||²+𝜂^{(𝑝(𝑥)−2)/2}∇𝑢_𝜀,𝜂∇𝑢_𝜀,𝜂−𝑢_𝜀𝑑𝑥𝑑𝑡=0.(4.23) By Hölder inequality and Lemma 4.3, we obtain _𝑄𝑇𝜕𝑢_𝜀,𝜂𝜕𝑡𝑢^𝜀,𝜂−𝑢_𝜀𝑢_𝜀,𝜂⟶0,(𝜂⟶0).(4.24) Hence, _𝑄𝑇||∇𝑢^𝜀,𝜂||²+𝜂^{(𝑝(𝑥)−2)/2}∇𝑢_𝜀,𝜂∇𝑢_𝜀,𝜂−𝑢_𝜀𝑑𝑥𝑑𝑡⟶0.(4.25) We divide the integral in (4.25) in the following way: _𝑄𝑇||∇𝑢^𝜀,𝜂=||²+𝜂^{(𝑝(𝑥)−2)/2}∇𝑢_𝜀,𝜂∇𝑢_𝜀,𝜂−𝑢_𝜀𝑑𝑥𝑑𝑡_𝑄𝑇||∇𝑢^𝜀,𝜂||²+𝜂^{(𝑝(𝑥)−2)/2}∇𝑢_𝜀,𝜂−||∇𝑢_𝜀||²+𝜂^{(𝑝(𝑥)−2)/2}∇𝑢_𝜀⋅∇𝑢^𝜀,𝜂+−𝑢_𝜀𝑑𝑥𝑑𝑡_𝑄𝑇||∇𝑢^𝜀||²+𝜂^{(𝑝(𝑥)−2)/2}−||∇𝑢_𝜀||^{(𝑝(𝑥)−2)}∇𝑢^𝜀+∇𝑢_𝜀,𝜂𝑑𝑥𝑑𝑡_𝑄𝑇||∇𝑢^𝜀||^{(𝑝(𝑥)−2)}−||∇𝑢_𝜀||²+𝜂^{(𝑝(𝑥)−2)/2}||∇𝑢^𝜀+||²𝑑𝑥𝑑𝑡_𝑄𝑇=𝐼₁||∇𝑢_𝜀||^{(𝑝(𝑥)−2)}∇𝑢_𝜀∇𝑢_𝜀,𝜂−𝑢_𝜀𝑑𝑥𝑑𝑡+𝐼₂+𝐼₃+𝐼₄.(4.26) From (4.20), we see that 𝐼₄⟶0,(𝜂⟶0).(4.27) Using Lemma 4.3, we have ||𝐼₃||≤_𝑄𝑇∇𝑢^𝜀||²+𝜂^𝑝(𝑥)/2−||∇𝑢_𝜀||^𝑝(𝑥)≤𝜂𝑑𝑥𝑑𝑡2_𝑄𝑇𝑝(𝑥)∇𝑢_𝜀||²+𝜂^{(𝑝(𝑥)−2)/2}𝑑𝑥𝑑𝑡⟶0,(𝜂⟶0).(4.28) Now we estimate 𝐼₂. If 𝑝(𝑥)∈(2,3], then (𝑝(𝑥)−1)/2∈(0,1]. Using |𝑎^𝑟−𝑏^𝑟|≤|𝑎−𝑏|^𝑟 (𝑟∈[0,1],𝑎,𝑏≥0) gives ||𝐼₂||≤_𝑄𝑇||||||∇𝑢_𝜀||²+𝜂^{(𝑝(𝑥)−1)/2}−||∇𝑢_𝜀||^{𝑝(𝑥)−1}||||||∇𝑢_𝜀,𝜂≤𝜂_{(𝑝−−1)/2}||𝑑𝑥𝑑𝑡_𝑄𝑇||∇𝑢_𝜀,𝜂||𝑑𝑥𝑑𝑡⟶0,(𝜂⟶0).(4.29) If 𝑝(𝑥)>3, we obtain ||𝐼₂||≤𝜂2_𝑄𝑇(𝑝(𝑥)−1)||∇𝑢_𝜀||²+𝜂^(𝑝−3)/2||∇𝑢_𝜀,𝜂||𝑑𝑥𝑑𝑡⟶0,(𝜂⟶0).