Recently, partial differential equations involving variable exponents, such as the p(x)-Laplace equation in (1.1), have been extensively investigated, owing to their physical importance and powerful application

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu

EXISTENCE OF SOLUTIONS TO A PARABOLIC p(x)-LAPLACE EQUATION WITH CONVECTION TERM VIA L^∞ ESTIMATES

ZHONGQING LI, BAISHENG YAN, WENJIE GAO

Abstract. This article is devoted to the study of the existence of weak so- lutions to an initial and boundary value problem for a parabolicp(x)-Laplace equation with convection term. Using the De Giorgi iteration technique, the authors establish the critical a prioriL^∞-estimates and thus prove the exis- tence of weak solutions.

1. Introduction

In this article, we consider the initial and boundary value problem for parabolic p(x)-Laplace equation

∂u

∂t −div |∇u|^p(x)−2∇u

=B(x, t)|∇u|^p(x)−div−→

F(x, t), (x, t)∈Q_T, u(x, t) = 0, (x, t)∈ΓT,

u(x,0) =u0(x)∈L^∞(Ω), x∈Ω.

(1.1)

Here, Ω⊂R^N is a bounded domain with smooth boundary∂Ω, QT = Ω×(0, T), ΓT =∂Ω×(0, T), T >0 is finite, and p(x),B(x, t), −→

F(x, t) are given quantities satisfying conditions to be specified later.

Recently, partial differential equations involving variable exponents, such as the p(x)-Laplace equation in (1.1), have been extensively investigated, owing to their physical importance and powerful application. The mathematical model of Problem (1.1) originates from heat and mass transfer in nonhomogeneous media and non- Newtonian fluids with thermo-convective effects [2]. Equations of this type also appear in the study of digital image recovery [4] and electrorheological fluids [16].

It describes the evolution diffusion and filtration process. In particular, the models like (1.1) with variable exponent provide a good mathematical interpretation for the mechanical properties of certain viscous electrorheological fluids characterized by their abilities to undergo significant changes when an electric field is applied.

We focus on mathematical analysis concerning the existence of solutions to Prob- lem (1.1). Similar problems with constant exponents orL¹ data have been studied by many authors; see, e.g., [3, 5, 13, 14, 15, 18, 21, 24]. To study our problem,

2000Mathematics Subject Classification. 35K65, 35K55, 46E35.

Key words and phrases. Parabolicp(x)-Laplace equation; convection term;

De Giorgi iteration;L^∞estimates.

c

2015 Texas State University - San Marcos.

Submitted November 28, 2014. Published February 17, 2015.

1

we encounter several difficulties arising from the variable exponents. To deal with (1.1), one must face the typical difficulty of how to define the solution space to (1.1).

Whenp(x) =pis a constant, it is well known thatL^p(0, T;W₀^1,p(Ω)) can be taken as the solution space. However, in the nonconstant case andp⁻ = infp(x)>1, if the solution space is defined to be L^p(x)(0, T;W₀^1,p(x)(Ω)), or L^p−(0, T;W₀^1,p(x)(Ω)), etc., then it leads to an unfavorable fact that the p(x)-Laplace operator is not bounded and not continuous from this space into its dual. To conquer this dif- ficulty, we adopt the appropriate solution space V as defined below, which helps us to define a weak solution to (1.1). However, other difficulties arise from it at the same time. On one hand, one must verify the chain rule in the variable ex- ponent space, as given in Lemma 2.2 with its proof in the Appendix, even if this is an obvious fact in the case when p is a constant [5, 13]. On the other hand, we will get the existence result for Problem (1.1) through a limit process in which Simon’s compactness theorem [17] plays a crucial role. Nevertheless, the solution spaceV prevents from directly employing the theorem. We take into account the properties associated withV and surmount this difficulty. There are other differ- ences between the variable exponent case and the constant exponent case. Some important properties and inequalities are no longer valid. For example, the variable exponent spaces are not translation invariant, Young’s inequality with convolution kf∗gk_p(·)≤Ckfk_p(·)kgk1holds if and only ifpis constant, and foru∈W₀^1,p(x)(Ω), R

Ω|u|^p(x)dx≤CR

Ω|∇u|^p(x)dxis not valid for the variable exponentp, etc.; we refer to monograph [7] for details and more references.

To define an appropriate solution space for Problem (1.1), we make the following hypotheses on the quantities appearing in (1.1).

(H1) p∈C(Ω), and p⁺ := max_Ωp(x), p⁻ := min_Ωp(x) satisfy 1 < p⁻ ≤p⁺ <

+∞; furthermore, there exists a positive constantCsuch that the following log-H¨older continuous condition holds:

|p(x)−p(y)| ≤ −C

log|x−y| for everyx, y∈Ω satisfying |x−y| ≤ 1

2. (1.2) (H2) B ∈ L^∞(Q_T) satisfies 0 ≤ B(x, t) ≤ b, where b > 0 is a constant, and

−

→F is a vector field satisfying|−→

F|^(p−)0 ∈L^r(QT), where (p⁻)⁰ = _p−^p−−1 and r > ^N+p_p−⁻. Hence,−→

F ∈ L^p0(x)(Q_T)N

as|−→

F| ∈L^(p−)0(Q_T),→L^p0(x)(Q_T);

see the relevant definitions below.

