A Periodic Problem of a Semilinear Pseudoparabolic Equation

and Applied Analysis 3 where α > 0, λ > 0, and p > 1. 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 addN ≤ 4 with p > 1 orN ≥ 5 with 1 < p ≤N/ N−4 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 ofMatahashi 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 ∂u ∂t − ∂Δu ∂t Δu f x, t 1.6 and the semilinear case ∂u ∂t − ∂Δu ∂t Δu − |u|p−1u f x, t 1.7 for 1 < p < 1 4/N with N 2, 3, 4 or 0 < p < 3 with N 1, respectively. As for onedimensional case with p > 1 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 N ≥ 1 and p ≥ 0. Certainly, some researches focus on the source which has the general form f u , but here we are quite interested in the special source u whichwas also studied bymany authors, see 9 e.g. and the existence and nonexistence of nontrivial nonnegative classical periodic solutions in different intervals divided by p. 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 p0 1 and pc N 2 / N − 2 for the exponent p. Precisely speaking, we have the following conclusions i There exist at least one positive classical periodic solution in the case 0 ≤ p < 1 ii When 1 < p < N 2 / N − 2 for N > 2, or 1 < p < ∞ for N ≤ 2 with convex domainΩ, there exist at least one nontrivial nonnegative classical periodic solution iii When p ≥ N 2 / N − 2 for N > 2 with star-shaped domain Ω and α x, t is independent of t, there is no nontrivial and nonnegative periodic solution iv For the singular case p 1, only for some special α x, t 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 p0 and pc are 4 Abstract and Applied Analysis 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 k ∂Δu/∂t , the proof is more complicated than the proof for the case k 0. Actually, the viscous term k ∂Δu/∂t 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 0 ≤ p < 1, and the existence of positive classical periodic solutions is established. Subsequently, in Section 4, we discuss the case p > 1, in which we will investigate the existence and nonexistence of nontrivial nonnegative classical periodic solutions. The singular case p 1 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 L∞, then it is just the classical solution. Let τ ∈ R be fixed, and set Q Ω × 0, T , Qω Ω × τ, τ ω , S inf Q α x, t , L sup Q α x, t . 2.1 In order to prove the existence of periodic solutions, we only need to consider the following problem: ∂u ∂t − k ∂t Δu α x, t |u|, x, t ∈ Qω, 2.2 u x, t 0, x ∈ ∂Ω, t ∈ τ, τ ω , 2.3 u x, τ u x, τ ω , x ∈ Ω. 2.4 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 E, E0 the reasonable weak solutions space, namely, E { u ∈ L 1 Ω ; ∂u ∂t ∈ L2 Qω , ∂∇u ∂t ∈ L2 Qω ,∇u ∈ L2 Qω } , E0 { u ∈ E;u x, t 0 for any x ∈ ∂Ω}. 2.5 Abstract and Applied Analysis 5 Definition 2.1. A function u ∈ E is called to be a weak ω-periodic upper solution of the problem 2.2 – 2.4 provided that for, any nonnegative function φ ∈ C1 0 Qω , there holds ∫∫ Qω ∂u ∂t φdxdt ∫∫and Applied Analysis 5 Definition 2.1. A function u ∈ E is called to be a weak ω-periodic upper solution of the problem 2.2 – 2.4 provided that for, any nonnegative function φ ∈ C1 0 Qω , there holds ∫∫ Qω ∂u ∂t φdxdt ∫∫ Qω k ∂∇u ∂t ∇φdxdt ≥ − ∫∫


