Existence of solutions to an initial Dirichlet problem of evolutional <span class="mathjax-formula">$ p(x)$</span>-Laplace equations

Ann. I. H. Poincaré – AN 29 (2012) 377–399

www.elsevier.com/locate/anihpc

Existence of solutions to an initial Dirichlet problem of evolutional p(x)-Laplace equations

Songzhe Lian, Wenjie Gao

, Hongjun Yuan, Chunling Cao

Institute of Mathematics, Jilin University, Changchun, Jilin 130012, People’s Republic of China Received 1 September 2011; received in revised form 20 December 2011; accepted 10 January 2012

Available online 16 January 2012

Abstract

The existence and uniqueness of weak solutions are studied to the initial Dirichlet problem of the equation ut=div

|∇u|^p(x)−2∇u

+f (x, t, u),

with infp(x) >2. The problems describe the motion of generalized Newtonian fluids which were studied by some other authors in which the exponentpwas required to satisfy a logarithmic Hölder continuity condition. The authors in this paper use a difference scheme to transform the parabolic problem to a sequence of elliptic problems and then obtain the existence of solutions with less constraint top(x). The uniqueness is also proved.

©2012 Elsevier Masson SAS. All rights reserved.

Keywords:Electrorheological fluids;p(x)-Laplace; Degenerate; Parabolic

1. Introduction

LetΩ⊂R^N be a bounded domain with Lipschitz continuous boundary∂Ω. Consider the following problem ut=div

|∇u|^p(x)−2∇u

+f (x, t, u), x∈Ω, 0< t < T , (1.1)

u|ΓT =0, u|t=0=u₀, (1.2)

whereΓ_T =∂Ω× [0, T]andp(x)is a measurable function.

In the case whenpis a constant, there have been many results about the existence, uniqueness and the regularity of the solutions. We refer the readers to the bibliography given in [5,11,12] and the references therein.

A new interesting kind of fluids of prominent technological interest has recently emerged: the so-called electrorhe- ological 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 evolutionp-Laplace equation with exponentpas a function of the external electromagnetic field (see [1,2,10] and the references therein).

✩ The project is supported by NSFC (10771085, 10571072) and by the 985 program of Jilin University.

* Corresponding author.

E-mail addresses:liansz@jlu.edu.cn (S. Lian), gaowj@jlu.edu.cn (W. Gao), hjy@jlu.edu.cn (H. Yuan), caocl@jlu.edu.cn (C. Cao).

0294-1449/$ – see front matter ©2012 Elsevier Masson SAS. All rights reserved.

doi:10.1016/j.anihpc.2012.01.001

In [13], Zhikov showed that W₀^1,p(x)(Ω)=

v∈W^1,p(x)(Ω)v|∂Ω=0

=W˚^1,p(x)(Ω).

Hence, the property of the space is different from the case whenpis a constant (see Section 2 for the definition of the function spaces).

As we have known, when p is a constant, the non-degenerate problems have classical solutions and hence the weak solutions exist. But to the case ofp(x)-Laplace type, there is no results to the corresponding non-degenerate problems. These will bring us some new difficulties in studying the weak solutions.

For more generalp(x, t )-Laplace equation, the authors of [3] established the existence and uniqueness results with the exponentp(x, t )satisfying the so-called logarithmic Hölder continuity condition, i.e.

p(x)−p(y)ω

|x−y|

, ∀x, y∈QT, |x−y|<1

2 (1.3)

with

s→lim0⁺ω(s)ln 1

s

=C <∞.

However ifp(x, t )satisfies (1.3), then (see [14]) W₀^1,p(x)(Ω)=W˚^1,p(x)(Ω).

Therefore, we can ask whether the logarithmic Hölder continuity top(x, t )is indispensable for the existence of solutions to the problem.

In the present work, we will study the existence of the solutions to problem (1.1)–(1.2) without the condition (1.3).