We remark that, when p is a constant, it is well known that W₀^1,p(Ω) (the clo- sure of C₀^∞(Ω) in W^1,p(Ω)) is identical to H₀^1,p(Ω) := {f ∈ L^p(Ω) : |∇f| ∈ L^p(Ω) with f|∂Ω = 0}. However, when p is a function, there exists an interest- ing Lavrentiev phenomenon [22], which shows that the above two space are not equivalent. The log-H¨older continuous condition (1.2) above guarantees an impor- tant fact that C₀^∞(Ω) is dense in W^1,p(x)(Ω) [23]. Under this condition, one can define variable Sobolev spaces with homogeneous boundary values,W₀^1,p(x)(Ω), as the closure ofC₀^∞(Ω) inW^1,p(x)(Ω); moreover, the condition makesp(x)-Poincar´e’s inequality hold [1, 10, 21].

We introduce the function space

V ={v∈L^p−(0, T;W₀^1,p(x)(Ω)) :|∇v| ∈L^p(x)(Q_T)},

endowed with the norm kukV = |∇u|_Lp(x)(Q_T), or the equivalent norm kukV =

|u|_Lp−(0,T;W₀^1,p(x)(Ω))+|∇u|_Lp(x)(Q_T); the equivalence follows from p(x)-Poincar´e’s inequality. ThenV is a separable and reflexive Banach space (see [3, 21]).

We now give the definition of weak solutions to Problem (1.1).

Definition 1.1. We say thatu∈V∩L^∞(QT) is a weak solution to (1.1), provided thatut∈V^∗+L¹(QT),u(x,0) =u0(x) inL^p−(Ω), and

Z T 0

hut, φidt+ Z T

0

Z

Ω

|∇u|^p(x)−2∇u· ∇φ dx dt

= Z T

0

Z

Ω

B|∇u|^p(x)φ dx dt+ Z T

0

Z

Ω

∇φ·−→ F dx dt

(1.3)

holds for everyφ(x, t)∈V ∩L^∞(QT). Here, withut=α⁽¹⁾+α⁽²⁾∈V^∗+L¹(QT), it is understood that

Z T 0

hut, φidt:=hut, φiV^∗+L¹(Q_T),V∩L^∞(Q_T)=hα⁽¹⁾, φiV^∗,V + Z T

0

Z

Ω

α⁽²⁾φ dx dt.

Whenp(x) =pis a constant, sup-/sub-solution method is powerful and direct to the existence results (see [13]). Nevertheless, it is not suitable to our problem be- cause, due to the complicated nonlinearities ofp(x)-Laplace, it may be quite difficult to construct a supsolutionuand a subsolutionuinV which simultaneously satisfy u≤u. Roughly speaking, in Equation (1.1), the growth power of|∇u|^p(x)−2∇uat the left-hand side of (1.1) is less than that of the convection term |∇u|^p(x) at the right-hand side, which leads us not to directly utilizing pseudo-monotone operator method [12]. Instead, to obtain the existence of weak solutions to Problem (1.1), we will employ theL^∞ estimate method and get the solution through a limit process to the approximate equations. We carry out the De Giorgi iteration, different from the classical constant exponent case (see [24, 5, 14] and the excellent and elegant argument therein), in the setting of variable exponent. We first give a general form of [5, Theorem 5.1] or [24, Lemma 1], as stated in (2.5), by which we obtain the L^∞ regularity under the classification when p⁻ ≥2 and when 1 < p⁻ <2, other than the classification appeared in [24]. It should be remarked that, we employ the infimum ofp(x), which facilitates this iteration, however, on the other side of the coin, it makes the iteration process more technical and complexity. By the way, our result in Theorem 2.3 shows an interesting phenomenon: the uniformly L^∞ bound of u can depend on p⁻ other than p(x) itself as in the constant exponent case [24]. In the limit process, the properties of solution space V and its related variable exponent space will be frequently used, which is one of the features in the equation with variable exponent.

The plan of this paper is as follows. In section 2, we apply the De Giorgi iteration to Problem (1.1) to obtain a uniform bound for the boundedweak solutionu∈V; this a prioriL^∞-assumption is crucial for such a uniform bound, as in [5, 14]. In section 3, we construct an approximation equation to Problem (1.1). Based on the uniform bound of u_n, we obtain the strong convergence ofu_n in the solution spaceV, by virtue of which we establish the existence of solutions. Section 4 is an Appendix in which we give some brief proofs to some lemmas in the paper.

To conclude this section, we recall some preliminary results on the Lebesgue and Sobolev spaces with variable exponents; for more details, see [9, 10] or monograph [7, 16]. Letpbe a continuous function defined in Ω,p(x)>1, for anyx∈Ω.

1. The space L^p(x)(Ω) :=

u:uis measurable in Ω and Z

Ω

|u(x)|^p(x)dx <∞ .