(4.30) By (4.25), (4.26), and 𝐼₂,𝐼₃,𝐼₄→0, we obtain 𝐼₁=_𝑄𝑇||∇𝑢^𝜀,𝜂||²+𝜂^{(𝑝(𝑥)−2)/2}∇𝑢_𝜀,𝜂−||∇𝑢_𝜀||²+𝜂^{(𝑝(𝑥)−2)/2}∇𝑢_𝜀⋅∇𝑢^𝜀,𝜂−𝑢_𝜀𝑑𝑥(𝑑𝜂𝑡⟶⟶00),.(4.31) Again by Lemma 2.5, we get 𝐼₁=_𝑄𝑇||∇𝑢^𝜀,𝜂||²+𝜂^{(𝑝(𝑥)−2)/2}∇𝑢_𝜀,𝜂−||∇𝑢_𝜀||²+𝜂^{(𝑝(𝑥)−2)/2}∇𝑢_𝜀⋅∇𝑢^𝜀,𝜂≥𝐶_𝑄𝑇−𝑢_𝜀𝑑𝑥𝑑𝑡||∇𝑢_𝜀,𝜂−∇𝑢_𝜀||^𝑝(𝑥)𝑑𝑥𝑑𝑡.(4.32) Letting 𝜂→0, we obtain (1). Again noticing that _𝑄𝑇|||||∇𝑢_𝜀,𝜂||^𝑝(𝑥)−||∇𝑢_𝜀||^𝑝(𝑥)≤|||𝑑𝑥𝑑𝑡_𝑄𝑇≤𝑝(𝑥)||∇𝑢_𝜀,𝜂||+||∇𝑢_𝜀||^{𝑝(𝑥)−1}||||∇𝑢_𝜀,𝜂||−||∇𝑢_𝜀||||𝑑𝑥𝑑𝑡_𝑄𝑇≤𝐶𝑝₊𝑝(𝑥)||∇𝑢_𝜀,𝜂||+||∇𝑢_𝜀||^𝑝−1||∇𝑢_𝜀,𝜂−∇𝑢_𝜀||𝑑𝑥𝑑𝑡||||∇𝑢_𝜀,𝜂||+||∇𝑢_𝜀||||_𝑝(𝑥)||∇𝑢_𝜀,𝜂−∇𝑢_𝜀||_𝑝(𝑥)≤𝐶||∇𝑢_𝜀,𝜂−∇𝑢_𝜀||_𝑝(𝑥),(4.33) by Proposition 2.4, we see that (2) holds. To prove (3), we have _𝑄𝑇||||||∇𝑢_𝜀,𝜂||²+𝜂^{(𝑝(𝑥)−2)/2}||∇𝑢_𝜀,𝜂||²−||∇𝑢_𝜀||^𝑝(𝑥)≤||||𝑑𝑥𝑑𝑡_𝑄𝑇||||||∇𝑢_𝜀,𝜂||²+𝜂^{(𝑝(𝑥)−2)/2}−||∇𝑢_𝜀,𝜂||^{𝑝(𝑥)−2}||||||∇𝑢_𝜀,𝜂+||²𝑑𝑥𝑑𝑡_𝑄𝑇|||||∇𝑢_𝜀,𝜂||^𝑝(𝑥)−||∇𝑢_𝜀||^𝑝(𝑥)≤|||𝑑𝑥𝑑𝑡_𝑄𝑇||||||∇𝑢_𝜀,𝜂||²+𝜂^𝑝(𝑥)/2−||∇𝑢_𝜀,𝜂||^𝑝(𝑥)||||𝑑𝑥𝑑𝑡+_𝑄𝑇|||||∇𝑢_𝜀,𝜂≤𝜂||^𝑝(𝑥)−||∇𝑢_𝜀||^𝑝(𝑥)|||𝑑𝑥𝑑𝑡2_𝑄𝑇𝑝(𝑥)||∇𝑢_𝜀,𝜂||²+𝜂^{(𝑝(𝑥)−2)/2}𝑑𝑥𝑑𝑡+_𝑄𝑇|||||∇𝑢_𝜀,𝜂||^𝑝(𝑥)−||∇𝑢_𝜀||^𝑝(𝑥)|||𝑑𝑥𝑑𝑡.(4.34) Using Lemma 4.3 and (2), we see that (3) holds.
