Journal of Applied Mathematics

VolumeΒ 2012Β (2012), Article IDΒ 835495, 21 pages

http://dx.doi.org/10.1155/2012/835495

## Existence of Solutions for the Evolution -Laplacian Equation Not in Divergence Form

Department of Mathematics, Jilin University, Changchun 130012, China

Received 31 October 2011; Accepted 6 December 2011

Academic Editor: Hui-ShenΒ Shen

Copyright Β© 2012 Changchun Liu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The existence of weak solutions is studied to the initial Dirichlet problem of the equation , with inf . 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 -Laplacian equation The equation is supplemented the boundary condition and the initial condition where , , is a bounded domain with smooth boundary and .

In the case when 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 [1β5] 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 -Laplace equation with exponent as a function of the external electromagnetic field (see [6β12] 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 -Laplace equations [13β15].

In the present work, we will study the existence of the solutions to problem (1.1)β(1.3). As we know, when is a constant, the nondegenerate problems have classical solutions, and hence the weak solutions exist. But in the case of -Laplace type, there are no results to the corresponding non-degenerate problems. Since (1.1) degenerates whenever and , 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 that

Throughout the paper, we assume that where are given constants.

To study our problems, we need to introduce some new function spaces. Denote that We use to denote the closure of in .

*Remark 2.1. *In [16, 17], Zhikov showed
Hence, the property of the space is different from the case when is a constant. This will bring us some difficulties in taking the limit of the weak solutions. Luckily, our approximating solutions are in , and hence the limit function is also in which avoids the above difficulties.

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

*Definition 2.2. *A nonnegative function , , and is said to be a weak solution of (1.1)β(1.3), if for all satisfies the following:

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 is a separable, uniform convex Banach space, and its conjugate space is , where . For any and , we have
*(ii)*If for any , then and the imbedding continuous.*(iii)*There is a constant , such that
This implies that and are equivalent norms of .*(iv)*We have .*

Proposition 2.4 (see [18]). *If we denote
**
then*(i)*,
*(ii)*, **,
*(iii)*.*

Lemma 2.5 (see [4]). *Let . Then
**
where is a positive constant depending on .*

#### 3. A General Quasilinear Equation

Here, we will consider the general quasilinear equations where .

Proposition 3.1 (see [19, Theoremββ2.9 of Chapter I]). *Let be a classical solution of (3.1) in . Suppose that the functions and take finite value for any finite , and , and that for and arbitrary **
where and are nonnegative constants. Then
**
where depends only on , and .*

We suppose that for and arbitrary the functions are continuous in , continuously differentiable with respect to , and , and satisfy the inequalities where is a nonnegative continuous function that tends to zero for and is an arbitrary function.

Lemma 3.2. *Let be a classical solution of (3.1) in . Suppose that the conditions of Proposition 3.1 hold and satisfy (3.5) with a sufficiently small determined by the numbers , , , , and
**
Then
*

The proof of Lemma 3.2 is quite similar to the Theoremββ4.1, chapter VI of [19]; one only has to replace with and remark that the constants in the proof are depending only on and ; we omit the details.

Theorem 3.3. *Suppose that the following conditions are fulfilled.*(a)*For and arbitrary either conditions (3.3) are fulfilled.*(b)*For , (where is taken from estimate (3.4)) and arbitrary , the functions and are continuous and differentiable with respect to , , and and satisfy inequalities (3.5) with a sufficiently small determined by the numbers , , , , and
*(c)*For and (where is taken from estimate (3.7)), the functions and are continuously differentiable with respect to all of their arguments.*(d)*The boundary condition (3.2) is given by a function belonging to and satisfying on (3.1), that is,
(in other words, the compatibility conditions of zero and first orders are assumed to be fulfilled).*(e)*We have .**Then there exists a unique solution of problem (3.1) and (3.2) in the space . This solution has derivatives from .*

*Proof. *We consider problem (3.1) and (3.2) along with a one-parameter family of problems of the same type
Define the Banach space
For any , let . Using Schauder theory, the linear problem
admits a unique solution . Let , clearly , and define the map such that . By [19], we know that is continuous and compact. By Proposition 3.1, Lemma 3.2, and the Leray-Schauder fixed point principle, the operator has a fixed point .