This space is equipped with the Luxemburg’s norm

|u|_Lp(x)(Ω):= inf λ >0 :

Z

Ω

|u(x)

λ |^p(x)dx≤1 . The space L^p(x)(Ω),| · |_Lp(x)(Ω)

is a separable, uniformly convex Banach space.

2. The space

W^1,p(x)(Ω) :=n

u∈L^p(x)(Ω) :|∇u| ∈L^p(x)(Ω)o , endowed with the norm

|u|_W1,p(x)(Ω):=|∇u|_Lp(x)(Ω)+|u|_Lp(x)(Ω).

We denote by W₀^1,p(x)(Ω) the closure of C₀^∞(Ω) inW^1,p(x)(Ω). In fact, the norm

|∇u|_Lp(x)(Ω) and|u|_W1,p(x)(Ω) are equivalent norms inW₀^1,p(x)(Ω). W^1,p(x)(Ω) and W₀^1,p(x)(Ω) are separable and reflexive Banach spaces.

3. Frequently used relationships for the estimates.

min

|u|^p_L⁻_p(x)(Ω), |u|^p_L⁺_p(x)(Ω) ≤ Z

Ω

|u(x)|^p(x)dx≤max

|u|^p_L⁻_p(x)(Ω), |u|^p_L⁺_p(x)(Ω) .

Consequently,

|uk−u|_Lp(x)(Ω)→0⇐⇒

Z

Ω

|uk−u|^p(x)dx→0.

4. p(x)-H¨older’s inequality: For any u ∈ L^p(x)(Ω) and v ∈ L^p0(x)(Ω), with

1

p(x)+_p0¹(x)= 1, we have

Z

Ω

uv dx ≤ 1

p⁻ + 1 (p⁰)⁻

|u|_Lp(x)(Ω)|v|_Lp0(x)(Ω)≤2|u|_Lp(x)(Ω)|v|_Lp0(x)(Ω). 5. Embedding relationships: Ifp1 and p2 are inC(Ω), and 1≤p1(x)≤p2(x), for anyx∈Ω, then there exists a positive constantC_p1(x),p2(x) such that

|u|_Lp1 (x)(Ω)≤C_p1(x),p2(x)|u|_Lp2 (x)(Ω).

i.e. the embedding L^p2(x)(Ω) ,→ L^p1(x)(Ω) is continuous. If q ∈ C(Ω) and 1 ≤ q(x)< p^∗(x), for anyx∈Ω, then the embeddingW₀^1,p(x)(Ω),→L^q(x)(Ω) is contin- uous and compact, where

p^∗(x) :=

( _{N p(x)}

N−p(x), p(x)< N, +∞, p(x)≥N.

6. p(x)-Poincar´e’s inequality: Under the condition (1.2), there exists a positive constantC_p such that

|u|_Lp(x)(Ω)≤C_p|∇u|_Lp(x)(Ω), for allu∈W₀^1,p(x)(Ω).

2. A priori bounds

First of all, we give some technical lemmas frequently used in the process of De Giorgi iteration. In particular, (2.5) can be seen as a general form of [5, Theorem 5.1] or [24, Lemma 1]. Their proofs will be given in the Appendix for the convenience of the readers.

Lemma 2.1. Assume thata, b, λ are positive constants, withλ≥ ¹₂+_a^b. Define ϕ(s) =

(e^λs−1, s≥0,

−e^−λs+ 1, s≤0. (2.1)

Then the following properties hold:

(1) For alls∈R,

|ϕ(s)| ≥λ|s|, aϕ⁰(s)−b|ϕ(s)| ≥a

2e^λ|s|. (2.2)

(2) There exist constantsd≥0 andM >1 such that, for alls≥d, ϕ⁰(s)≤λM

ϕ s p⁻

^p−

, ϕ(s)≤M ϕ s

p^{− p−}

. (2.3)

(3) Let Φ(s) =Rs

0 ϕ(σ)dσ. If p⁻ ≥2, then there exists a positive constant c^∗ such that

Φ(s)≥c^∗ ϕ s

p⁻ p⁻

, ∀s≥0; (2.4)

if1< p⁻ <2, then there existd≥0 andc^∗=c^∗(p⁻, d)such that Φ(s)≥c^∗

ϕ s p⁻

^p−

, ∀s≥d, Φ(s)≥c^∗

ϕ s p⁻

²

, ∀0≤s≤d.

(2.5)

Lemma 2.2. Assume that function π : R → R is piecewise C¹ with π(0) = 0 and π⁰ = 0 outside a compact set. Let Π(s) = Rs

0 π(σ)dσ. If u ∈ V with u_t ∈ V^∗+L¹(Q_T), then

Z T 0

hut, π(u)idt=hut, π(u)iV^∗+L¹(Q_T),V∩L^∞(Q_T)= Z

Ω

Π(u(T))dx− Z

Ω

Π(u(0))dx.

(2.6) Using the lemmas above, we begin the De Giorgi iteration to get the a prioriL^∞ estimate.

Theorem 2.3. Let u∈L^∞(QT)∩V be a weak solution to Problem (1.1). Then kuk_L^∞_(QT)≤ ku0k_L^∞_(Ω)+C,

whereCis a constant depending onp⁻, N, T, r, b,Ω,|||−→

F|^(p−)0kL^r(Q_T), but indepen- dent of u.