Finally, we prove (4). We have _𝑄𝑇||||𝑢_𝜀,𝜂||∇𝑢^𝜀,𝜂||²+𝜂^{(𝑝(𝑥)−2)/2}∇𝑢_𝜀,𝜂−𝑢_𝜀||∇𝑢_𝜀||^{𝑝(𝑥)−2}∇𝑢_𝜀≤||||𝑑𝑥𝑑𝑡_𝑄𝑇+||𝑢_𝜀,𝜂−𝑢_𝜀||||∇𝑢_𝜀,𝜂||²+𝜂^{(𝑝(𝑥)−2)/2}||∇𝑢_𝜀,𝜂||𝑑𝑥𝑑𝑡_𝑄𝑇+𝑢_𝜀||∇𝑢^𝜀,𝜂||²+𝜂^{(𝑝(𝑥)−2)/2}||∇𝑢_𝜀,𝜂−∇𝑢_𝜀||𝑑𝑥𝑑𝑡_𝑄𝑇𝑢_𝜀||||||∇𝑢_𝜀,𝜂||²+𝜂^{(𝑝(𝑥)−2)/2}−||∇𝑢_𝜀||^{𝑝(𝑥)−2}||||||∇𝑢_𝜀=𝐼_𝑎||𝑑𝑥𝑑𝑡+𝐼_𝑏+𝐼_𝑐.(4.35) Equation (4.19) implies that 𝐼_𝑎⟶0,(𝜂⟶0).(4.36) To estimate 𝐼_𝑏, notice that 𝐼_𝑏≤𝐶|||||∇𝑢_𝜀,𝜂||²+𝜂|||_{(𝑝(𝑥)−2)𝑝(𝑥)/2(𝑝(𝑥)−1)}||∇𝑢_𝜀,𝜂−∇𝑢_𝜀||_𝑝(𝑥)⟶0,(𝜂⟶0).(4.37) As 𝑝(𝑥)∈(2,3], we have (𝑝(𝑥)−2)/2∈(0,1], 𝐼_𝑐≤𝐶_𝑄𝑇|||||∇𝑢_𝜀,𝜂||²≤𝐶_𝑄𝑇−||∇𝑢_𝜀||²+𝜂|||^{(𝑝(𝑥)−2)/2}||∇𝑢_𝜀||𝑑𝑥𝑑𝑡2^{(𝑝(𝑥)−2)/2}|||||∇𝑢^𝜀,𝜂||²−||∇𝑢_𝜀||²|||^{(𝑝(𝑥)−2)/2}+𝜂||^(𝑝−2)/2||∇𝑢^𝜀≤𝐶_𝑄𝑇||𝑑𝑥𝑑𝑡+𝐶_𝑄𝑇2^{(𝑝(𝑥)−2)/2}||∇𝑢_𝜀,𝜂−∇𝑢_𝜀||^{(𝑝(𝑥)−2)/2}||∇𝑢_𝜀,𝜂||+||∇𝑢_𝜀||^𝑝(𝑥)/2𝑑𝑥𝑑𝑡(2𝜂)^{(𝑝(𝑥)−2)/2}||∇𝑢_𝜀||𝑑𝑥𝑑𝑡.(4.38) By Hölder inequality, we have 𝐼_𝑐⟶0,(𝜂⟶0).(4.39) If 𝑝(𝑥)>3, we have 𝐼_𝑐≤𝐶_𝑄𝑇(𝑝(𝑥)−2)||∇𝑢_𝜀,𝜂||²+𝜂^(𝑝−3)/2+||∇𝑢_𝜀||^{𝑝(𝑥)−3}⋅||||||∇𝑢_𝜀,𝜂≤𝐶_𝑄𝑇||²+𝜂^1/2−||∇𝑢_𝜀||||||||∇𝑢_𝜀||𝑑𝑥𝑑𝑡(𝑝(𝑥)−2)||∇𝑢_𝜀,𝜂||²+𝜂^(𝑝−3)/2+||∇𝑢_𝜀||^{𝑝(𝑥)−3}⋅|||||∇𝑢_𝜀,𝜂≤𝐶2_1/2||²+𝜂−||∇𝑢_𝜀||²|||^1/2||∇𝑢_𝜀||𝑑𝑥𝑑𝑡_𝑄𝑇(𝑝(𝑥)−2)||∇𝑢_𝜀,𝜂||²+𝜂^(𝑝−3)/2+||∇𝑢_𝜀||^{𝑝(𝑥)−3}⋅||∇𝑢_𝜀,𝜂+𝐶(2𝜂)_1/2||+||∇𝑢_𝜀||^3/2||∇𝑢_𝜀,𝜂−𝑢_𝜀||𝑑𝑥𝑑𝑡_𝑄𝑇(𝑝(𝑥)−2)||∇𝑢_𝜀||²+𝜂^{(𝑝(𝑥)−3)/2}+||∇𝑢_𝜀||^{𝑝(𝑥)−3}||∇𝑢^𝜀||𝑑𝑥𝑑𝑡.(4.40) Hence, 𝐼_𝑐⟶0,(𝜂⟶0).(4.41) Thus (4) is proved, and the proof of Lemma 4.4 is complete.