Introduction
The purpose of this paper is to give a complete classification of the exponent p, in terms of the existence and nonexistence of nontrivial nonnegative classical periodic solutions for the pseudoparabolic equation with nonlinear periodic sources ∂u ∂t − k ∂Δu ∂t Δu α x, t u p , x, t ∈ Ω × R, 1.1 subject to the homogeneous boundary value condition and periodic condition where Ω ⊂ R N is a bounded domain with smooth boundary, k > 0 and p ≥ 0 are all constants, and α x, t is an appropriately smooth and positive function which is periodic in time with periodicity ω > 0.
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-ofequilibrium 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 pseudoparabolic 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 k 0 and p 0 of 1.1 , namely, where the function α x, t is periodic in t.From then on, many authors dealt with semilinear equations of the form ∂u ∂t Δu α x, t u p , p > 0. 1.4 It were Beltramo and Hess 12 who first considered the case p 1 of 1.4 and showed that only for some special α x, t can the equation have nontrivial periodic solutions.It seems that the exponent p 1 of the source is a singular value.Indeed, this interesting phenomenon was verified by Esteban 13,14 .Her results imply that, for p in a neighborhood of 1 except for p 1, nontrivial periodic solutions exist definitely for any α x, t > 0. Her results also imply the existence of positive periodic solutions when p > 1 with N ≤ 2, or 1 < p < N/ N − 2 with N > 2, for any positive α x, t .At the same time, she also indicated that if p ≥ N 2 / N − 2 with N > 2, 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 p N 2 / N − 2 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 1 < p < N 2 / N − 2 with N > 2, although there are still some restrictions on the structure of α x, t .
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 ∂Δu/∂t 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: Abstract and Applied Analysis 3 where α > 0, λ > 0, and p > 1.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 N ≤ 4 with p > 1 or N ≥ 5 with 1 < p ≤ N/ N −4 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., for 1 < p < 1 4/N with N 2, 3, 4 or 0 < p < 3 with N 1, respectively.As for onedimensional case with p > 1 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 N ≥ 1 and p ≥ 0. Certainly, some researches focus on the source which has the general form f u , but here we are quite interested in the special source u p 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 p.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 p 0 1 and p c N 2 / N − 2 for the exponent p. Precisely speaking, we have the following conclusions i There exist at least one positive classical periodic solution in the case 0 ≤ p < 1 ii When 1 < p < N 2 / N − 2 for N > 2, or 1 < p < ∞ for N ≤ 2 with convex domain Ω, there exist at least one nontrivial nonnegative classical periodic solution iii When p ≥ N 2 / N − 2 for N > 2 with star-shaped domain Ω and α x, t is independent of t, there is no nontrivial and nonnegative periodic solution iv For the singular case p 1, only for some special α x, t 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 p 0 and p c are Abstract and Applied Analysis 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 k ∂Δu/∂t , the proof is more complicated than the proof for the case k 0. Actually, the viscous term k ∂Δu/∂t 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 0 ≤ p < 1, and the existence of positive classical periodic solutions is established.Subsequently, in Section 4, we discuss the case p > 1, in which we will investigate the existence and nonexistence of nontrivial nonnegative classical periodic solutions.The singular case p 1 will be discussed in Section 5.

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 L ∞ , then it is just the classical solution.
Let τ ∈ R be fixed, and set

2.1
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 E, E 0 the reasonable weak solutions space, namely,

2.5
Definition 2.1.A function u ∈ E is called to be a weak ω-periodic upper solution of the problem 2.2 -2.4 provided that for, any nonnegative function ϕ ∈ C 1 0 Q ω , there holds

2.6
Replacing "≥" by "≤" in the above inequalities, it follows the definition of a weak lower solution.Furthermore, if u 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 L ∞ Q ω .Meanwhile, the classical solution u is nonnegative which implies that we can throw off the symbol of absolute value of |u|.

Theorem 2.2. If the weak solution u in Definition 2.1 also belongs to
Proof.We lift the regularity of the weak solution u 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,

2.8
Since the regularity of the weak solution u is not sufficient at the beginning, we start our proof from the abstract form in a weaker sense.From Definition 2.1, if u in E 0 is a weak solution, then, after a small deformation, it also satisfies Then the weak solution u in E 0 is just the weak solution of the following equation:

2.11
We can also relate the extended operator M 0 to the operator M 1 which are just the extension of I − kΔ to the domain and u ∈ L ∞ Q ω , there hold

2.13
Multiplying e t/k on both sides of 2.12 , we get

2.14
For any t ∈ 0, ω , integrating the above equation in t, t ω and using the periodicity of u yield which with 2.13 imply that Thus, the weak solution u is just the strong solution which satisfies 2.2 almost everywhere in Q ω .Furthermore, following the discussion in 32, 33 , we can introduce a linear operator 2.17 Similar to the above discussion, we can deduce that From the Isotropic Embedding Theorem 34 , we know that

2.20
In the same way, we have which implies that u is the classical solution.
Once u is the classical solution, we conclude that u ≥ 0. Inspired by the method in 24, 37 , we suppose, to the contrary, that there exists a pair of points x 0 , t 0 ∈ Ω × 0, ω such that u x 0 , t 0 < 0.