Proof. Letkbe a real number such thatk >ku0k_L^∞_(Ω) and letϕbe the function defined in (2.1) with constantλ≥¹₂+ 2b, whereb >0 is the constant in Hypothesis (H2). (We shall use (2.2) witha= 1 anda= 1/2 below.) Define

Gk(u) =







u−k, ifu > k, u+k, ifu <−k, 0, if|u| ≤k.

Note thatu∈L^∞(QT)∩V; so does ϕ(Gk(u)). Then, for eachτ∈[0, T], one may choosev=ϕ(Gk(u))χ[0,τ]as a test function in (1.3) (whereχAis the characteristic function on the setA). Noting that∇v=χ[0,τ]χ{|u|> k}ϕ⁰(Gk(u))∇u, we have

Z τ 0

hut, ϕ(Gk(u))idt+ Z τ

0

Z

Ω

|∇u|^p(x)ϕ⁰(Gk(u))χ{|u|> k}dx dt

= Z τ

0

Z

Ω

B|∇u|^p(x)ϕ(G_k(u))dx dt+ Z τ

0

Z

Ω

χ{|u|> k}ϕ⁰(G_k(u))∇u·−→ F dx dt.

(2.7) Denote Ak(t) = {x∈Ω :|u(x, t)|> k}. In what follows, we write ϕ =ϕ(Gk(u)) andϕ⁰=ϕ⁰(Gk(u)) for simplicity. Thanks to the choice ofk, one has

Z τ 0

hut, ϕ(Gk(u))idt= Z

Ω

Φ(Gk(u))(τ)dx− Z

Ω

Φ(Gk(u0))dx

= Z

Ak(τ)

Φ(Gk(u))(τ)dx− Z

Ak(0)

Φ(Gk(u0))dx

= Z

A_k(τ)

Φ(G_k(u))(τ)dx.

(2.8)

From Young’s inequality with, it follows that Z τ

0

Z

A_k(t)

ϕ⁰∇u·−→ F dx dt

≤ Z τ

0

Z

A_k(t)

|∇u|^p−ϕ⁰dx dt+C() Z τ

0

Z

A_k(t)

|−→

F|^(p−)0ϕ⁰dx dt.

(2.9)

Substituting (2.8) and (2.9) in (2.7) yields Z

Ak(τ)

Φ(Gk(u))(τ)dx+ Z τ

0

Z

Ak(t)

|∇u|^p(x)(ϕ⁰−B|ϕ|)dx dt

≤ Z τ

0

Z

A_k(t)

|∇u|^p−ϕ⁰dx dt+C() Z τ

0

Z

A_k(t)

|−→

F|^(p−)0ϕ⁰dx dt.

(2.10)

Note thatϕ⁰−B|ϕ| ≥ϕ⁰−b|ϕ| ≥¹₂e^λ|Gk(u)|>0 by (2.2) (witha= 1). By utilizing

|∇u|^p(x)≥ |∇u|^p−−1 and choosing= ¹₂, we get from (2.10) that Z

Ak(τ)

Φ(Gk(u))(τ)dx+ Z τ

0

Z

Ak(t)

|∇u|^p− 1

2ϕ⁰−B|ϕ|

dx dt

≤C Z τ

0

Z

A_k(t)

|−→

F|^(p−)0ϕ⁰dx dt+ Z τ

0

Z

A_k(t)

(ϕ⁰−B|ϕ|)dx dt

≤ Z τ

0

Z

A_k(t)

C|−→

F|^(p−)0+ 1

ϕ⁰dx dt.

(2.11)

Using (2.2) witha=¹₂, we have¹₂ϕ⁰−B|ϕ| ≥ ¹₂ϕ⁰−b|ϕ| ≥ ¹₄e^λ|Gk(u)|>0. Denoting w_k=ϕ_|G

k(u)|

p⁻

, we proceed to estimate (2.11), Z τ

0

Z

Ak(t)

|∇u|^p−1

2ϕ⁰−B|ϕ|

dx dt≥ 1 4

Z τ 0

Z

Ak(t)

|e^λ

|Gk(u)|

p− ∇u|^p−dx dt

≥ 1 4

1 λ

^p− Z τ

0

Z

A_k(t)

|∇w_k|^p−dx dt.

(2.12)

By definition, Ak(t)\Ak+d(t) = t{x ∈ Ω : k < |u(x, t)| ≤ k+d}; hence 0 <

|Gk(u)| ≤dandϕ⁰(Gk(u)) =λe^λ|Gk(u)|≤λe^λd onAk(t)\Ak+d(t). So, from (2.3), it follows that

Z τ 0

Z

Ak(t)

C|−→

F|^(p−)0+ 1

ϕ⁰dx dt

≤λM Z τ

0

Z

A_k+d(t)

C|−→

F|^(p−)0+ 1

|wk|^p−dx dt

+ Z τ

0

Z

Ak(t)\Ak+d(t)

C|−→

F|^(p−)0+ 1

ϕ⁰dx dt

≤λM Z τ

0

Z

A_k+d(t)

h|wk|^p−dx dt+λe^λd Z τ

0

Z

A_k(t)\Ak+d(t)

h dx dt,

(2.13)

whereh=C|−→

F|^(p−)0+ 1. Putting (2.11), (2.12) and (2.13) together, we deduce Z

Ak(τ)

Φ(Gk(u))(τ)dx+1 4

1 λ

^p− Z τ

0

Z

Ak(t)

|∇wk|^p−dx dt

≤λM Z τ

0

Z

A_k+d(t)

h|wk|^p−dx dt+λe^λd Z τ

0

Z

A_k(t)\Ak+d(t)

h dx dt.