Proposition 4.5. We obtain that 𝑢_𝜀 is a weak solution of the problem 𝜕𝑢𝜕𝑡=𝑢div||∇𝑢||^{𝑝(𝑥)−2}𝑢|_𝑆𝑇∇𝑢,=𝜀,𝑢|_𝑡=0=𝑢₀+𝜀,(4.42) then 𝜀≤𝑢_𝜀≤||𝑢₀||_∞+𝜀,𝑢_𝜀1≤𝑢_𝜀2,𝜀₁≤𝜀₂,a.e.in𝑄_𝑇,(4.43)_𝑄𝑇||∇𝑢_𝜀||^𝑝𝑑𝑥𝑑𝑡≤𝐶,_𝑄𝑇𝜕𝜕𝑢𝑡^𝜀²𝑑𝑥𝑑𝑡≤𝐶,(4.44) where 𝐶 is independent of 𝜀.

Proof. Obviously, for all 𝜀∈(0,1), 𝑢_𝜀−𝜀∈𝐿^𝑝(𝑥)(0,𝑇;𝑊^1,𝑝(𝑥)(Ω)). By Proposition 4.2 and (4.19)–(4.21), we know that (4.43) holds. (4.44) follows from (4.5), (4.19)–(4.21), and Lemma 4.3. To prove that 𝑢 satisfies the integral equality in Definition 2.2, we multiply (4.1) by 𝜑∈𝐶^∞0(𝑄_𝑇), integrate both sides of the equality on 𝑄_𝑇, and integrate by parts to derive _𝑄𝑇−𝑢^𝜀,𝜂+||∇𝜑𝑢_𝑡+_𝜀,𝜂𝑢_𝜀,𝜂||∇𝑢^𝜀,𝜂||²+𝜂^{(𝑝(𝑥)−2)/2}∇𝑢_𝜀,𝜂∇𝜑||²+𝜂^{(𝑝(𝑥)−2)/2}||∇𝑢_𝜀,𝜂||²𝜑𝑑𝑥𝑑𝑡=0.(4.45) Letting 𝜂=𝜂_𝑘→0 to pass to limit and using (4.19) and Lemma 4.4 show that _𝑄𝑇−𝑢^𝜀𝜑_𝑡+𝑢_𝜀||∇𝑢_𝜀||^{(𝑝(𝑥)−2)}∇𝑢_𝜀∇𝜑+||∇𝑢_𝜀||^𝑝(𝑥)𝜑𝑑𝑥𝑑𝑡=0.(4.46) Applying Lemma 4.3, we derive _Ω||𝑢_𝜀,𝜂−𝑢₀(𝑥)−𝜀||𝑑𝑥≤𝐶𝑡^1/2,(4.47) where 𝐶 is independent of 𝜀 and 𝜂. Hence, _Ω||𝑢_𝜀−𝑢₀(𝑥)−𝜀||𝑑𝑥⟶0,(𝑡⟶0).(4.48)

From (4.43), we see that 𝑢 is bounded and increasing in 𝜀, which implies the existence of a function 𝑢, such that, as 𝜀→0,𝑢_𝜀⟶𝑢,a.e.in𝑄_𝑇,(4.49)∇𝑢_𝜀⟶∇𝑢,weaklyin𝐿^𝑝(𝑥)𝑄_𝑇,(4.50)𝜕𝑢_𝜀𝜕𝑡⟶𝜕𝜕𝑢𝑡,weaklyin𝐿²𝑄_𝑇.(4.51)