Unlike [3] , we will, in this paper, adopt a method of difference in time. Note that the author in [9] considered the p-Laplace equation without the termf (x, t, u)by using a similar method. To overcome the difficulties caused byp(x), we will develop some new ideas and new techniques.

The outline of this paper is the following: In Section 2, we introduce some basic Lebesgue and Sobolev spaces and state our main theorems. In Section 3, we give the existence of weak solutions to a difference equation (approximating problem). In Section 4, we will prove the global existence of solutions to the problem (1.1)–(1.2). Section 5 will be devoted to the proof of the local existence and the existence of weak solutions under some weaker conditions to the initial functionu0.

2. Basic spaces and the main results

To study our problems, we need to introduce some new function spaces.

Denote

p⁺=ess sup

Ω

p(x), p⁻=ess inf

Ω

p(x).

Throughout the paper we assume that

2< p⁻p(x)p⁺<∞, ∀x∈Ω. (2.1)

Set

L^p(x)(Ω)= uuis a measurable real-valued function,

Ω

u(x)^p(x)dx <∞

,

|u|L^p(x)(Ω)=inf λ >0

Ω

u(x) λ

^p(x)dx1

,

W^1,p(x)(Ω)=

u∈L^p(x)(Ω)|∇u| ∈L^p(x)(Ω) ,

|u|W^1,p(x)= |u|L^p(x)(Ω)+ |∇u|L^p(x)(Ω), ∀u∈W^1,p(x)(Ω).

We useW₀^1,p(x)(Ω)to denote the closure ofC₀^∞(Ω)inW^1,p(x).

In the following, we state some of the properties of the function spaces introduced as above (see [6] and [7]).

Proposition 2.1.(i) The space (L^p(x)(Ω),| · |_L^p(x)_(Ω)),(W^1,p(x)(Ω),| · |_W^1,p(x)_(Ω))and W₀^1,p(x)(Ω) are reflexive Banach spaces.

(ii)Letq₁(x)andq₂(x)be real functions with1/q1(x)+1/q2(x)=1andq₁(x) >1. Then, the conjugate space of L^q1(x)(Ω)isL^q2(x)(Ω). And for anyu∈L^q1(x)(Ω)andv∈L^q2(x)(Ω), we have

Ω

uv dx

2|u|_L^q1(x)(Ω)|v|_L^q2(x)(Ω). (iii)

|u|L^p(x)(Ω)=1, then

Ω

|u|^p(x)dx=1,

|u|L^p(x)(Ω)>1, then|u|^p_L⁻p(x)(Ω)

Ω

|u|^p(x)dx|u|^p_L⁺p(x)(Ω),

|u|L^p(x)(Ω)<1, then|u|^p_L⁺p(x)(Ω)

Ω

|u|^p(x)dx|u|^p_L⁻p(x)(Ω).

(iv)Ifp1(x)p2(x), thenL^p1(x)⊃L^p2(x).

Proposition 2.2.Ifp(x)∈C(Ω), then there is a constantC >0, such that

|u|L^p(x)(Ω)C|∇u|L^p(x)(Ω), ∀u∈W₀^1,p(x)(Ω).

This implies that|∇u|L^p(x)(Ω)and|u|W^1,p(x)(Ω)are equivalent norms ofW₀^1,p(x). We now give the definition of the solutions to our problem.

Definition 2.1.A functionuis said to be a weak solution of (1.1)–(1.2), ifusatisfies the following:

u∈L²(QT), f (x, t, u)∈L¹(QT), Diu∈L^p(x)(QT), u=0 on∂Ω×(0, T )in the sense of traces,

T 0

Ω

u∂ϕ

∂t − |∇u|^p(x)−2∇u· ∇ϕ+f ϕ

dx dt=0, (2.2)

for allϕ∈C₀^∞(Q_T)and

tlim→0

Ω

u(x, t )−u₀(x)

ψ (x) dx=0, (2.3)

for allψ∈C₀^∞(Ω), whereQ_T =Ω×(0, T ).