(2.14)

Case 1. p⁻≥2. In this case, by (2.4), one has Z

A_k(τ)

Φ(Gk(u))(τ)dx≥c^∗ Z

A_k(τ)

|wk|^p−dx. (2.15) Substituting (2.15) in (2.14) and taking the supremum for τ∈[0, t1], witht1≤T to be determined later, we have

c^∗ sup

τ∈[0,t1]

Z

A_k(τ)

|wk|^p−dx+1 4

1 λ

p⁻Z t1

0

Z

A_k(t)

|∇wk|^p−dx dt

≤λM Z t₁

0

Z

Ak(t)

h|wk|^p−dx dt+λe^λd Z t₁

0

Z

Ak(t)\Ak+d(t)

h dx dt.

(2.16)

By the embedding inequality (see [6, 11]), we have Z t₁

0

Z

Ak(t)

|wk|^p−N+p

−

N dx dt_N+p^N−

≤γ sup

τ∈[0,t1]

Z

A_k(τ)

|w_k|^p−dx+ Z t1

0

Z

A_k(t)

|∇w_k|^p−dx dt ,

(2.17)

whereγis a constant depending onN, p⁻, but independent oft₁≤T. Hence, from (2.16), it follows that

Jk_t1 :=Z t₁ 0

Z

A_k(t)

|wk|^p−N+p

−

N dx dt_N+p^N−

≤CZ t₁ 0

Z

Ak(t)

h|wk|^p−dx dt+ Z t₁

0

Z

Ak(t)\Ak+d(t)

h dx dt ,

where C is a constant independent of t1. Consequently, by H¨older’s inequality (thanks to the assumption|−→

F|^(p−)0 ∈L^r(Q_T) withr > ^N+p_p−⁻), we deduce

Jk_t1 ≤CZ t₁ 0

Z

A_k(t)

|wk|^p−N+p

−

N dx dt_N+p^N−Z t₁ 0

Z

A_k(t)

h

N+p−

p− dx dt ^p

− N+p−

+CZ t₁ 0

Z

Ak(t)

h^rdx dt1/rZ t₁ 0

µ(Ak(t))dt1−¹_r

≤CZ t1

0

Z

A_k(t)

|w_k|^p−N+p

−

N dx dt_N+p^N−

khk_Lr(Q_t1) t₁µ(Ω) ^p

− N+p−−¹_r

+CkhkL^r(Q_t1)

Z t₁ 0

µ(Ak(t))dt1−¹_r

,

whereµ(Ω) represents the Lebesgue measure of Ω. Choosingt₁ small enough such that

CkhkL^r(Q_t1)(t₁µ(Ω))

p− N+p−−¹_r

≤ 1

2 (2.18)

and we obtain

Jk_t1 ≤Ckhk_Lr(QT)

Z t₁ 0

µ(Ak(t))dt1−¹_r

. (2.19)

For anyl > k≥ ku0kL^∞(Ω), using (2.2), we conclude that

Jk_t1 ≥Z t₁ 0

Z

Ak(t)

|λGk(u) p⁻ |^p−N+p

−

N dx dt_N+p^N−

≥ λ p⁻

^p−Z t1

0

Z

A_k(t)

|u| −k^p−N+p

−

N dx dt_N+p^N₋

≥ λ p⁻

p⁻

(l−k)^p−Z t1

0

µ(A_l(t))dt_N+p^N−

.

(2.20)

Letψk=Rt1

0 µ(Ak(t))dt. It follows from (2.19) and (2.20) that

ψl≤ C

(l−k)^p

−(N+p−) N

ψ⁽¹⁻

1 r)^N+p_N⁻

k . (2.21)

Case 2. 1 < p⁻ < 2. In this case, from (2.5) (it should be remarked that the constant din (2.3) and (2.5) could be the same if we choosedsuitably large), we have

Z

Ak(τ)

Φ(Gk(u))(τ)dx≥c^∗ Z

Ak+d(τ)

|wk|^p−dx+c^∗ Z

Ak(τ)\Ak+d(τ)

|wk|²dx. (2.22)

Substituting (2.22) into (2.14) and taking the supremum for τ ∈ [0, t1], where t1≤T to be chosen later, we derive

c^∗ sup

τ∈[0,t1]

Z

A_k+d(τ)

|wk|^p−dx+1 4

1 λ

^p− Z t1

0

Z

A_k+d(t)

|∇wk|^p−dx dt

+c^∗ sup

τ∈[0,t1]

Z

Ak(τ)\Ak+d(τ)

|wk|²dx+1 4

1 λ

p⁻Z t₁ 0

Z

Ak(t)\Ak+d(t)

|∇wk|^p−dx dt

≤λM Z t1

0

Z

A_k+d(t)

h|w_k|^p−dx dt+λe^λd Z t1

0

Z

A_k(t)\A_k+d(t)

h dx dt.