Lemma 4.6. As 𝜀→0, we have ((12))_{𝑄𝑇𝑄𝜀𝑐}||∇𝑢^2𝜀−∇𝑢²||^𝑝(𝑥)𝑑𝑥𝑑𝑡⟶0,(3)_𝑄𝑐||∇𝑢_𝜀−∇𝑢||^𝑝(𝑥)𝑑𝑥𝑑𝑡⟶0,𝑄^𝜀𝑐=(𝑥,𝑡)∈𝑄_𝑇,𝑢_𝜀≥𝑐,𝑐>0,||∇𝑢_𝜀−∇𝑢||^𝑝(𝑥)𝑑𝑥𝑑𝑡⟶0,𝑄_𝑐=(𝑥,𝑡)∈𝑄_𝑇,𝑢≥𝑐,𝑐>0.(4.52)

Proof. We may take 𝜑=𝑢^{𝜀𝑝(𝑥)−2}(𝑢^2𝜀−𝜀²−𝑢²), in the integral equality satisfied by 𝑢. Then it is easy to see that _𝑄𝑇𝜕𝑢_𝜀𝜕𝑡𝑢^{𝜀𝑝(𝑥)−2}𝑢^2𝜀=−_𝑄𝑇−𝜀²−𝑢²𝑑𝑥𝑑𝑡−𝑢_𝜀||∇𝑢_𝜀||^{𝑝(𝑥)−2}∇𝑢_𝜀∇𝑢^{𝜀𝑝(𝑥)−2}𝑢^2𝜀−𝜀²−𝑢²𝑑𝑥𝑑𝑡_𝑄𝑇=−_𝑄𝑇||∇𝑢_𝜀||^𝑝(𝑥)𝑢^{𝜀𝑝(𝑥)−2}𝑢^2𝜀−𝜀²−𝑢²𝑑𝑥𝑑𝑡+𝜀₂(𝑝(𝑥)−1)||∇𝑢_𝜀||^𝑝(𝑥)𝑢^{𝜀𝑝(𝑥)−2}𝑢^2𝜀−𝑢²𝑑𝑥𝑑𝑡_𝑄𝑇−(𝑝(𝑥)−1)||∇𝑢_𝜀||^𝑝(𝑥)𝑢^{𝜀𝑝(𝑥)−2}𝑑𝑥𝑑𝑡_𝑄𝑇−||∇𝑢_𝜀||^{𝑝(𝑥)−2}∇𝑢_𝜀∇𝑢^2𝜀−𝑢²𝑢^{𝜀𝑝(𝑥)−1}𝑑𝑥𝑑𝑡_𝑄𝑇||∇𝑢_𝜀||^{𝑝(𝑥)−2}∇𝑢_𝜀𝑢^{𝜀𝑝(𝑥)−1}∇𝑝(𝑥)ln𝑢_𝜀𝑢^2𝜀−𝜀²−𝑢²𝑑𝑥𝑑𝑡.(4.53) Hence, by (4.43), we have _𝑄𝑇𝜕𝑢_𝜀𝜕𝑡𝑢^{𝜀𝑝(𝑥)−2}𝑢^2𝜀≤𝜀₂−𝜀²−𝑢²𝑑𝑥𝑑𝑡_𝑄𝑇−(𝑝(𝑥)−1)||∇𝑢_𝜀||^𝑝(𝑥)𝑢^{𝜀𝑝(𝑥)−2}𝑑𝑥𝑑𝑡_𝑄𝑇−||∇𝑢_𝜀||^{𝑝(𝑥)−2}∇𝑢_𝜀∇𝑢^2𝜀−𝑢²𝑢^{𝜀𝑝(𝑥)−1}𝑑𝑥𝑑𝑡_𝑄𝑇||∇𝑢_𝜀||^{𝑝(𝑥)−2}∇𝑢_𝜀𝑢^{𝜀𝑝(𝑥)−1}∇𝑝(𝑥)ln𝑢_𝜀𝑢^2𝜀−𝜀²−𝑢²𝑑𝑥𝑑𝑡.(4.54) Notice that _𝑄𝑇||=∇𝑢_𝜀||^{𝑝(𝑥)−2}∇𝑢_𝜀∇𝑢^2𝜀−𝑢²𝑢^{𝜀𝑝(𝑥)−1}𝑑𝑥𝑑𝑡_𝑄𝑇21^{𝑝(𝑥)−1}||∇𝑢^2𝜀||^𝑝−2∇𝑢^2𝜀∇𝑢^2𝜀−𝑢²𝑑𝑥𝑑𝑡.(4.55) So _𝑄𝑇𝜕𝑢_𝜀𝜕𝑡𝑢^{𝜀𝑝(𝑥)−2}𝑢^2𝜀−𝜀²−𝑢²𝑑𝑥𝑑𝑡+_𝑄𝑇12^{𝑝(𝑥)−1}≤𝜀₂||∇𝑢^2𝜀||^{𝑝(𝑥)−2}∇𝑢^2𝜀∇𝑢^2𝜀−𝑢²𝑑𝑥𝑑𝑡_𝑄𝑇−(𝑝(𝑥)−1)||∇𝑢_𝜀||^𝑝(𝑥)𝑢^{𝜀𝑝(𝑥)−2}𝑑𝑥𝑑𝑡_𝑄𝑇||∇𝑢_𝜀||^{𝑝(𝑥)−2}∇𝑢_𝜀𝑢^{𝜀𝑝(𝑥)−1}∇𝑝(𝑥)ln𝑢_𝜀𝑢^2𝜀−𝜀²−𝑢²𝑑𝑥𝑑𝑡.