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 ≥ − ∫∫