Abstract and Applied Analysis

VolumeΒ 2011Β (2011), Article IDΒ 363579, 27 pages

http://dx.doi.org/10.1155/2011/363579

## A Periodic Problem of a Semilinear Pseudoparabolic Equation

^{1}School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China^{2}School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China

Received 23 April 2011; Revised 2 October 2011; Accepted 13 October 2011

Academic Editor: IrenaΒ Rachůnková

Copyright Β© 2011 Yang Cao 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

A class of periodic problems of pseudoparabolic type equations with nonlinear periodic sources are investigated. A rather complete classification of the exponent is given, in terms of the existence and nonexistence of nontrivial and nonnegative periodic solutions.

#### 1. Introduction

The purpose of this paper is to give a complete classification of the exponent , in terms of the existence and nonexistence of nontrivial nonnegative classical periodic solutions for the pseudoparabolic equation with nonlinear periodic sources subject to the homogeneous boundary value condition and periodic condition where is a bounded domain with smooth boundary, and are all constants, and is an appropriately smooth and positive function which is periodic in time with periodicity .

Pseudo-parabolic equations are characterized by the occurrence of a time derivative appearing in the highest-order term [1] and arise in applications from radiation with time delay [2], dynamic capillary pressure in unsaturated flow [3], and heat conduction involving two temperatures [4], and so forth. They can also be used as a regularization of ill-posed transport problems, especially as a quasi-continuous approximation to discrete models for population dynamics [5]. Actually, comparing with another regularized method, the Cahn-Hilliard equations, pseudo-parabolic equations are more incorporated with the out-of-equilibrium viscoelastic relaxation effects according to experimental results [6]. Furthermore, pseudo-parabolic equations are closely related to the well-know, BBM equations [7] which are advocated as a refinement of KdV equations.

Since the last century, pseudo-parabolic equations have been studied in different aspects, such as the integral representations of solutions [8], long-time behavior of solutions [9], Riemann problem and Riemann-Hilbert problem [10], and nonlocal boundary value problems [11]. However, as far as we know, the researches on periodic problems for pseudo-parabolic equations are far from those of parabolic equations [12–18]. Among the earliest works for periodic parabolic equations, Seidman’s work [18] caused much attention, in which one can find the existence of nontrivial periodic solutions, for the case and of (1.1), namely, where the function is periodic in . From then on, many authors dealt with semilinear equations of the form It were Beltramo and Hess [12] who first considered the case of (1.4) and showed that only for some special can the equation have nontrivial periodic solutions. It seems that the exponent of the source is a singular value. Indeed, this interesting phenomenon was verified by Esteban [13, 14]. Her results imply that, for in a neighborhood of 1 except for , nontrivial periodic solutions exist definitely for any . Her results also imply the existence of positive periodic solutions when with , or with , for any positive . At the same time, she also indicated that if with , then the equation might have no positive periodic solution. In fact, at least for star-shaped domains, there is definitely no such solution. So, this is another interesting phenomenon, and it is imaginable that should be a critical value. In fact, until 2004, this guess was solved by Quittner [17] who proved the existence of positive periodic solutions for the case with , although there are still some restrictions on the structure of .

Looking back to periodic problems of pseudo-parabolic equations, to our knowledge, most works are devoted to space periodic problems. For instance, the existence and uniqueness for regular solutions of the well-known BBM equation with were proved by the differential-difference method in [19]. The existence and blowup of solutions to the initial and periodic boundary value problem for the Camassa-Holm equation were considered in [20]. In [21], Kaikina et al. considered the periodic boundary value problem for the following pseudo-parabolic equation: where , , and . Their proof revealed that if the initial data is small enough, then there exists a unique solution. Once removing the assumption that the initial data is small, then one should add with or with to assure the existence of a unique solution. Further, from their results, one can also find that the solutions of (1.5) exhibit power-law decay in time or dichotomous large-time behavior which unlike the usual exponentially decay in time arose in periodic problems.