(4.56) Hence, _𝑄𝑇𝜕𝑢_𝜀𝜕𝑡𝑢^{𝜀𝑝(𝑥)−2}𝑢^2𝜀+−𝜀²−𝑢²𝑑𝑥𝑑𝑡_𝑄𝑇21^{𝑝(𝑥)−1}||∇𝑢^2𝜀||^{𝑝(𝑥)−2}∇𝑢^2𝜀−||∇𝑢²||^{𝑝(𝑥)−2}∇𝑢²≤−_𝑄𝑇∇𝑢^2𝜀−𝑢²𝑑𝑥𝑑𝑡21^{𝑝(𝑥)−1}+𝜀₂||∇𝑢²||^{𝑝(𝑥)−2}∇𝑢²∇𝑢^2𝜀−𝑢²𝑑𝑥𝑑𝑡_𝑄𝑇−(𝑝(𝑥)−1)||∇𝑢_𝜀||^𝑝(𝑥)𝑢^{𝜀𝑝(𝑥)−2}𝑑𝑥𝑑𝑡_𝑄𝑇≤−||_𝑄𝑇∇𝑢_𝜀||^{𝑝(𝑥)−2}∇𝑢_𝜀𝑢^{𝜀𝑝(𝑥)−1}∇𝑝(𝑥)ln𝑢_𝜀𝑢^2𝜀−𝜀²−𝑢²𝑑𝑥𝑑𝑡21^{𝑝(𝑥)−1}+𝜀₂||∇𝑢²||^{𝑝(𝑥)−2}∇𝑢²∇𝑢^2𝜀−𝑢²𝑑𝑥𝑑𝑡_𝑄𝑇+𝐶_𝑄𝑇(𝑝(𝑥)−1)||∇𝑢_𝜀||^𝑝(𝑥)𝑢^{𝜀𝑝(𝑥)−2}𝑑𝑥𝑑𝑡||∇𝑢_𝜀||^{𝑝(𝑥)−1}𝑢^{𝜀𝑝(𝑥)}𝑢^2𝜀−𝜀²−𝑢²𝑑𝑥𝑑𝑡.(4.57) By (4.43), (4.44), and (4.49), we see that as 𝜀→0, ∇_𝑄𝑇𝑢^2𝜀⟶∇𝑢²,weaklyin𝐿^𝑝(𝑥)𝑄_𝑇,𝜕𝑢_𝜀𝜕𝑡𝑢^{𝜀𝑝(𝑥)−2}𝑢^2𝜀−𝜀²−𝑢²𝑑𝑥𝑑𝑡⟶0,_𝑄𝑇||∇𝑢_𝜀||^{𝑝(𝑥)−1}𝑢^{𝜀𝑝(𝑥)}𝑢^2𝜀−𝜀²−𝑢²𝑑𝑥𝑑𝑡⟶0,(𝜀⟶0).(4.58) Therefore, lim_𝜀→0_𝑄𝑇||∇𝑢^2𝜀||^{𝑝(𝑥)−2}∇𝑢^2𝜀−||∇𝑢²||^{𝑝(𝑥)−2}∇𝑢²∇𝑢^2𝜀−𝑢²𝑑𝑥𝑑𝑡≤0.(4.59) By Lemma 2.5, _𝑄𝑇||∇𝑢^2𝜀||^{𝑝(𝑥)−2}∇𝑢^2𝜀−||∇𝑢²||^{𝑝(𝑥)−2}∇𝑢²∇𝑢^2𝜀−𝑢²𝑑𝑥𝑑𝑡≥_𝑄𝑇||∇𝑢^2𝜀−∇𝑢²||^𝑝(𝑥)𝑑𝑥𝑑𝑡.(4.60) Hence, lim_𝜀→0_𝑄𝑇||∇𝑢^2𝜀−∇𝑢²||^𝑝(𝑥)𝑑𝑥𝑑𝑡≤0,(4.61) that is, _𝑄𝑇||∇𝑢^2𝜀−∇𝑢²||^𝑝(𝑥)𝑑𝑥𝑑𝑡⟶0,(𝜀⟶0).(4.62) Applying (𝑎+𝑏)^𝑝≤2^𝑝(𝑎^𝑝+𝑏^𝑝), (𝑎,𝑏>0), and 2𝑢_𝜀∇(𝑢_𝜀−𝑢)=∇(𝑢^2𝜀−𝑢²)−2(𝑢_𝜀−𝑢)∇𝑢, we have _𝑄𝑇𝑢≤^{𝜀𝑝(𝑥)}||∇𝑢_𝜀−∇𝑢||^𝑝(𝑥)𝑑𝑥𝑑𝑡_𝑄𝑇||∇𝑢^2𝜀−∇𝑢²||^𝑝𝑑𝑥𝑑𝑡+_𝑄𝑇2^𝑝(𝑥)||∇𝑢_𝜀−∇𝑢||^𝑝(𝑥)||∇𝑢||^𝑝(𝑥)𝑑𝑥𝑑𝑡.(4.63) Equation (4.50) and (1) imply that the right side tends to zero as 𝜀→0. Since 𝑢_𝜀≥𝑐 in 𝑄^𝜀𝑐, (2) is proved. (3) is an immediate consequence of (2).