2.22
Since u is continuous, then there exists a domain Ω 0 such that u x, t 0 < 0 in Ω 0 and u x, t 0 0 on ∂Ω 0 .Multiplying 2.2 by ϕ, which is the principle eigenfunction of −Δ in Ω 0 with homogeneous Dirichlet boundary condition, and integrating on Ω 0 , we get where λ 0 is the first eigenvalue.Integrating the above inequality from 0 to ω and using the periodicity of u, we have By the mean value theorem, there exists a point t * ∈ 0, ω such that Actually 2.23 is equivalent to Integrating the above inequality from t * to ω implies that Recalling the periodicity of u, we see that Then integrating 2.26 over 0, t implies that which contradicts with u x, t 0 < 0 in Ω 0 .

The Case 0 ≤ p < 1
In this section, we consider the case 0 ≤ p < 1, in which we will show that there exists at least one positive periodic solution.

3.1
It is well known see e.g., 38 that φ x > 0 for x ∈ Ω and ψ > 0 for x ∈ B R .Therefore, there exists a constant γ > 0 such that ψ x > γ for x ∈ Ω. Set where κ 1 and κ 2 are constants which are to be determined later.
Actually, if we choose κ 1 S/λ 1 1/ 1−p , then a simple calculation yields that Then Φ is a lower positive classical periodic solution of 2.2 -2.4 .Moreover, Ψ 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 u Φ and u Ψ be the coupled bounded lower and upper classical periodic solutions of 2.2 -2.4 .We get a function sequences {u m } ∞ m 0 via the following iteration process Abstract and Applied Analysis for m 1, 2, . .., where u 0 u.The existence and uniqueness of classical solutions for the above problem can be proved by the method in 36 and Theorem 2.2, so u m is well defined.Then we have that the above sequence is monotone and bounded, that is, 3.9 Since u 0 u is the lower solution, we get

3.10
By using the comparison principle of pseudo-parabolic equation 35, 39 , we have that u 1 ≥ u 0 , and which means that u 1 is a lower periodic solution.Furthermore, for u − u 1 , there also holds 3.12 which indicates that u ≥ u 1 by the comparison principle.Repeating the above procedures, there holds 3.9 .Due to that u m is monotone of m, then there exists a function u such that u m → u in Q ω , and u x, t u x, t ω .Multiplying both sides of 3.6 by u m , integrating the result over Q ω , and recalling 3.7 , 3.8 yields

3.13
Next, multiplying both sides of 3.7 by ∂u m /∂t, we have

3.14
Using the Young inequality to the above equality yields Hence, when n → ∞, it follows that, for any r > 0,

3.16
which imply that u ∈ E 0 is the periodic solution of 2.2 -2.4 .Furthermore, the above weak periodic solution u 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 3.17 namely, that u ∈ L ∞ Q ω .Thus, from Theorem 2.2, u is the positive classical periodic solution of 1.1 and 1.2 .

The Case p > 1
In what follows, we pay our attention to the case p > 1, in which we will determine an exponent p c , such that 1 < p < p c and p ≥ p c 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 .
where 2N/ N 2 p < 1 p ≤ p < 2N/ N − 2 is a constant.Let u 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 G.
Proof.To verify the compactness of the operator G, we first need to make some a priori estimates.Multiplying 4.4 by u and integrating over Ω yield Integrating the above inequality from τ to τ ω and combining with the H ölder inequality and the Isotropic Embedding Theorem 34 , we conclude that

4.8
Abstract and Applied Analysis 13 Thus, we have where where C 4 depends on l, p, p, α L ∞ Q ω , Q ω , and sup t∈ τ,τ ω v L p Ω .Moreover, by means of 4.13 , 4.14 , and 4.18 , we obtain the compactness of the operator G, while, for the continuity of G, 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.
If Ω is a convex domain and

4.19
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. Recalling the Poincaré inequality, we see that which implies that u 0 a.e. in Q ω .Secondly, we will show that there exists an R > 0 such that if u G 0, u and then u ≡ 0. Taking l 0, replacing v by u in 4.4 , and then multiplying the equation by u on both sides and integrating over Ω yield