For time periodic problems of pseudo-parabolic equations, according to our survey, expect the early works of Matahashi and Tsutsumi and the recent research of Li et al., there are no other investigations. In [22, 23], Matahashi and Tsutsumi have established the existence theorems of time periodic solutions for the linear case and the semilinear case for with or with , respectively. As for one-dimensional case with for (1.1) and (1.2), we refer to the joint work with two authors of this paper for the existence of nontrivial and nonnegative periodic solutions; see [24].

In this paper, we consider the time periodic problem (1.1) and (1.2) when and . Certainly, some researches focus on the source which has the general form , but here we are quite interested in the special source (which was also studied by many authors, see [9] e.g.) and the existence and nonexistence of nontrivial nonnegative classical periodic solutions in different intervals divided by . It will be shown that, as an important aspect of good viscosity approximation to the corresponding periodic problem of the semilinear heat equation, there still exist two critical values and for the exponent . Precisely speaking, we have the following conclusions(i)There exist at least one positive classical periodic solution in the case (ii)When for , or for with convex domain , there exist at least one nontrivial nonnegative classical periodic solution(iii)When for with star-shaped domain and is independent of , there is no nontrivial and nonnegative periodic solution(iv)For the singular case , only for some special can the problem have positive classical periodic solutions.

From the existing investigations, we can see that, not only for space periodic problem but also for time periodic problem of pseudo-parabolic equations, the results are still far from complete. Specially, notice that pseudo-parabolic equations can be used to describe models which are sensitive to time periodic factors (e.g., seasons), such as aggregating populations [5, 25], and there are some numerical results and analysis of stabilities of solutions [26–28] which indicate that time periodic solutions should exist, so it is reasonable to study the periodic problem (1.1) and (1.2). Our results reveal that the exponents and are consistent with the corresponding semilinear heat equation [12–14, 17]. This fact exactly indicates that the viscous effect of the third-order term is not strong enough to change the exponents. However, due to the existence of the third-order term , the proof is more complicated than the proof for the case . Actually, the viscous term seems to have its own effect [29], and our future work will be with a particular focus on this. Moreover, comparing with the previous works of pseudo-parabolic equations, our conclusions not only coincide with those of [22] but also contain the results of [24].

The content of this paper is as follows. We describe, in Section 2, some preliminary notations and results for our problem. Section 3 concerns with the case , and the existence of positive classical periodic solutions is established. Subsequently, in Section 4, we discuss the case , in which we will investigate the existence and nonexistence of nontrivial nonnegative classical periodic solutions. The singular case will be discussed in Section 5.

#### 2. Preliminaries

In this section, we will recall some standard definitions and notations needed in our investigation. Specially, we will prove that if the weak solution under consideration belongs to , then it is just the classical solution.

Let be fixed, and set In order to prove the existence of periodic solutions, we only need to consider the following problem:

Though the final existence results in this paper are established for the classical solutions, but due to the proof procedure, we first need to consider solutions in the distribution sense. Denote by , the reasonable weak solutions space, namely,

*Definition 2.1. *A function is called to be a weak -periodic upper solution of the problem (2.2)–(2.4) provided that for, any nonnegative function , there holds
Replacing “≥” by “≤” in the above inequalities, it follows the definition of a weak lower solution. Furthermore, if is a weak upper solution as well as a weak lower solution, then we call it a weak solution of (2.2)–(2.4).

In what follows, we show that the weak solution defined above is classical if it belongs to . Meanwhile, the classical solution is nonnegative which implies that we can throw off the symbol of absolute value of .