Lemma 4.7. For any 𝛼∈[0,1), we have _𝑄𝑇||∇𝑢_𝜀||^𝑝(𝑥)𝑢^𝜀−𝛼𝑑𝑥𝑑𝑡≤𝐶,(4.64) where 𝐶 is independent of 𝜀.

Proof. From Lemmas 4.3 and 4.4, it is easily seen that _𝑄𝑇||∇𝑢_𝜀,𝜂||^𝑝(𝑥)𝑢^{−𝛼𝜀,𝜂}𝑑𝑥𝑑𝑡≤𝐶,(4.65)_𝑄𝑇|||||∇𝑢_𝜀,𝜂||^𝑝(𝑥)−||∇𝑢_𝜀||^𝑝(𝑥)|||𝑑𝑥𝑑𝑡⟶0,𝜂=𝜂_𝑘⟶0.(4.66) Using 𝑢_𝜀,𝜂,𝑢_𝜀≥𝜀, (4.66), and Proposition 4.5, we have _𝑄𝑇|||||∇𝑢_𝜀,𝜂||^𝑝(𝑥)𝑢^{−𝛼𝜀,𝜂}−||∇𝑢_𝜀||^𝑝(𝑥)𝑢^𝜀−𝛼≤|||𝑑𝑥𝑑𝑡_𝑄𝑇|||||∇𝑢_𝜀,𝜂||^𝑝(𝑥)−||∇𝑢_𝜀||^𝑝(𝑥)|||𝑢_{−𝛼𝜀,𝜂}≤1𝑑𝑥𝑑𝑡+_𝑄𝑇||∇𝑢_𝜀||^𝑝(𝑥)||𝑢^{−𝛼𝜀,𝜂}−𝑢^𝜀−𝛼||𝑢^{−𝛼𝜀,𝜂}𝑢^𝜀−𝛼𝑑𝑥𝑑𝑡𝜀^𝛼_𝑄𝑇|||||∇𝑢_𝜀,𝜂||^𝑝(𝑥)−||∇𝑢_𝜀||^𝑝(𝑥)|||𝑑𝑥𝑑𝑡+1𝜀^2𝛼_𝑄𝑇||∇𝑢_𝜀||^𝑝(𝑥)||𝑢^{−𝛼𝜀,𝜂}−𝑢^𝜀−𝛼||𝑑𝑥𝑑𝑡⟶0.(4.67) The proof of Lemma 4.7 is completed by combining (4.67) with (4.65).