4.26
By virtue of the Isotropic Embedding Theorem, we get where μ is the constant in the Isotropic Embedding Theorem; that is, where which means u 0 a.e. in Q ω .Next, we check the condition b in Lemma 4.1, namely, there exists an R < R 0 , such that Consider the following problem: where σ ∈ 0, 1 .Construct a homotopic mapping

4.34
Similar to Lemma 4.2, we see that T is completely continuous.Assume that where R ≤ R 0 is to be determined.Multiplying the first equation of the above problem by u and integrating over Ω yield

4.36
Integrating from τ to τ ω gives Then, we further have By means of the integral mean value theorem, we see that there exists a t σ ∈ τ, τ ω such that By the periodicity of u and a similar process as Lemma 4.2, we obtain that sup Using the Isotropic Embedding Theorem gives 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 E in Lemma 4.1.Otherwise, the set of solutions to the problem u G l, u is unbounded.Therefore, there exist l n , u n such that u n G l n , u n and which implies that If this were true, then we would have Suppose the contrary, and note that if

4.49
Noticing the density of Ω , then it is sound to take ϕ v n thus, we have 4.50 In addition, for any 0 ≤ ϕ x ∈ C 1 0 Ω , we also have

4.52
Clearly, it is a contradiction since l n → ∞.Therefore, 4.46 holds, which also implies that By the convexity of Ω, we see that there exists a δ 0 > 0 such that dist x n , ∂Ω ≥ δ 0 ; see, for example, 41, 42 .Then, there exists a subsequence, and for simplicity, we still denote it by x n such that x n → x 0 , t n → t 0 with dis x n , ∂Ω ≥ δ 0 .Let α n y ω n l n ρ −1 n p ωη 4 dy

4.68
Then, for sufficiently large R > 0, we have where C is independent of n and R.Then, there exists a function ω ∈ W 1,2 loc R N such that, passing to a subsequence if necessary, as n → ∞ 4.70 fact, if α x, t ≡ α |x| , 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.

The Singular Case p 1
In this section, we consider the case p 1, in which the problem is written as 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 α x, t , any solution of the initial boundary value problem decays to zero as time goes to infinity, while, for large α x, t , all positive solutions blow up at finite time.These imply that there is no positive periodic solution.However, when α x, t is independent of t, there may exist positive periodic solution.Here we consider the problem 5.1 , 5.2 with the initial value condition u x, 0 u 0 x , 5.3 where u 0 x ≥ 0 for x ∈ Ω and satisfies some compatibility conditions.We have the following theorem.
Theorem 5.1.Assume that p 1. Let λ 1 be the first eigenvalue of −Δ with homogeneous boundary condition on Ω.
1 If L sup α x, t < λ 1 , then all the solutions of the problem 5.1 -5.3 go to 0 as t → ∞, which means that 1.1 and 1.2 admits no nontrivial nonnegative periodic solutions.
2 If S inf α x, t > λ 1 , then all the positive solutions of the problem 5.1 -5.3 go to ∞ as t → ∞; that is, 1.1 and 1.2 admits no positive periodic solutions.
3 If α x, t is independent of t and α x ≥ λ 1 , then there exists a nontrivial and nonnegative periodic solution for 1.1 and 1.2 .Moreover, if α ≡ λ 1 , then there is a positive periodic solution for 1.1 and 1.2 .
Proof.Firstly, we consider the case L < λ 1 .We note that there exist λ with L < λ < λ 1 and domain Ω with Ω ⊂ Ω such that λ is the first eigenvalue of −Δ with homogeneous boundary condition on Ω. Correspondingly, ψ is the first eigenfunction with ψ ∞ 1.A simple calculation yields that K ψ is an upper solution of the problem 5.1 -5.3 for appropriately large K > 0.Then, we have u ≤ K ψ.Let w x, t be the solution of the following problem: Δu α x, t u, x, t ∈ Ω × R , u x, t K ψ e −t , x ∈ ∂Ω, u x, 0 K ψ x , x ∈ Ω.

5.4
We conclude that w x, t is decreasing on t, and u x, t ≤ w x, t .Thus, there exists a function w such that w x lim t → ∞ w x, t .

5.5
It follows that w x is a steady state of the first equation of 5.4 with homogeneous Dirichlet boundary condition.Clearly, we have w x 0 since α x, t < λ 1 , which means that u x, t goes to 0 uniformly as t → ∞.

