International Journal of Partial Differential Equations

Volume 2014, Article ID 436369, 9 pages

http://dx.doi.org/10.1155/2014/436369

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

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*

*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*

*Conflict of Interests*

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

*Acknowledgments*

*Acknowledgments*

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