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

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).