(2.23) Again, recall the following embedding estimates [6, 11]:

Z t₁ 0

Z

Ak+d(t)

|wk|^p−N+p

− N dx dt

≤γ^p−N+p

− N

sup

τ∈[0,t₁]

Z

A_k+d(τ)

|wk|^p−dx+ Z t₁

0

Z

A_k+d(t)

|∇wk|^p−dx dt^1+p

− N ,

(2.24) Z t₁

0

Z

Ak(t)\Ak+d(t)

|wk|^p−N+2N dx dt

≤γ^p−N+2N sup

τ∈[0,t₁]

Z

A_k(τ)\Ak+d(τ)

|wk|²dx

+ Z t1

0

Z

A_k(t)\A_k+d(t)

|∇w_k|^p−dx dt1+^p_N⁻

.

(2.25)

Combining (2.24), (2.25) with (2.23), we obtain J_k⁽¹⁾

t1

:=Z t1

0

Z

A_k+d(t)

|wk|^p−N+p

−

N dx dt_N+p^N−

+Z t₁ 0

Z

Ak(t)\Ak+d(t)

|wk|^p−N+2N dx dt_N+p^N−

≤C Z t1

0

Z

A_k+d(t)

h|wk|^p−dx dt+C Z t1

0

Z

A_k(t)\Ak+d(t)

h|wk|^p−dx dt

+C Z t₁

0

Z

Ak(t)

h dx dt:= (E1) + (E2) + (E3).

We estimate (E1) as follows.

(E1)

≤CZ t₁ 0

Z

Ak+d(t)

|wk|^p−N+p

−

N dx dt_N+p^N−Z t₁ 0

Z

Ak+d(t)

h

N+p−

p− dx dt ^p

− N+p−

≤CZ t1

0

Z

A_k+d(t)

|wk|^p−N+p

−

N dx dt_N+p^N−

khkL^r(Q_t1)(t₁µ(Ω))

p− N+p−−¹_r

.

Using H¨older’s inequality and Young’s inequality with, we have (E2)

≤CZ t₁ 0

Z

Ak(t)\Ak+d(t)

|wk|^p−N+2N dx dt_N+2^N Z t₁ 0

Z

Ak(t)\Ak+d(t)

h^N+22 dx dt_N+2²

≤ 1 2

Z t1

0

Z

A_k(t)\A_k+d(t)

|w_k|^p−N+2N dx dt_N+p^N₋

+CZ t₁ 0

Z

Ak(t)\Ak+d(t)

h^N+22 dx dt_2−p²−

≤ 1 2

Z t1

0

Z

A_k(t)\A_k+d(t)

|w_k|^p−N+2N dx dt_N+p^N₋

+Ckhk

N+2 2−p−

L^r(Q_t1)

Z t₁ 0

µ(Ak(t))dt_2−p²−(1−^N+2_2r )

. For (E3), we have

(E3)≤Ckhk_Lr(Qt1)

Z t1

0

µ(Ak(t))dt1−¹_r

. Now selectt1∈(0,(µ(Ω))⁻¹] sufficiently small so that

Ckhk_Lr(Q_t1)(t₁µ(Ω))

p− N+p−−¹_r

≤ 1

2. (2.26)

From the above estimates, we have J_k⁽¹⁾

t1

≤Ckhk_Lr(Q_t1)

Z t1

0

µ(A_k(t))dt¹⁻¹_r +Ckhk

N+2 2−p−

L^r(Q_t1)

Z t1

0

µ(Ak(t))dt_2−p²₋(1−^N+2_2r )

.

(2.27)

Noticing that r > ^N+p_p−⁻, after a straightforward computation, we have _2−p²−(1−

N+2

2r )>1−¹_r. Meanwhile, the choice oft₁ ensuresψ_k ≤t₁µ(Ω)≤1. As a result, (2.27) becomes

J_k⁽¹⁾

t1 ≤CZ t₁ 0

µ(Ak(t))dt1−¹_r

. (2.28)

For anyl > k≥ ku0k_L^∞_(Ω), using (2.2), we deduce that J_k⁽¹⁾

t1

≥Z t1

0

Z

A_k+d(t)

|λGk(u) p⁻ |^p−N+p

−

N dx dt_N+p^N₋

+Z t₁ 0

Z

A_k(t)\Ak+d(t)

|λGk(u)

p⁻ |^p−N+2N dx dt_N+p^N−

≥ λ p⁻

^p−Z t1

0

Z

A_k+d(t)

|u| −k^p−N+p

−

N dx dt_N+p^N₋

+ λ

p⁻

p^{− N+2}

N+p−Z t₁ 0

Z

A_k(t)\Ak+d(t)

|u| −kp^−N+2_N

dx dt_N+p^N−

≥ λ p⁻

^p−

(l−k)^p−Z t1

0

µ(Al(t)∩Ak+d(t))dt_N+p^N₋

+ λ

p⁻

p^{− N+2}

N+p−(l−k)^p

− N+2 N+p−Z t₁

0

µ(Al(t)\Ak+d(t))dt_N+p^N−

.