Theorem 2.2. *If the weak solution in Definition 2.1 also belongs to , then there holds and , namely, is just the classical solution. Furthermore, is nonnegative.*

*Proof. *We lift the regularity of the weak solution step by step, via using the following abstract setting of pseudo-parabolic equation:
which can be derived by Fourier transform [9] or by reducing the pseudo-parabolic equation to a system of second-order equations [30], namely,

Since the regularity of the weak solution is not sufficient at the beginning, we start our proof from the abstract form in a weaker sense. From Definition 2.1, if in is a weak solution, then, after a small deformation, it also satisfies
As what Showalter et al. have done in [1, 31], by using the Lax-Milgram theorem on bounded positive-definite bilinear forms in Hilbert space, we obtain the corresponding Friedrichs extensions of , denoted by , with domain dense in , satisfying the identity
whenever and . The range of is all of , and has an inverse which is a bounded mapping of into . Then the weak solution in is just the weak solution of the following equation:
We can also relate the extended operator to the operator which are just the extension of to the domain in the sense of generalized derivatives. has a continuous inverse operator from to . Thus, from
and , there hold
Multiplying on both sides of (2.12), we get
For any , integrating the above equation in and using the periodicity of yield
which with (2.13) imply that
Thus, the weak solution is just the strong solution which satisfies (2.2) almost everywhere in . Furthermore, following the discussion in [32, 33], we can introduce a linear operator by , whose inverse is continuous. Then, we have
Similar to the above discussion, we can deduce that
From the Isotropic Embedding Theorem [34], we know that
As in [35, 36], is bounded from to , then satisfies
In the same way, we have
which implies that is the classical solution.

Once is the classical solution, we conclude that . Inspired by the method in [24, 37], we suppose, to the contrary, that there exists a pair of points such that
Since is continuous, then there exists a domain such that in and on . Multiplying (2.2) by , which is the principle eigenfunction of in with homogeneous Dirichlet boundary condition, and integrating on , we get
where is the first eigenvalue. Integrating the above inequality from 0 to and using the periodicity of , we have
By the mean value theorem, there exists a point such that
Actually (2.23) is equivalent to
Integrating the above inequality from to implies that
Recalling the periodicity of , we see that
Then integrating (2.26) over implies that
which contradicts with in .

#### 3. The Case

In this section, we consider the case , in which we will show that there exists at least one positive periodic solution.

Theorem 3.1. *Assume that . Then the problem (1.1) and (1.2) admits at least one positive periodic classical solution .*

*Proof. *We prove the theorem by constructing monotone sequence. Just as what we have done in Section 2, we may as well consider the problem (2.2)–(2.4). First, we construct a coupled upper and lower classical solution of (2.2)–(2.4). Choose to be appropriately large such that . Let , be the first principle eigenvalues of with homogeneous Dirichlet boundary value conditions on and of , respectively. Furthermore, we let and be the corresponding principle eigenfunctions normalized by and . Precisely speaking, and satisfy
It is well known (see e.g., [38]) that for and for . Therefore, there exists a constant such that for . Set
where and are constants which are to be determined later.

Actually, if we choose , then a simple calculation yields that
Then is a lower positive classical periodic solution of (2.2)–(2.4). Moreover, is an upper positive classical periodic solution of (2.2)–(2.4) if and only if
which is ensured by
Clearly we also have .

Set and be the coupled bounded lower and upper classical periodic solutions of (2.2)–(2.4). We get a function sequences via the following iteration process
for , where . The existence and uniqueness of classical solutions for the above problem can be proved by the method in [36] and Theorem 2.2, so is well defined. Then we have that the above sequence is monotone and bounded, that is,
Since is the lower solution, we get
By using the comparison principle of pseudo-parabolic equation [35, 39], we have that , and
which means that is a lower periodic solution. Furthermore, for , there also holds
which indicates that by the comparison principle. Repeating the above procedures, there holds (3.9). Due to that is monotone of , then there exists a function such that in , and .

Multiplying both sides of (3.6) by , integrating the result over , and recalling (3.7), (3.8) yields
Next, multiplying both sides of (3.7) by , we have
Using the Young inequality to the above equality yields
Hence, when , it follows that, for any ,
which imply that is the periodic solution of (2.2)–(2.4).

