**Advances in Pure Mathematics** Vol.3 No.1A(2013), Article ID:27566,5 pages DOI:10.4236/apm.2013.31A028

Existence of Weak Solutions for a Class of Quasilinear Parabolic Problems in Weighted Sobolev Space^{*}

^{1}School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, China

^{2}Center for Computational Geosciences, Xi’an Jiaotong University, Xi’an, China

Email: ^{#}lqmei@mail.xjtu.edu.cn

Received November 23, 2012; revised December 27, 2012; accepted January 7, 2013

**Keywords:** Weighted Sobolev Space; Energy Estimates; Compact Imbedding; Sobolev Interpolation Inequalities

ABSTRACT

In this paper, we investigate the existence and uniqueness of weak solutions for a new class of initial/boundary-value parabolic problems with nonlinear perturbation term in weighted Sobolev space. By building up the compact imbedding in weighted Sobolev space and extending Galerkin’s method to a new class of nonlinear problems, we drive out that there exists at least one weak solution of the nonlinear equations in the interval for the fixed time.

1. Introduction

Now we consider the initial/boundary-value problem [1] as following

(1.1)

where, , is a real positive parameter and is (uniformly) parabolic, for some fixed time, is an open bounded subset with smooth boundary in, is given, is the unknown, , , are functions satisfying some suitable conditions [2-4].

The main purpose of this paper is to establish the existence of weak solutions for the parabolic initial/boundary-value problem (1.1) in a weighted Sobolev space. For this purpose, we assume for now that

1) is a positive measurable sufficiently smooth function2) is a non-negative smooth function which may change sign3) is a weighted Sobolev space [5-8] with a weight function, its norm defined as

.

For convenience, we will denote by Xnote by, and unless otherwise statedintegrals are over.

Similar problems have been studied by Evans [9], he investigated the solvability of the initial/bondary-value problem for the reaction-diffusion system

(1.2)

Here, , and as usual, is open and bounded with smooth boundary. Via the techniques of Banach's fixed point theorem method, he obtained the existence and uniqueness and some estimates of the weak solution under the assumer that the initial function belongs to and is Lipschitz continuous. He also studied the nonlinear heat equation with a simple quadratic nonlinearity

(1.3)

The Blow-up solution has been established under the assumer that and are large enough in an appropriate sense.

The main results of this paper can be stated as followsTheorem 1.1. There exists a unique weak solution of problem (1.1) on the interval for the fixed time.

For the further argument, we need the following Lemma.

Lemma 1.1. If, then1) are the compact imbedding [6]2) are also compact imbedding.

Proof. 1) Since, and is a positive sufficiently smooth function, there exists a positive constant C, such that. Hence

for all, and a.e. time. We used the poincare’s inequality in the last inequality above. Thus1) Holds and is compact.

2) The proof of 2) is almost the same as 1). This completes the proof of Lemma 1.1.

2. Weak Solutions

According to Lemma 1.1, it suffices to consider the initial/boundary-value problem (1.1) in spaces and. We will employ the Galerkin’s method to prove our results.

Definition 2.1. We say a function

is a weak solution of the parabolic initial/boundary-value problem (1.1) provided

1), for each, and a.e. time, and 2).

Here denotes the time-dependent bilinear form

for each and a.e. time.

is the nonlinearity term. the pairing denoting inner product in, being the pairing of and.

By the Definition 2.1, we see, and thus the equality 2) makes sense.

We now switch our view point, by associating with u a mapping

defined by

More precisely, assume that the functions are smooth1) is an orthogonal basis of and 2) is an orthogonal basis of, (0 ≤ t ≤ T, i = 1, 2, ∙∙∙, m) taken with the inner product

S_{m} is the finite dimensional subspace spanned by. Fix a positive integer m, we will look for a function of the form

(2.1)

Here we hope to select the coefficients, (0 ≤ t ≤ T, i = 1, 2, ∙∙∙, m) such that

(2.2)

That is

(2.3)

This amounts to our requiring that u_{m} solves the “projection” of problem (1.1) onto the finite dimensional subspace.

Theorem 2.1. (construction of approximate solutions)

For each integer there exists a function u_{m} of the form (2.1) satisfying the identities (2.3).