5.7
After a direct calculation, we see that w is a lower solution of 5.1 -5.3 for any nontrivial and nonnegative initial value on Ω.Furthermore, is a lower solution of 5.1 -5.3 for any nontrivial and nonnegative initial value on Ω, while by comparison we see that u x, t ≥ u on Ω.Thus, we have that u ∞ goes to infinity as t → ∞ since g t → ∞ as t → ∞.

5.9
Then, φ is the periodic solution.Specially, when α x λ 1 , then clearly the first eigenfunction is a periodic solution of 1.1 and 1.2 .
Remark 5.2.For p 1 and spatially constant α t with α t α t ω , as a fairly obvious observation, which is with no-flux boundary conditions, a spatially constant solution of the following ODE: u t α t u t , u t u t ω , 5.10 is a periodic solution to our problem.However, a simple calculation indicates that the above problem has periodic solutions provided that ω 0 α s ds 0, which does not coincide with our assumption α x, t > 0.

Lemma 4 . 1 .
Let R : 0, ∞ , and let E, • be a real Banach-space.Let G : R × E → E be continuous and map-bounded subsets on relatively compact subsets.Suppose moreover that G satisfiesa G 0, 0 0, b there exist R > 0 such that i u ∈ E, u ≤ R and u G 0, u implies u 0, ii deg id − G 0, • , B 0, R , 0 0.Let J denote the set of solutions to the problemu G l, u 4.1 in R × E.Let E denote the component (closed connected subset maximal with respect to inclusion) of J to which 0, 0 belongs.Then if there are no other investigations.In 22, 23 , Matahashi and Tsutsumi have established the existence theorems of time periodic solutions for the linear case 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 I − kΔ, denoted by M 0 , with domain t |u| p ϕ dx dt.
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.principle eigenfunctions normalized by φ L ∞ Ω 1 and ψ L ∞ B R 1. Precisely speaking, φ and ψ satisfy Theorem 3.1.Assume that 0 ≤ p < 1.Then the problem 1.1 and 1.2 admits at least one positive periodic classical solution u.Proof. 2 -2.4 .Choose R to be appropriately large such that Ω ⊂ B R/2 .Let λ 1 , λ 1 be the first principle eigenvalues of −Δ with homogeneous Dirichlet boundary value conditions on Ω and B R of R n , respectively.Furthermore, we let φ and ψ be the corresponding and sup t∈ τ,τ ω v L p Ω .By the mean value theorem, we see that there exists a point t ∈ τ, τ ω such that Integrating the above equality from τ to τ ω and using the H ölder inequality and the Isotropic Embedding Theorem 34 , we get Ω|u| 2 k|∇u| 2 dx ≤ C 2 , t ∈ t, τ ω .4.11Noticing the periodicity of u, we arrive atΩ |u x, τ | 2 k|∇u x, τ | 2 dx ≤ C 2 , 4.12 from which it is easy to obtain that sup t∈ τ,τ ω Ω |u| 2 k|∇u| 2 x, t dx ≤ C 2 , 4.13 where C 2 depends only on l, p, p, k, α L ∞ Q ω , Q ω , and sup t∈ τ,τ ω v L p Ω .Using the Isotropic Embedding Theorem 34 yields Ω |u| r dx ≤ C 3 , Ω ∂u ∂t α x, t |v| l p dx. 4.16 1. Recalling the definition of the operator G and Lemma 4.2, we see that the operator G is completely continuous.In what follows, we first need to check the condition a in Lemma 4.1.Let u G 0, 0 that is, u 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 u and integrating over Q ω yield which means 4.46 .Thus, without loss of generality, we may assume that 0 < l n → ∞.Making change of variable By Lebesgue differential theorem, there exists s j ∈ τ − t j /j, τ ω − t j /j such that where C 5 is a constant independent of j, n, Ω n .By the periodicity of ω nj , we further have2 dy ≤ C 5 |Ω n |.4.60Repeating the process above, we finally obtain that for any s ∈ τ − t j /j, σ j , with 0 ≤ η ≤ 1 is sufficiently smooth and |∇η| ≤ C/R.Then, for sufficiently large n, we have B 2R ⊂ Ω n and n ∇ω n ∇η dyB 2R