Furthermore, the above weak periodic solution we find is just the positive classical periodic solution of the problem (1.1) and (1.2). From (3.9) and the convergent procedures, we have that
namely, that . Thus, from Theorem 2.2, is the positive classical periodic solution of (1.1) and (1.2).

#### 4. The Case

In what follows, we pay our attention to the case , in which we will determine an exponent , such that and are corresponding to the existence and nonexistence of nontrivial and nonnegative periodic solutions, respectively. To prove the existence of periodic solutions, we need the following lemma, which can be found in [40].

Lemma 4.1. *Let , and let be a real Banach-space. Let be continuous and map-bounded subsets on relatively compact subsets. Suppose moreover that satisfies*(a)*,
*(b)*there exist such that(i) , and implies ,(ii).*

*Let denote the set of solutions to the problem*

*in . Let denote the component (closed connected subset maximal with respect to inclusion) of to which belongs. Then if*

*then is unbounded in .*

Define an operator by where is a constant. Let be a solution of the following problem

We aim to apply Lemma 4.1 to get the existence of nontrivial weak periodic solutions and then by lifting the regularity of the weak solutions (Theorem 2.2) to get the existence of classical solutions. For these purposes, firstly, we need to verify the compactness and continuity of the operator .

Lemma 4.2. *When with or with , the operator is completely continuous.*

*Proof. *To verify the compactness of the operator , we first need to make some a priori estimates. Multiplying (4.4) by and integrating over yield
Integrating the above inequality from to and combining with the Hölder inequality and the Isotropic Embedding Theorem [34], we conclude that
Thus, we have
where depends only on , , , , , and . By the mean value theorem, we see that there exists a point such that
Integrating (4.7) from to gives
Noticing the periodicity of , we arrive at
from which it is easy to obtain that
where depends only on , , , , , , and . Using the Isotropic Embedding Theorem [34] yields
where
and depends on , , , , , , , and . Multiplying (4.4) by and integrating over yield
Integrating the above equality from to and using the Hölder inequality and the Isotropic Embedding Theorem [34], we get
Then one has
where depends on , , , , , and . Moreover, by means of (4.13), (4.14), and (4.18), we obtain the compactness of the operator , while, for the continuity of , it is easy to obtain just by a simple and cumbersome real analysis process, so we omit it. The proof is complete.

By using the above lemmas, we obtain the following results.

Theorem 4.3. *Assume . If is a convex domain and
**
then the problem (1.1) and (1.2) admits at least one nontrivial nonnegative classical periodic solution.*

*Proof. *We will complete the proof by using Lemma 4.1. Recalling the definition of the operator and Lemma 4.2, we see that the operator is completely continuous. In what follows, we first need to check the condition (a) in Lemma 4.1. Let that is, is a solution of the following problem:
Clearly, the above problem admits only zero solution. In fact, multiplying the first equation of (4.20) by and integrating over yield
Recalling the Poincaré inequality, we see that
which implies that a.e. in .

Secondly, we will show that there exists an such that if and
then . Taking , replacing by in (4.4), and then multiplying the equation by on both sides and integrating over yield
By virtue of the Isotropic Embedding Theorem, we get
where is the constant in the Isotropic Embedding Theorem; that is,
Thus, if
where
then we have
which means a.e. in .

Next, we check the condition (b) in Lemma 4.1, namely, there exists an , such that
Consider the following problem:
where . Construct a homotopic mapping
Similar to Lemma 4.2, we see that is completely continuous. Assume that
where is to be determined. Multiplying the first equation of the above problem by and integrating over yield
Integrating from to gives
Then, we further have
By means of the integral mean value theorem, we see that there exists a such that
By the periodicity of and a similar process as Lemma 4.2, we obtain that
Using the Isotropic Embedding Theorem gives
If with is appropriately small, such that , therefore,
which means that

