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

Changchun Liu, Junchao Gao, and Songzhe Lian Department of Mathematics, Jilin University, Changchun 130012, China Correspondence should be addressed to Changchun Liu, liucc@jlu.edu.cn Received 31 October 2011; Accepted 6 December 2011 Academic Editor: Hui-Shen Shen Copyright q 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. The existence of weak solutions is studied to the initial Dirichlet problem of the equation ut udiv |∇u| 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.


Introduction
In this paper, we investigate the existence of solutions for the p x -Laplacian equation The equation is supplemented the boundary condition u x, t 0, x ∈ ∂Ω, 1.2 and the initial condition where Q T Ω × 0, T , inf p x > 2, Ω ⊂ R N is a bounded domain with smooth boundary ∂Ω and 0 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.

Denote that
Throughout the paper, we assume that where p − , p are given constants.
To study our problems, we need to introduce some new function spaces.Denote that

2.3
We use W Remark 2.1.In 16, 17 , Zhikov showed 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 1, p x 0 , and hence the limit function is also in W 1, p x 0 which avoids the above difficulties.
We now give the definition of the solutions to our problem.
and u t ∈ L 2 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:

2.5
In the following, we state some of the properties of the function spaces introduced as above.
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 ii If p 1 , p 2 ∈ C Ω , p 1 x ≤ p 2 x for any x ∈ Ω, then L p 2 x → L p 1 x and the imbedding continuous.

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iii There is a constant C > 0, such that This implies that |∇u| p x and u 1, p x are equivalent norms of W where C C p is a positive constant depending on p.

3.3
where b 1 and b 2 are nonnegative constants.Then max where M depends only on b 1 , b 2 , T, and max Γ T |u|.
We suppose that for x, t ∈ Q T , max Q T |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 inequalities

3.5
where P ρ is a nonnegative continuous function that tends to zero for ρ → ∞ and 1 < m x, t ∈ C 1 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, μ 1 , and 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 inf m x, t and sup m x, t ; we omit the details.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, Journal of Applied Mathematics u, and p and satisfy inequalities 3.5 with a sufficiently small ε determined by the numbers M, ν, μ, μ 1 , and where M 1 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 S 0 { x, t : x ∈ ∂Ω, t 0} 3.1 , that is, (in other words, the compatibility conditions of zero and first orders are assumed to be fulfilled).
Then there exists a unique solution of problem 3.1 and 3.2 in the space Proof.We consider problem 3.1 and 3.2 along with a one-parameter family of problems of the same type

3.10
Define the Banach space For any w ∈ X, let v w ψ.Using Schauder theory, the linear problem 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.

Existence
In this section, we are going to prove the existence of solutions of the problem 1.1 -1.3 .
Then the problem 1.1 -1.3 admits a weak solution u.
Consider the following problem: 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 form where

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 Proof.By the maximum principle, we know that where a ij x, t, u, u x and a x, t, u, u x are defined as 4.4 .

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It is easy to prove that Hence, we have

4.8
Let Using the mean value theorem, we have

4.11
Since w ≤ 0 on Γ T , by comparison principle of linear parabolic equation, we have w ≤ 0.

4.22
Proof.Observe that u ε,η − u ε /u ε,η ∈ L p x 0, T; W 1,p x 0 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 By H ölder inequality and Lemma 4.3, we obtain Hence, We divide the integral in 4.25 in the following way:

4.26
From 4.20 , we see that Using Lemma 4.3, we have

4.28
Now we estimate

4.33
Journal of Applied Mathematics 13 by Proposition 2.4, we see that 2 holds.To prove 3 , we have

4.34
Using Lemma 4.3 and 2 , we see that 3 holds.Finally, we prove 4 .We have

4.38
By H ölder inequality, we have If p x > 3, we have

4.40
Hence, 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 where C is independent of ε.

4.45
Letting η η k → 0 to pass to limit and using 4.19 and Lemma 4.4 show that where C is independent of ε and η.Hence, 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,

4.56
Hence,   where C is independent of ε.

4.73
Thus the proof of Theorem 4.1 is complete.

Theorem 3 . 3 .
Suppose that the following conditions are fulfilled.aFor x, t ∈ Q T and arbitrary u either conditions 3.3 are fulfilled.
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