In the study of the global existence of solutions, we need the following hypotheses to the functionf: f (x, t, z)∈C¹

Ω× [0, T] ×R

and f (x, t, z)C₀

φ(x, t)+ |z|^α

, (A)

whereφ0, φ∈L^r(Ω×(0, T )), r > (N+p⁻)/p⁻andC₀>0, α0 are constants.

Our main results are the following.

Theorem 2.1.Letu₀∈L^∞(Ω)∩W₀^1,p(x)(Ω)and(A)hold.

Assume that α < p⁻−1 or

α=p⁻−1 and |Ω|is sufficiently small, (B)

where|Ω|denotes the Lebesgue measure ofΩ. Then there exists a weak solution of (1.1)–(1.2)such that u∈L^∞(Q_T)∩L^∞

0, T;W₀^1,p(x)(Ω)

, u_t∈L²

Ω×(0, T ) .

Remark 2.1.In certain sense, the constrains toαin (B) is necessary even to the case whenpis a constant (see [11]).

Theorem 2.2.Ifu0∈L^∞(Ω)∩W₀^1,p(x)(Ω)andf (x, t, z)∈C¹(Ω× [0, T] ×R). Then there exists aT^∗>0such that(1.1)–(1.2)has a solutionuinQ_T∗.

Theorem 2.3.Iff (x, t, z)∈C¹(Ω× [0, T] ×R), then the solution of (1.1)–(1.2)with u∈L^∞(QT), ut∈L²

Ω×(0, T ) , is unique.

Remark 2.2.Combining Theorems 2.1 and 2.3, we can obtain the existence of global solutions.

We also consider the problem under a weaker condition foru₀. Theorem 2.4.Letu0∈L^∞(Ω).

(i)If (A)and(B)hold, then there exists a weak solutionuof(1.1)–(1.2)such that u∈L^∞(Q_T)∩L^∞

, T;W₀^1,p(x)(Ω)

, u_t∈L²

Ω×(, T )

, (2.4)

where0< < T is a constant.

(ii)Iff (x, t, z)∈C¹(Ω× [0, T] ×R), then there exists aT^∗>0 such that(1.1)–(1.2)has a solution uinQ_T∗

satisfying(2.4).

3. Existence of weak solutions to a difference equation

Let

Fⁱ(x, u)= u ui−1

1 h

(i+1)h ih

f (x, τ, s) dτ

ds, (3.1)

and

ψⁱ(u)=

Ω

1

p(x)|∇u|^p(x)dx−

Ω

Fⁱ(x, u) dx+ 1 2h

Ω

(u−u_i−1)²dx, i=1,2, . . . . (3.2) Denote

p^∗=

⎧⎨

⎩

Np⁻

N−p−, ifN > p⁻, (^N+_N^p−)p⁻, ifNp⁻.

(3.3)

Lemma 3.1.Assume thatp(x)∈C(Ω), u_i−1(x)∈L^p∗(Ω)and(A), (B)hold. Then the functionalψⁱ(u)achieves its minimum on the set

S=

u∈W₀^1,p(x)(Ω)

. (3.4)

Proof. We will show, in three steps, thatψⁱ(u)satisfies the conditions which assure the existence of a minimum on the set.

Step1.Sis weakly closed.

By Proposition 2.1(i) we know thatW₀^1,p(x)(Ω)is a reflexive Banach space and then by Mazur theorem it is weakly closed.

Step2.ψⁱ(u)satisfies the coerciveness conditions.

By (A) we have

ψⁱ(u)

Ω

1

p(x)|∇u|^p(x)dx−C₀

Ω

1 h

(i+1)h ih

φ(x, τ ) dτ

|u−u_i−1|dx

−C₁

Ω

|u|^α+1+ |u_i−1|^α+1 dx+ 1

2h

Ω

(u−u_i−1)²dx, (3.5)

whereC1>0 is a constant.