Lemma 4.8. As 𝜀→0, we have(1)_𝑄𝑇|||||∇𝑢_𝜀||^𝑝(𝑥)−||∇𝑢||^𝑝(𝑥)(2)_𝑄𝑇|||𝑑𝑥𝑑𝑡⟶0,|||𝑢_𝜀||∇𝑢_𝜀||^{𝑝(𝑥)−2}∇𝑢_𝜀−𝑢||∇𝑢||^{𝑝(𝑥)−2}∇𝑢|||𝑑𝑥𝑑𝑡⟶0.(4.68)

Proof. Let 𝜒_𝜌 and 𝜒_𝜀𝜌 be the characteristic functions of {(𝑥,𝑡)∈𝑄_𝑇;𝑢(𝑥,𝑡)<𝜌} and {(𝑥,𝑡)∈𝑄_𝑇;𝑢_𝜀(𝑥,𝑡)<𝜌}, respectively. Then _𝑄𝑇|||||∇𝑢_𝜀||^𝑝(𝑥)−||∇𝑢||^𝑝(𝑥)≤|||𝑑𝑥𝑑𝑡_𝑄𝑇|||||∇𝑢_𝜀||^𝑝(𝑥)𝜒_𝜀𝜌−||∇𝑢||^𝑝(𝑥)𝜒_𝜌|||𝑑𝑥𝑑𝑡+_𝑄𝑇|||||∇𝑢_𝜀||^𝑝(𝑥)≤1−𝜒_𝜀𝜌−||∇𝑢||^𝑝(𝑥)1−𝜒_𝜌|||𝑑𝑥𝑑𝑡_𝑄𝑇+||∇𝑢_𝜀||^𝑝(𝑥)𝜒_𝜀𝜌𝑑𝑥𝑑𝑡+_𝑄𝑇||∇𝑢||^𝑝(𝑥)𝜒_𝜌𝑑𝑥𝑑𝑡+_𝑄𝑇||∇𝑢||^𝑝(𝑥)𝜒_𝜀𝜌−𝜒_𝜌𝑑𝑥𝑑𝑡_𝑄𝑇|||||∇𝑢_𝜀||^𝑝(𝑥)−||∇𝑢||^𝑝(𝑥)|||1−𝜒_𝜀𝜌=𝐼₁𝑑𝑥𝑑𝑡+𝐼₂+𝐼₃+𝐼₄.(4.69) Taking 𝛼=1/2 in Lemma 4.7, we obtain that 𝐼₁≤𝐶𝜌^1/2. Since 𝐼₂→0 (𝜌→0), for any 𝛿>0, we can choose 𝜌>0 such that 𝐼₁+𝐼₂<𝛿/3. For fixed 𝜌>0, 𝜒_𝜀𝜌→𝜒_𝜌 (𝜀→0) a.e. in 𝑄_𝑇, so there exists 𝜀₁>0 such that 𝐼₃<𝛿/3 as 𝜀<𝜀₁. By Lemma 4.6, 𝐼₄→0 (𝜀→0), so there exists 𝜀₂∈(0,𝜀₁) such that 𝐼₄<𝛿/3 as 𝜀∈(0,𝜀₂). Summing up, we have _𝑄𝑇|||||∇𝑢_𝜀||^𝑝(𝑥)−||∇𝑢||^𝑝(𝑥)|||𝑑𝑥𝑑𝑡<𝛿,∀𝜀<𝜀₂,(4.70) thus (1) holds. To prove (2), observe that _𝑄𝑇|||𝑢_𝜀||∇𝑢_𝜀||^{𝑝(𝑥)−2}≤1∇𝑢_𝜀−𝑢||∇𝑢||^{𝑝(𝑥)−2}∇𝑢|||𝑑𝑥𝑑𝑡2_𝑄𝑇||∇𝑢_𝜀||^{𝑝(𝑥)−2}||∇𝑢^2𝜀−∇𝑢²||𝑑𝑥𝑑𝑡+12_𝑄𝑇|||||∇𝑢_𝜀||^{𝑝(𝑥)−2}−||∇𝑢||^{𝑝(𝑥)−2}=1|||||∇𝑢₂||𝑑𝑥𝑑𝑡2𝐼^𝑎+𝐼_𝑏.(4.71) Using Höder’s inequality and Lemma 4.7, we obtain that 𝐼_𝑎≤||∇𝑢_𝜀||_{𝑝(𝑥)(𝑝(𝑥)−2)/(𝑝(𝑥)−1)}||∇𝑢^2𝜀−∇𝑢²||_𝑝(𝑥)⟶0.(4.72) By means of the inequality |𝑎^𝑟−𝑏^𝑟|≤|𝑎−𝑏|^𝑟(𝑟∈[0,1],𝑎,𝑏≥0), Höder’s inequality and (1), we have 𝐼_𝑏=_𝑄𝑇||||||∇𝑢_𝜀||^𝑝(𝑥)^{(𝑝(𝑥)−2)/𝑝(𝑥)}−||∇𝑢||^𝑝(𝑥)^{(𝑝(𝑥)−2)/𝑝(𝑥)}||||||∇𝑢₂≤𝐶_𝑄𝑇||𝑑𝑥𝑑𝑡|||||∇𝑢_𝜀||^𝑝(𝑥)−||∇𝑢||^𝑝(𝑥)≤𝐶|||||∇𝑢_𝜀|||^{(𝑝(𝑥)−2)/𝑝(𝑥)}||∇𝑢²||𝑑𝑥𝑑𝑡||^𝑝(𝑥)−||∇𝑢||^𝑝(𝑥)|||||∇𝑢₂||_𝑝(𝑥)/2⟶0.(4.73) Thus the proof of Theorem 4.1 is complete.

Acknowledgment

This work is supported by the National Science Foundation of China (no. J1030101).