Proof. Taking arbitrary, then

Thus, , and

Hence,

(2.4)

Since is random, therefore, system (2.4) becomes

(2.5)

This is a nonlinear system of ordinary differential equation, according to the existence theory for nonlinear ODE, there exists a unique local solution on interval for fixed time T > 0. That is, the initial/boundaryvalue problem (1.1) has a unique local weak solution on the interval.

3. Energy Estimates

Theorem 3.1. There exists a constant C, depending only on and, such that

(3.1)

for

Proof. We separate this proof into 3 steps.

Step 1. Multiply equality (2.2) by and sum for, and then recall to (2.1) to find

(3.2)

Whereas,

and

for a.e. time, here, , since is a smooth function.

Consequently (3.2) yields the inequality

(3.3)

Since, that is, then by Sobolev imbedding theorem, we obtain, and moreover,

here k is the best Sobolev constant [10-13].

Thus, we can write inequality (3.3) as

(3.4)

For a.e. time, and appropriate constant.

In addition, since, by Sobolev interpolation inequality, we find

here, and we have used the Young’s inequality with in the last inequality. Thus

(3.5)

By Lemma 1.1 2) and Sobolev’s inequality, we have

, is the best Sobolev imbedding constant, insert the inequality above and (3.5) into inequality (3.4) yields

(3.6)

for a.e. time, and appropriate constants and.

Furthermore, we rewrite inequality (3.6) as

(3.7)

for a.e. time, and appropriate constants and.

By Gronwall’s inequality, (3.7) yields the estimate

(3.8)

for a.e. time, and appropriate constant.

Step 2. Returning once more to inequality (3.7), we integrate from 0 to T and employ the inequality (3.8) to obtain

(3.9)

for a.e. time, and appropriate constant C.

Step 3. Fix any, with, and write, where and

. Since functions are orthogonal in,. Utilizing (2.2) we deduce for a.e. time, that

Then (2.1) implies

Consequently,

since. Thus

(3.10)

for a.e. time, and appropriate constant C.

Combing (3.8), (3.9) and (3.10) we complete the proof of Theorem 3.1.

4. Existence of Weak Solutions

Next we pass to limits as, to build a weak solution of our initial/boundary-value problem (1.1).

Theorem 4.1. There exists a local weak solution of problem (1.1).

Proof. According to the energy estimates (3.1), we see that the sequence is bounded in

, and is bounded in. Consequently there exists a subsequence and a function

with, such that

1) weakly in, and strongly in.

2) weakly in.

Now we fix an integer and choose a function having the form

(4.1)

here are given smooth functions. We choose

, multiply (2.2) by, sum, and then integrate with respect to t, we find

(4.2)

We set, and recall 1), 2) to find upon passing to weak limits that

(4.3)

This equality then holds for all functions , as functions of the form (4.1) are dense in this space. Hence in particular

(4.4)

for each and a.e. time.

In order to prove, we first note from (4.3) that

(4.5)

for each with. Similarly, from (4.2) we deduce

(4.6)

We set and once again employ 1), 2), we obtain

(4.7)

since in. As is arbitrary, comparing (4.5) and (4.7), we conclude. This completes the proof of theorem 4.1.

5. Uniqueness of Weak Solutions

In this part, we will prove Theorem 1.1.

Proof. Let and are two weak solutions for the initial/boundary-value problem, put, and insert it into the origin equation, we discover

Taking, we obtain the energy estimates inequality

Since,. So we have

for a.e. time. This completes the proof of Theorem 1.1.

6. Conclusion

In this paper, we established the existence and uniqueness of weak solutions for initial/boundary-value parabolic problems with nonlinear perturbation term in weighted Sobolev space. First, we investigated the compact imbedding in weighted Sobolev space, which can be imbedded compactly into and spaces. By exploiting Sobolev interpolation inequalities and extending Galerkin’s method to a new class of nonlinear problems, we proofed the energy estimates of the equations and furthermore obtained the unique weak solution of the problem.

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NOTES

^{*}The project is supported by NSF of China (10971164).

^{#}Corresponding author.