We first estimate the second term on the right-hand side of the inequality.

Denoter1=(N+p⁻)/p⁻andr2=(N+p⁻)/N.

By (A) and the Hölder inequality, we get

I₁=C₀

Ω

1 h

(i+1)h ih

φ(x, τ ) dτ

|u−u_i−1|dx

C0 Ω

1 h

(i+1)h ih

φ(x, τ ) dτ r1

dx 1/r₁

Ω

|u−u_i−1|^r2dx 1/r₂

C

1 h

Ω (i+1)h

ih

φ^r1(x, τ ) dτ dx 1/r1

Ω

|u−u_i−1|^r2dx 1/r₂

Cu−ui−1L^r2(Ω)C

uL^r2(Ω)+ ui−1L^r2(Ω)

.

Notice thatr2=(N+p⁻)/N < p^∗ for N > p⁻. By the imbedding inequality and Young’s inequality, for all N1, we have

I₁1 4

Ω

|∇u|^p−dx+C1 4

Ω

|∇u|^p(x)

p(x) dx+C.

Now, we estimateC1

Ω(|u|^α+1+ |ui−1|^α+1) dxin the following two cases.

(i)α < p⁻−1.

By Young’s inequality and the Poincaré inequality, we get C₁

Ω

|u|^α+1+ |u_i−1|^α+1 dx

Ω

|u|^p−dx+C

1

4

Ω

|∇u|^p−dx+C1 4

Ω

|∇u|^p(x)

p(x) dx+C.

(ii)α=p⁻−1, but the Lebesgue measure ofΩ is sufficiently small.

By thePoincaréinequality, we get C₁

Ω

|u|^p−+ |u_i−1|^p−

dxC₁

Ω

|u|^p−dx+C

1

4

Ω

|∇u|^p−dx+C1 4

Ω

|∇u|^p(x)

p(x) dx+C.

Summarizing up the above estimates and combining Proposition 2.2, we get

ψⁱ(u) 1

2p⁺|∇u|^p_L⁻p(x)(Ω)−C

1

2Cp⁺|u|^p_W⁻1,p(x)(Ω)−C→ ∞, as|u|_W^1,p(x)_(Ω)→ ∞. Step3.ψⁱ(u)is weakly lower semicontinuous.

At first, by the convexity of the functional, we know that for

Ω 1

p(x)|∇u|^p(x)dx, weakly lower semicontinuous is equivalent to lower semicontinuous (see [4]).

Let

v_l→v, inW₀^1,p(x) asl→ ∞. (3.6)

Then by Proposition 2.1(iii), we have|v_l|L^p(x)(Ω),

Ω|∇v_l|^p(x)dxC,l=1,2, . . .. Now

Ω

1

p(x)|∇vl|^p(x)dx−

Ω

1

p(x)|∇v|^p(x)dx

Ω

1 0

s∇v_l+(1−s)∇v^p(x)−1· |∇v_l− ∇v|ds dx

= 1 0

Ω

s∇v_l+(1−s)∇v^p(x)−1· |∇v_l− ∇v|dx ds.

Then combining Proposition 2.1(ii), (iii), we know that

Ω 1

p(x)|∇u|^p(x)dx is a continuous functional. Therefore,

Ω 1

p(x)|∇u|^p(x)dxis weakly lower semicontinuous.

Now, consider the functional I₂= −

Ω

Fⁱ(x, u) dx+ 1 2h

Ω

(u−u_i−1)²dx.

By (3.6), using (ii) and (iv) of Proposition 2.1, for any 0< < p⁻, we have v_l^weak v, inW₀^1,p−−,

and then using the Sobolev compact imbedding theorem we get vl→v, inL^p∗,

where p^∗=

⎧⎨

⎩

N (p⁻−)

N−(p⁻−), ifN > p⁻−,

N+(p⁻−)

N (p⁻−), ifNp⁻−.