In fact, we have

J_k⁽¹⁾

t1

^N+p

−

N ≥ λ

p⁻

p^−N+p_N⁻

(l−k)^p−N+p

− N

Z t₁ 0

µ(Al(t)∩Ak+d(t))dt

+ λ

p^{− p−N+2}_N

(l−k)^p−N+2N Z t1

0

µ(A_l(t)\A_k+d(t))dt.

(2.29)

Consequently, combining (2.29) and (2.28), with ψ_k = Rt₁

0 µ(A_k(t))dt, we have again

ψ_l≤ C

min

(l−k)^p

−(N+p−)

N ,(l−k)^p

−(N+2) N

ψ^(1−1r)

N+p− N

k . (2.30)

Now we have proved (2.30) and (2.21). Our hypothesis r > ^N_p^+p−⁻ guarantees 1−¹_rN+p⁻

N >1. Therefore, thanks to the iteration lemma in [24], we eventually obtain that ψ_(ku0kL∞(Ω)+D) = 0, where D > 0 is a constant depending only on p⁻, N, t1, r, b,Ω,k|−→

F|^(p−)0k_Lr(Qt1). This proves that, for a fixedλvalidating Lemma 2.1,

ku(x, t)k_L^∞_(Qt

1)≤ ku0k_L^∞_(Ω)+D. (2.31) Finally, partition the time interval [0, T] into finite subintervals [0, t1], [t1, t2]· · · [t_n−1, T] such that the conditions similar to those in (2.18) and (2.26) are available for each subinterval [ti, ti+1]; then, using the same method, we deduce an inequality of the form (2.31). Eventually, we conclude thatku(x, t)k_L^∞_(QT)≤ ku0k_L^∞_(Ω)+C, where the constantC depends only onp⁻, N, T, r, b,Ω,k|−→

F|^(p−)0kL^r(Q_T). 3. Application to the existence of solutions to(1.1)

With theL^∞-estimate obtained above, we can prove the existence of solutions to Problem (1.1). First, we recall a lemma from [13], which plays an important role in our estimates.

Lemma 3.1. Let θ(s) = se^ηs2, s ∈ R, where η ≥ _4a^b22 is fixed, and let Θ(s) = Rs

0 θ(τ)dτ. Then θ(0) = 0 and

Θ(s)≥0, aθ⁰(s)−b|θ(s)| ≥ a

2, ∀s∈R. (3.1)

We are now in a position to prove the existence of solutions to (1.1) based on theL^∞ estimate.

Theorem 3.2. Under the hypotheses (H1) and(H2), there exists a solution u∈ L^∞(QT)∩V to (1.1).

Proof. Step 1: The approximation equation. We introduce the following approximation equation of Problem (1.1).

∂u_n

∂t −div

|∇u_n|^p(x)−2∇u_n

=B(x, t) min{|∇u_n|^p(x), n} −div−→ F(x, t), (x, t)∈QT,

un(x, t) = 0, (x, t)∈ΓT, u_n(x,0) =u₀(x)∈L^∞(Ω), x∈Ω.

(3.2)

For each fixedn∈N, since min

|∇un|^p(x), n is bounded, the existence of a weak solutionu_n∈L^∞∩V to (3.2) follows from the standard methods (for instance, the

pseudo-monotonicity operator theory in [12, 10, 20], or the difference and variation methods in [21]).

We write the term B(x, t) min{|∇un|^p(x), n} in (3.2) asBn(x, t)|∇un|^p(x), with Bn(x, t) defined by

B_n(x, t) =







0, if|∇un(x, t)|= 0,

B(x, t)^min{|∇u_|∇u^{n(x,t)|p(x),n}}

n(x,t)|^p(x) , if|∇un(x, t)| 6= 0.

ThenBn ∈L^∞(QT) satisfies 0≤Bn(x, t)≤B(x, t)≤b. Hence, by Theorem 2.3, we have the uniform bound

ku_n(x, t)k_L∞(Q_T)≤ ku₀k_L∞(Ω)+C, (3.3) where C depends only on p⁻, N, T, r, b,Ω,k|−→

F|^(p−)0k_Lr(Q_T) and it is independent ofn. Our goal is to show that a subsequence of the approximate solution sequence {u_n} converges to a measurable function u, which coincides with a weak solution of Problem (1.1).

Step 2: The weak convergence u_n * u in L^p−(0, T;W₀^1,p(x)(Ω)). Choosing θ(u_n) as a testing function in (3.2), we have

Z T 0

h∂un

∂t , θ(un)idt+ Z Z

Q_T

|∇un|^p(x)θ⁰(un)dx dt

= Z Z

Q_T

Bmin{|∇un|^p(x), n}θ(un)dx dt+ Z Z

Q_T

θ⁰(un)∇un·−→ F dx dt.