#### 4. Existence

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

Theorem 4.1. *Assume that , and . Then the problem (1.1)β(1.3) admits a weak solution .*

Consider the following problem: where , , and . Roughly speaking, here we use to regularize the initial-boundary value and use to regularize the equation. Thus, we have to carry out two limit processes, that is, first let (along a certain subsequence) and then let .

We first change (4.1) into the form where It is easily seen that (4.4) satisfies (3.3) and (3.5), where instead of . By Theorem 3.3, we know that (4.1)-(4.2) has a classical solution .

Proposition 4.2. *We have
*

*Proof. *By the maximum principle, we know that .

A simple calculation shows that
where and are defined as (4.4).

It is easy to prove that
Hence, we have
Let . Using the mean value theorem, we have
Similarly, we get
where , and are bounded functions. Hence, we see that
Since on , by comparison principle of linear parabolic equation, we have .

Lemma 4.3. *For all and , there hold
*

*Proof. *Multiplying (4.1) by , integrating both sides of the equality over and integrating by parts, we derive
where denotes the outward normal to . Since from (4.5), , we have on . Hence
where .

Using and Youngβs inequality, we have

Combining (4.14) with (4.15) yields
Hence,
Multiplying (4.1) by , integrating both sides of the equality over and integrating by parts and noticing that on , we derive

Equation (4.5), Lemma 4.3, and Proposition 2.3 imply that, for any , there exists a subsequence of , denoted by , and a function ββ, such that, as ,

Lemma 4.4. *As , we have
*

*Proof. *Observe that . Multiplying (4.1) by , integrating both sides of the equality over and integrating by parts, we derive
By HΓΆlder inequality and Lemma 4.3, we obtain
Hence,
We divide the integral in (4.25) in the following way:
From (4.20), we see that
Using Lemma 4.3, we have
Now we estimate . If , then . Using () gives
If , we obtain
By (4.25), (4.26), and , we obtain
Again by Lemma 2.5, we get
Letting , we obtain (1). Again noticing that
by Proposition 2.4, we see that (2) holds. To prove (3), we have
Using Lemma 4.3 and (2), we see that (3) holds.

Finally, we prove (4). We have
Equation (4.19) implies that
To estimate , notice that
As , we have ,
By HΓΆlder inequality, we have
If , we have
Hence,
Thus (4) is proved, and the proof of Lemma 4.4 is complete.

Proposition 4.5. *We obtain that is a weak solution of the problem
**
then
**
where is independent of .*

*Proof. *Obviously, for all , . By Proposition 4.2 and (4.19)β(4.21), we know that (4.43) holds. (4.44) follows from (4.5), (4.19)β(4.21), and Lemma 4.3. To prove that satisfies the integral equality in Definition 2.2, we multiply (4.1) by , integrate both sides of the equality on , and integrate by parts to derive
Letting to pass to limit and using (4.19) and Lemma 4.4 show that
Applying Lemma 4.3, we derive
where is independent of and . Hence,

From (4.43), we see that is bounded and increasing in , which implies the existence of a function , such that, as ,

Lemma 4.6. *As , we have
*

*Proof. *We may take , in the integral equality satisfied by . Then it is easy to see that
Hence, by (4.43), we have
Notice that
So
Hence,
By (4.43), (4.44), and (4.49), we see that as ,
Therefore,
By Lemma 2.5,
Hence,
that is,
Applying , (), and , we have
Equation (4.50) and (1) imply that the right side tends to zero as . Since in , (2) is proved. (3) is an immediate consequence of (2).

Lemma 4.7. *For any , we have
**
where is independent of .*

*Proof. *From Lemmas 4.3 and 4.4, it is easily seen that
Using , (4.66), and Proposition 4.5, we have
The proof of Lemma 4.7 is completed by combining (4.67) with (4.65).

Lemma 4.8. *As , we have*

*Proof. *Let and be the characteristic functions of and , respectively. Then
Taking in Lemma 4.7, we obtain that . Since (), for any , we can choose such that . For fixed ,