For small enough , we have L^p∗ >max{r/(r−1),2}. Combining (A), we may prove that the functional I₂ is continuous inL^p∗. HenceI₂is weakly lower semicontinuous.

Obviously, the sum of two weakly lower semicontinuous functionals is weakly lower semicontinuous functional and our conclusion follows.

By above results and a standard argument (see [4]), we know that the functionalψⁱ(u)achieves its minimum on the setS. 2

Lemma 3.2.Letu₊=max{0, u}. Assume thatuis a minima obtained in Lemma3.1. Then for any constantk1, bothuand−usatisfy

1 h

Ω

(u−k)²₊dx+

Dk

|∇u|^p(x)dx

1

h

Ω

(u−k)₊ (i+1)h

ih

f (x, τ, u)dτ+ |u_i−1|

dx, (3.7)

whereD_k= {x∈Ω:u(x) > k}.

Proof. For 0 <1, we haveu−(u−k)₊∈Sand then g(−)=ψⁱ

u−(u−k)₊

ψⁱ(u)=g(0).

Therefore,

→lim0, >0

g(−)−g(0)

− 0.

Plugging into the definition ofg, we get 1

h

Ω

(u−k)₊(u−u_i−1) dx+

Ω

|∇u|^p(x)−2∇u∇(u−k)₊dx

1

h

Ω

(u−k)₊ (i+1)h

ih

f (x, τ, u) dτ

dx.

Notice that

(u−k)₊(u−ui−1)=(u−k)²₊−(u−k)₊(ui−1−k)(u−k)²₊−(u−k)₊(ui−1−k)₊, the conclusion of the lemma can be proved easily.

Also, by ψⁱ

u+(−u−k)₊

ψⁱ(u),

we know that the conclusion of the lemma holds for−u. 2 Remark 3.1.Moreover, ifu∈L^∞(Ω), we have

Ω

(u−k)^(q₊⁺¹⁾ h dx+q

Ω

|∇u|^p(x)(u−k)^(q₊⁻¹⁾dx

Ω

(ui−1−k)₊(u−k)^q₊

h dx+1

h

Ω

(u−k)^q₊

(i+1)h ih

f (x, τ, u) dτ dx, (3.8)

whereq1 is a constant.

Now we consider the following problem.

1

h(u_i−u_i−1)=div

|∇u_i|^p(x)−2∇u_i +1

h

(i+1)h ih

f (x, τ, u_i) dτ, x∈Ω, (3.9)

u_i|∂Ω=0, i=1,2, . . . , (3.10)

whereh >0 is a constant.

By Lemma 3.1, similarly to Lemma 3.2, we get

Lemma 3.3.Let(A), (B)hold. Assume thatp(x)∈C(Ω)andu_i−1(x)∈L^p∗(Ω). Then there exists a weak solutionu_i of (3.9)–(3.10)such thatui∈W₀^1,p(x)(Ω).

4. Global existence of weak solutions In the following we assume that

lhT < (l+1)h, wherelis an integer.

Defineu^h:Ω× [0,∞)→Rsuch that u^(h)(·, t )=ui, fort∈

ih, (i+1)h

, i=0,1, . . . , l (4.1)

whereui is a solution obtained in Lemma 3.3.

We will prove that a subsequence ofu^(h)converges and the limiting function is a solution of (1.1)–(1.2).