(3.4)

Lemma 2.2 yields RT

0 h^∂u_∂tⁿ, θ(un)idt = R

Ω[Θ(un(T))−Θ(u0)]dx. Using Young’s inequality within the last term of the right-hand side, (3.4) becomes

Z

Ω

Θ(un(T))dx+ Z Z

QT

|∇un|^p(x)θ⁰(un)dx dt

≤ Z

Ω

Θ(u0)dx+ Z Z

QT

B|∇un|^p(x)|θ(un)|dx dt +

Z Z

QT

|∇un|^p(x)θ⁰(un)dx dt+ Z Z

QT

^−p(x)−11 |−→

F|^p0(x)θ⁰(un)dx dt.

Taking= 1/2, we rewrite the above inequality as Z

Ω

Θ(u_n(T))dx+ Z Z

Q_T

1

2θ⁰(u_n)−B|θ(u_n)|

|∇u_n|^p(x)dx dt

≤ Z

Ω

Θ(u0)dx+ 1

2

−_{p− −1}¹ Z Z

Q_T

|−→

F|^p0(x)θ⁰(un)dx dt.

(3.5)

With the aid of (3.1) in Lemma 3.1 (witha= ¹₂, and¹₂θ⁰(un)−B|θ(un)| ≥ ¹₂θ⁰(un)−

b|θ(un)| ≥ ¹₄), we deduce that 1

4 Z Z

Q_T

|∇un|^p(x)dx dt≤ Z

Ω

Θ(u₀)dx+ 1

2

−_{p− −1}¹ Z Z

Q_T

|−→

F|^p0(x)θ⁰(u_n)dx dt.

(3.6) Sinceun is uniformly bounded with respect tonandu0∈L^∞(Ω), it follows that

Z Z

Q_T

|∇un|^p(x)dx dt≤C

|−→

F|_Lp0(x)(Q_T),ku0k_L^∞_(Ω),sup

n

kunk_L^∞_(QT)

. (3.7)

This implies thatunis uniformly bounded inV. By the way, obviously, the following inequality holds

|un|^p−

L^p−(0,T;W₀^1,p(x)(Ω))

= Z T

0

|∇un|^p_L⁻_p(x)(Ω)dt

≤maxnZ Z

QT

|∇un|^p(x)dx dt^p

− p+

T¹⁻

p− p+,

Z Z

QT

|∇un|^p(x)dx dto ,

which implies

|un|_Lp−

(0,T;W₀^1,p(x)(Ω))≤C

|−→

F|_Lp0(x)(Q_T),ku0k_L^∞_(Ω),sup

n

kunk_L^∞_(QT), p⁻, p⁺, T . (3.8) Therefore, un is bounded in the space L^∞(QT)∩L^p−(0, T;W₀^1,p(x)(Ω)). We can extract a subsequence of un, still denoted by un, such that un * u, weakly in L^p−(0, T;W₀^1,p(x)(Ω)). Simultaneously,un * u, weakly* in L^∞(QT).

Step 3: The strong convergence u_n →u in L^p−(0, T;L^p(x)(Ω)). From (3.2), we deduce that

∂un

∂t = div

|∇u_n|^p(x)−2∇u_n−−→ F

+Bmin{|∇u_n|^p(x), n} ∈V^∗+L¹(Q_T). (3.9) For each v ∈V, by the definition of the norm on V and p(x)-H¨older’s inequality, we have

sup

kvkV≤1

|hdiv

|∇un|^p(x)−2∇un−−→ F

, viV^∗,V|

= sup

kvkV≤1

Z Z

Q_T

−|∇u_n|^p(x)−2∇u_n· ∇v+−→ F · ∇v

dx dt

≤ sup

kvkV≤1

2||∇u_n|^p(x)−2∇u_n|_Lp0(x)(QT)|∇v|_Lp(x)(QT)+ 2|−→

F|_Lp0(x)(QT)|∇v|_Lp(x)(QT)

≤2 maxnZ Z

QT

|∇un|^p(x)dx dt_(p0¹)+

,Z Z

QT

|∇un|^p(x)dx dt_(p0¹)−o

+ 2|−→

F|_Lp0(x)(QT). It follows from (3.7) that

div |∇un|^p(x)−2∇un−−→ F

V^∗≤C, (3.10)

whereC is independent ofn. Thanks to the embedding relationship L^(p−)0(0, T;W^−1,p0(x)(Ω)),→V^∗

,→L^(p+)0(0, T;W^−1,p0(x)(Ω)) =L^(p0)−(0, T;W^−1,p0(x)(Ω)),

(3.11) from (3.10), (3.7) and (3.9), we conclude that ^∂u_∂tⁿ is bounded in the space

L^(p0)−(0, T;W^−1,p0(x)(Ω)) +L¹(Q_T).

For a fixeds such thats > ^N₂ + 1, the following embedding relationships hold 1 s > ^N₂, we have H₀^s(Ω),→L^∞(Ω), and then L¹(Ω) ,→H^−s(Ω); 2 s−1> ^N₂,