International Journal of Partial Differential Equations
Volume 2014 (2014), Article ID 436369, 9 pages
http://dx.doi.org/10.1155/2014/436369

Research Article

Existence of Solutions for a Class of Quasilinear Parabolic Equations with Superlinear Nonlinearities

Zhong-Xiang Wang, Gao Jia, and Xiao-Juan Zhang

College of Science, University of Shanghai for Science and Technology, Shanghai 200093, China

Received 13 August 2014; Accepted 23 November 2014; Published 21 December 2014

Academic Editor: Yuncheng You

Copyright © 2014 Zhong-Xiang Wang 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.

Working in a weighted Sobolev space, this paper is devoted to the study of the boundary value problem for the quasilinear parabolic equations with superlinear growth conditions in a domain of . Some conditions which guarantee the solvability of the problem are given.

1. Introduction

In this paper, we deal with the existence of solutions for the quasilinear parabolic problem:

where , is an open set in , , is a weighted Sobolev space, is the first eigenvalue of , and is a singular quasilinear operator defined by

where . The nonlinear part satisfies the superlinear growth condition

where , is a nonnegative constant, , and .

There are a number of results concerning solvability of different boundary problems for quasilinear equations (elliptic and parabolic) in which the nonlinearities satisfy sublinear or linear conditions in the weighted Sobolev space, for example, [1–6].

In [1], Shapiro established a new weighted compact Sobolev embedding theorem and proved a series of existence problems for weighted quasilinear elliptic equations and parabolic equations.

In [2], working in Sobolev space only for the first eigenvalue, Rumbos and Shapiro on the basis of [3] by using the generalized Landesman-Lazer conditions discussed the existence of the solutions for weighted quasilinear elliptic equations

where

In [4], Jia and Zhao obtained the existence of a nontrivial solution for a class of singular quasilinear elliptic equations in weighted Sobolev spaces.

However, past research results regarding this kind of parabolic equations on superlinearity in the weighted Sobolev space like (1) are very limited. Two notable exceptions are found in [7, 8], where they discuss the periodic solutions for quasilinear parabolic equations when the nonlinearity may grow superlinearly.

Our goal here is to extend these results to the case of quasilinear parabolic operators.

In fact, (1) is one of the most useful sets which describe the motion of viscous fluid substances. They are widely used in the design of aircrafts and cars, the study of blood flow, the design of power stations, and so forth. Furthermore, coupled with Maxwell’s equations, the Navier-Stokes equations can also be used to model and study magnetohydrodynamics.

The main tools applied in our approaches consist of Galerkin method, Brouwer’s theorem, and a new weighted compact Sobolev-type embedding theorem due to Shapiro.

This paper is organized as follows. In Section 2, we introduce some necessary assumptions and basic results. In Section 3, five fundamental lemmas are established. The subsequent Section 4 contains proofs of the main results.

2. Basic Assumptions and Main Theorem

In this section, we introduce some assumptions and give the main results in this paper.

Let be an open (possibly unbounded) set, a fixed closed set ( may be an empty set), , and . We assume throughout this paper that , are positive functions, function is a nonnegative function ( may be zero), and

We define , , , and is a set of real-valued functions defined as

We consider the following pre-Hilbert spaces (see [1]):

with inner product , and

with inner product

We denote which is the Hilbert space obtained from the completion of with the norm , and denote which is the Hilbert space obtained from the completion of the space with the norm . Similarly, we have and .

It is assumed throughout the paper that , , and meet the following assumptions:

(-1): is weakly sequentially continuous;(-2)there are , s.t. , and is measurable, for ;(-1), where , for ;(-2), for ;(-3)there is a constant , for and , such that

Definition 1. For the quasilinear differential operator , the two-form is

Defining

for (as described in [5]), and

the two-form of is

and the two-form of is

Definition 2. One says that is -related to if the following condition holds:

Definition 3. is a simple if

()there is a complete orthonormal system in . Also for all ;()there is a sequence of eigenvalues with such that for all . Also in ;, where is an open set for ;for each , there are positive functions satisfying , and , ;for each , , and , there exist and for , such that

where

There are many examples to illustrate the simple region. One can refer to [1].

Remark 4. We give an example to establish existence results for and which meet Definition 2. Taking , (a bounded open connected set), and , we set , for , , and . Then, it is easy to see from [9] that is a simple region. We set

Also, we define

We observe with that

Clearly, and meet Definition 2.

Remark 5. If (as (12)) meets , (-1)–(-3), then

meets the following conditions:

satisfies the Carathéodory conditions;(Superlinear condition) there exists with such that , , , where . is a nonnegative constant and .

Now we state our main results in this paper.

Theorem 6. Assume that is a simple . Let given by (13) satisfy the conditions of simple , and let given by (12) satisfy (-1)–(-3), (-1)-(-2) and be -related to . Suppose that (the dual of ) and that - hold. Suppose furthermore there is a nonnegative function and a constant such that

Then problem (1) has at least one nontrivial weak solution, that is, , such that

Remark 7. Observing that, for , where is a positive function and meets and (24).