Denote

∂^(−h)u^(h)(·, t )= 1

−h

u^(h)(·, t−h)−u^(h)(·, t )

= ^h1(u_i−u_i−1)(·), fort∈

ih, (i+1)h

, 1il,

0, fort∈ [0, h). (4.2)

Define the following new functionsf^(h)(x, t )andφ^(h)(x, t )as

f^(h)(x, t )=1 h

(i+1)h ih

f

x, τ, u_i(x)

dτ, fort∈

ih, (i+1)h

, i=0,1, . . . , l (4.3)

φ^(h)(x, t )=1 h

(i+1)h ih

φ(x, τ ) dτ, fort∈

ih, (i+1)h

, i=0,1, . . . , l. (4.4)

By (A) we have f^(h)(x, t )C₀

φ^(h)+u^(h)α

. (A)

Lemma 4.1.Ifφ∈L^r(Q_T), thenφ^(h)∈L^r(Q_T)and

Q_T

φ^(h)r

dx dt

Q_T

φ^rdx dt,

whereris given in(A).

Proof. By Hölder’s inequality

Q_T

φ^(h)r

dx dt=

i (i+1)h

ih

Ω

1 h

(i+1)h ih

φ(x, τ ) dτ r

dx

=h

Ω

i

1 h

(i+1)h ih

φ(x, τ ) dτ r

dxh

Ω

i

1 h

(i+1)h ih

φ^r(x, τ ) dτ

dx

=

Ω

i

(i+1)h ih

φ^r(x, τ ) dτ

dx=

QT

φ^rdx dt. 2

In the following, we will give the estimate to the maximum norm of the solution by adopting the method in [11].

Lemma 4.2.Let (A), (B)hold. Assume that p(x)∈C(Ω)and u₀∈L^∞(Ω)∩W₀^1,p(x)(Ω). Then for any integer 1q <∞, there is a constantC(q) >0independent ofhsuch that

u^(h)

L^q+1(QT)C(q), ∀h >0.

Proof. Let u₊=max{0, u}andk be chosen so thatu₀L^∞(Ω)k. Multiplying (3.9) by (q+1)(ui −k)^q₊ and integrating overΩ we get

(q+1)

Ω

(ui−k)^(q₊⁺¹⁾

h dx+q(q+1)

Ω

|∇ui|^p(x)(ui−k)^(q₊⁻¹⁾dx

=(q+1)

Ω

(u_i−1−k)(u_i−k)^q₊

h dx+(q+1)

Ω

(u_i−k)^q₊f^(h)(x, ih) dx (q+1)

Ω

(u_i−1−k)₊(u_i−k)^q₊

h dx+(q+1)

Ω

(u_i−k)^q₊f^(h)(x, ih) dx. (4.5) By Young’s inequality

(u_i−1−k)₊(u_i−k)^q₊ q

q+1(u_i−k)^(q₊⁺¹⁾+ 1

q+1(u_i−1−k)^(q₊⁺¹⁾. Hence we have

Ω

(u_i−k)^(q₊⁺¹⁾

h dx+q(q+1)

Ω

|∇u_i|^p(x)(u_i−k)^(q₊⁻¹⁾dx

Ω

(ui−1−k)^(q₊⁺¹⁾

h dx+(q+1)

Ω

(ui−k)^q₊f^(h)(x, ih) dx, i=1, . . . , l. (4.6)

Summing overiin (4.6) and considering the definition ofu^(h), we have

Ω

u^(h)−k(q+1)

+ (·, t ) dx+q(q+1)

(l+1)h h

Ω

∇u^(h)p(x)

u^(h)−k(q−1) + dx dt

Ω

(u₀−k)^(q₊⁺¹⁾dx+(q+1)

(l+1)h h

Ω

u^(h)−kq

+f^(h)dx dt, (4.7)

wheret∈ [h, (l+1)h). By Young’s inequality ∇u^(h)p−∇u^(h)p(x)+C.

Usingu0L∞(Ω)k, we get

sup

t∈(h,(l+1)h)

Ω

u^(h)−k(q+1)

+ (·, t ) dx+q(q+1)

(l+1)h h

Ω

∇u^(h)p−

u^(h)−k(q−1) + dx dt

Cq(q+1)

(l+1)h h

Ω

u^(h)−k(q−1)

+ dx dt+(q+1)

(l+1)h h

Ω

u^(h)−kq

+f^(h)dx dt

=Cq(q+1)I1+(q+1)I2. (4.8)

Denote

μ(k)=(x, t )∈Ω×

0, (l+1)h

:u^(h)k.