To show that the problem (1.1) and (1.2) admits at least one nontrivial periodic solution, it remains to check the boundedness of the set in Lemma 4.1. Otherwise, the set of solutions to the problem is unbounded. Therefore, there exist , such that and
which implies that
If this were true, then we would have
Suppose the contrary, and note that if is bounded, then , which means (4.46). Thus, without loss of generality, we may assume that . Making change of variable
we have
If are bounded uniformly, that is, there is a constant such that , then, for any with , we have
Noticing the density of in , then it is sound to take thus, we have
In addition, for any , we also have
that is,
Clearly, it is a contradiction since . Therefore, (4.46) holds, which also implies that . Let . By the convexity of , we see that there exists a such that ; see, for example, [41, 42]. Then, there exists a subsequence, and for simplicity, we still denote it by such that , with . Let
and let
Then, with on satisfies
Similar to the proof of Theorem 2.2, we can deduce that which admits throwing off the symbol of absolute value of . Therefore, for any with on , we have
Taking , we have
which means that there exists such that
For any , taking in (4.56) yields
where is a constant independent of , , . By the periodicity of , we further have
Repeating the process above, we finally obtain that for any ,
Summing up, we finally obtain that
In addition, we note that, for any , we have
By Lebesgue differential theorem, there exists such that
Then, there exists a function with such that as (passing to a subsequence if necessary)
we obtain that
Take , where
with is sufficiently smooth and . Then, for sufficiently large , we have and
Then, for sufficiently large , we have
where is independent of and . Then, there exists a function such that, passing to a subsequence if necessary, as
Then, we have
Moreover, since , we have for all by the strong maximum principle [43]. Taking balls larger and larger and repeating the argument for the subsequence obtained at the previous step, we get a Cantor diagonal subsequence, and for simplicity, we still denote it by which converges in to a function ; namely,
which means that (4.72), is a contradiction. Indeed, for the case with and the case with , thanks to a Liouville-type theorem, Theorem II in [44], and [45, Lemma 3.6], we see that the problem (4.72), has no solution, which is a contradiction and implies that is bounded uniformly. By means of Lemma 4.1, we conclude that the problem (1.1) and (1.2) admits at least one nontrivial periodic solution.

Since we have prove, the boundedness of the solution, then, from Theorem 2.2, is the nontrivial nonnegative classical periodic solution.

In what follows, we consider the nonexistence of periodic solutions.

Theorem 4.4. *Assume and . If and is star shaped, then there is no nontrivial and nonnegative periodic solution.*

*Proof. *If is independent of , then we deduce that the periodic solution of the problem (1.1) and (1.2) must be a steady state. In fact, multiplying (1.1) by on both sides and integrating over yield
which means that is a steady state and satisfies the steady-state equation
However, by [46, 47], if , , and is star shaped, then the above equation subject to the homogeneous Dirichlet boundary condition has no nontrivial and nonnegative solution. It is a contradiction, whence (1.1) and (1.2) has no nontrivial and nonnegative periodic solution.

*Remark 4.5. *Here, it is worth mentioning that, for the case , if the domain is an annulus domain, then there may exist nontrivial and nonnegative periodic solution. In fact, if , then the periodic solution is a steady state; namely, it is a solution of the corresponding elliptic equation, while, from the results in [48], there exists radial solutions for this case in an annulus domain.

#### 5. The Singular Case

In this section, we consider the case , in which the problem is written as subject to

We are going to show the specialty of this case. It is quite different from other cases, in which positive periodic solutions definitely exist or definitely not exist. It will be shown that, for small , any solution of the initial boundary value problem decays to zero as time goes to infinity, while, for large , all positive solutions blow up at finite time. These imply that there is no positive periodic solution. However, when is independent of , there may exist positive periodic solution. Here we consider the problem (5.1), (5.2) with the initial value condition where for and satisfies some compatibility conditions.

We have the following theorem.

Theorem 5.1. *Assume that . Let be the first eigenvalue of with homogeneous boundary condition on .*