3. Preliminary Lemmas

In this section, we introduce some lemmas. First, we introduce some notions.

If the conditions of simple hold, we get

where

Obviously, both and are in . Defining

it is clear that is an inner product on . From (10), (16), and (28), there are such that

For , setting

and from , (16), and (28), we see that, for ,

Lemma 8. If is a CONS for defined by (27), setting

then

Lemma 9. (i) If , then

(ii) If and , then .

Lemma 10. Let be as Theorem 6 and assume that is a simple . Then is compactly imbedded in .

For the proofs of Lemmas 8–10, one can refer to [5].

We define

Remark 11. If , then .

Lemma 12. Let be given as (13) and suppose that is a simple . Then is continuously imbedded in for every satisfying , that is, , such that

Proof . To establish the lemma, we need to only prove the case that

For fixed is a function only depending on in . Hence, with (18) and (19), one gets

where . Applying the same kind of reasoning to for fixed , we have

where

By (39), it follows that

with . Taking and and applying the generalized version of Hölder inequality on , we deduce

Integrating the last inequality on both sides with respect to , we get

We continue integrating with respect to and eventually find

Furthermore, we have

with

From (18) we see that and . Hence, we obtain from (46) that there is a constant such that

By (41), (47), and , we see that

Therefore for , and .
By using (48), we get (36). The proof of Lemma 12 is complete.

Lemma 13. Let be given by (13) and suppose that is a simple . Then, for , is compactly imbedded in for . For , is compactly imbedded in for .

Lemma 13 is an immediate byproduct of [1, Theorem 9].

4. Proof of Theorem 6

In this section, we will give the proof of Theorem 6. In order to do this, we divide the proof into three parts.

Lemma 14. Let all the assumptions in Theorem 6 hold. Then, for , such that

Proof. To prove the lemma, we first observe from (35) that

Let be an enumeration of , and set

So is an enumeration of , where .
For , setting

where

from (52)–(54), we have

Further, (29) yields

For , we set

and we claim that such that

By (-2) and the above,

where and . Therefore, and is well defined for by Lemma 12.
To establish (58), we set

for , and we observe from (51) that

where

Now, from (62),

and according to Lemma 12, (56), (59), (63), and the fact that is -related to ,

We see from (61) that such that

Therefore, by virtue of generalized Brouwer’s theorem [10], there exists such that , for . We set and see from (60) that (58) does indeed hold.
By the definition of and (24), for ,

where .
We claim

Suppose that (68) is false. Then there is a subsequence (which for ease of notation, we take to be the full sequence) such that

Inserting in place of in (58), we find

and the left-hand side of (70) is

So we see from (67) and Hölder inequality that

From (36), (56), (69), and the fact that is -related to , dividing both sides of (72) by and taking the limit as , we obtain that . But is a positive integer. So we get a contradiction. Therefore, (68) does indeed hold.
Hence, from (56), (68), and , for , there is a subsequence and a [11] such that

It is easy to check that as , , and from Lemma 12 we obtain that . Consequently, there exist a function and a subsequence such that

By using (-2), Hölder inequality, and the Lebesgue dominated convergence theorem, we conclude that

Hence replacing by in (58) and passing to the limit as , we consequently obtain that

and the proof of the lemma is complete.

Lemma 15. Let the conditions in Lemma 14 hold. Then the sequence obtained in Lemma 14 is uniformly bounded in with respect to the norm .

Proof . According to Lemma 14, we obtain a sequence , where

and satisfies

We claim there is a constant such that

Suppose that the assertion is false. Then, without loss of generality, we assume

Putting in place of in (78), we have

Next, we observe from (77) and Lemma 8 that

and from (78), (24), and Hölder inequality that

Hence it follows from Lemma 12, , being -related to , being strictly positive, and (83) that

Next, we observe from (12) and (16) that there is a positive constant such that

Putting for in (78), we see that there exists such that

But then it follows from (81), (24), and that

So, we see from (28), (29), and (87) that there is a positive constant such that

Consequently, dividing both sides of (88) by and passing to the limit as , from Lemma 12, the fact that , (80), and (84), we obtain that . This is a contradiction. Hence (80) does not hold and (79) is true.

Proof of Theorem 6. Since is a separable Hilbert space, we see from (79) and Lemmas 9 and 13 that there exist a subsequence (for the sake of simplicity, we take to be a full sequence) and a function with the following properties [11]:

We let . Then it follows from (78) that, for ,

Now, from and (89), we obtin that

Next, we observe from (-2) and (89)(2) that

where and . From (-1) and (89)(3), we further obtain that

Note that . Hence we conclude from Hölder inequality, the Lebesgue dominated convergence theorem, Lemma 12, and (92) that

Then passing to the limit on both sides of (90), we obtain from (91), (94), and (89) that

Next, for given , we replace with (defined as (32)) in (95). From (29) and Lemma 8, as . From (95) and Lemma 12, it is then an easy matter to obtain that

and the proof of Theorem 6 is established.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This research is supported by the National Natural Science Foundation of China (11171220) and Shanghai Leading Academic Discipline Project (XTKX2012).

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