By Young’s inequality and the Poincaré inequality, we get I₁

(l+1)h h

Ω

u^(h)−k(q+p⁻−1)

+ dx dt+C(q)μ(k)

C

|Ω| ^(l+1)h

h

Ω

∇

u^(h)−k(q+p⁻−1)/p^−p−dx dt+C(q)μ(k). (4.9) Now we estimateI₂.

By the Hölder inequality and the Poincaré inequality

(l+1)h h

Ω

u^(h)−kq+α + dx dt

C

(l+1)h

h Ω

u^(h)−kq+p⁻−1

+ dx

(q+α)/(q+p⁻−1)

dt

C

|Ω| ^(l+1)h

h Ω

∇

u^(h)−k(q+p⁻−1)/p⁻ + ^p−dx

(q+α)/(q+p⁻−1)

dt. (4.10)

Similarly to the above, using the imbedding theorem, we may prove (see Lemma 3.1 in [11]) that

(l+1)h h

Ω

u^(h)−kq

+φ^(h)dx dt

C

sup

t∈(h,(l+1)h)

Ω

u^(h)−k(q+1) + (·, t ) dx +

(l+1)h h

Ω

∇

u^(h)−k(q+p⁻−1)/p⁻

+ ^p−dx dt

q₁

, (4.11)

whereq₁=q(N+p⁻)/(q(N+p⁻)+N (p⁻−1+p⁻/N )) <1.

Combining (4.10), (4.11), Lemma 4.1 and (A), we can obtain the estimate forI2. Substituting it into (4.8), by|μ(k)|2|QT|and Young’s inequality, we get

sup

t∈(0,(l+1)h)

Ω

u^(h)−k(q+1)

+ (·, t ) dxC. (4.12)

Here we used the fact that(u^(h)−k)₊(·, t )=0, fort∈ [0, h).

Ifα=p⁻−1 and|Ω|is sufficiently small, by the Poincaré inequality C

|Ω|

→0, as|Ω| →0,

in (4.9) and (4.10). Thus we can also obtain the estimate forI₂. Substituting it into (4.8), we may prove (4.12).

Similarly, we may prove sup

t∈(0,(l+1)h)

Ω

−u^(h)−k(q+1)

+ (·, t ) dxC.

Thus u^(h)

L^q+1(Q_T)C(q). 2

Remark 4.1.If we takek=0 andq=1 in Lemma 4.2, then

(l+1)h h

Ω

∇u^(h)p(x)dx dtC. (4.13)

Remark 4.2.For studying above problems, we have to study the terms like

|∇u_i|^p(x)|u_i|^qdx(q0) for|∇u_i| ∈ L^p(x) (see (4.5)). To insure that integrals to be well-defined, we needu_i∈L^∞. Actually, the solution that we get in Lemma 3.1 can be considered as a bounded function (see Lemma 4.4).

Now, we give a uniform estimate to the maximum norm of the solution.

We shall need the following proposition.

Proposition 4.1.(See [5, p. 12].) Let{Y_n},n=0,1,2, . . ., be a sequence of positive numbers, satisfying the recursive inequalities

Y_n+1BbⁿY_n^1+β

whereB, b >1andβ >0are given numbers. If

Y0B^−1/βb^−1/β2, (4.14)

then{Y_n}converges to zero asn→ ∞.

Lemma 4.3.Let the assumptions of Lemma4.2hold. Then there is a constantM₁>0depending only onT,|Ω|,N, p⁻,r,u₀L^∞(QT)such that

u^(h)

L^∞(QT)M1, ∀h >0.

Proof. Letkbe chosen so thatu₀L^∞(Ω)kand denote J_k= sup

t∈(0,(l+1)h)

Ω

u^(h)−k2

+(·, t ) dx+

(l+1)h 0

Ω

∇

u^(h)−k

+^p−dx dt.

Takeq=1 in (4.8), then by (A) JkC1

(l+1)h 0

Ω

φ^(h)+u^(h)α

u^(h)−k

+dx dt+μ(k)

. (4.15)

Now we estimate the integral of the right-hand side of the inequality.

By Lemma 4.1 and Hölder’s inequality, we get

(l+1)h 0

Ω

φ^(h)

u^(h)−k

+dx dt

C₂ (l+1)h

0

Ω

u^(h)−kr/(r−1)

+ dx dt

(r−1)/r

C3

(l+1)h 0

Ω

u^(h)−kp⁻+2p⁻/N

+ dx dt

N/(p⁻N+2p⁻)

μ(k)^{(r−1)/r−N/(p−N+2p−)}.

Hence, by imbedding inequality (see [5, p. 7] or [8, p. 62]) we have

(l+1)h 0

Ω

u^(h)−k

+φ^(h)dx dtC₄J_k^{(N+p−)/(p−N+2p−)}μ(k)^{(r−1)/r−N/(p−N+2p−)}. (4.16) Also, by Lemma 4.2 we have

(l+1)h 0

Ω

u^(h)α

u^(h)−k

+dx dt

(l+1)h

0

Ω

u^(h)αr

dx dt

1/r(l+1)h 0

Ω

u^(h)−kr/(r−1)

+ dx dt

(r−1)/r

C5

(l+1)h 0

Ω

u^(h)−kr/(r−1) + dx dt

(r−1)/r

C₆J_k^{(N+p−)/(p−N+2p−)}μ(k)^{(r−1)/r−N/(p−N+2p−)}. (4.17) Substituting (4.16), (4.17) into (4.15), we get

J_kC₇J_k^{(p−+N )/(p−N+2p−)}μ(k)^{(r−1)/r−N/(p−N+2p−)}+C₇μ(k).

By Young’s inequality J_kC₈

μ(k)^{1+(r−N−2)p−/(rN (p−−1)+rp−)}+μ(k) . Hence, for allk⁽²⁾k⁽¹⁾, we have

k⁽²⁾−k⁽¹⁾ μ

k⁽²⁾N/(p⁻N+2p⁻)

C9

^(l+1)h 0

Ω

u^(h)−k⁽¹⁾(p⁻+2p⁻/N ) +

N/(p⁻N+2p⁻)

C₉γ^{N/(p−N+2p−)}J^{(N+p−)/(p−N+2p−)}

k⁽¹⁾

C₁₀ μ

k⁽¹⁾1+(r−N−2)p⁻/(rN (p⁻−1)+rp⁻)

+μ

k⁽¹⁾(N+p⁻)/(p⁻N+2p⁻)

, (4.18)

whereγ is a constant, depending only onN, p⁻, T, comes from imbedding inequality (see [5, p. 7] or [8, p. 62]).

If we takek⁽²⁾= u₀L^∞(Ω)+j(j >1) andk⁽¹⁾= u₀L^∞(Ω)+1, then μ

k⁽²⁾N/(p⁻N+2p⁻)

C₁₀

j−1

(T+1)|Ω|1+(r−N−2)p⁻/(rN (p⁻−1)+rp⁻)

+(T+1)|Ω|(N+p⁻)/(p⁻N+2p⁻)

. Hence, there exists a constantj₀>1 depending only onT,|Ω|,N,p⁻,rsuch that

μ k⁽²⁾

1, asjj₀.

We takek_m= ˜M(2−2^−m),m=0,1,2, . . ., whereM˜ u₀L^∞(Ω)+j₀is a constant.

Then it is easy to see that

μ(k_m)1, form=0,1, . . . . (4.19)

Now we consider the following two cases.