A Periodic Problem of a Semilinear Pseudoparabolic Equation

Abstract

A class of periodic problems of pseudoparabolic type equations with nonlinear periodic sources are investigated. A rather complete classification of the exponent 𝑝 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 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𝜕𝜕Δ𝑡𝑢=Δ𝑢+𝛼(𝑥,𝑡)𝑢^𝑝,(𝑥,𝑡)∈Ω×ℝ,(1.1) subject to the homogeneous boundary value condition and periodic condition𝑢(𝑥,𝑢𝑡)(𝑥=,𝑡𝑢)(=𝑥,0𝑡,+𝜔𝑥)∈,𝜕𝑥Ω,∈𝑡Ω∈,ℝ𝑡,∈ℝ,(1.2) 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-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 𝑘=0 and 𝑝=0 of (1.1), namely,𝜕𝑢𝜕𝑡=Δ𝑢+𝛼(𝑥,𝑡),(1.3) 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:𝜕𝑢𝜕𝑡−𝜕𝜕Δ𝑡𝑢=𝛼Δ𝑢−𝜆|𝑢|^𝑝−1𝑢,(1.5) 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., 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−𝜕𝜕Δ𝑡𝑢=Δ𝑢+𝑓(𝑥,𝑡)(1.6) and the semilinear case𝜕𝑢𝜕𝑡−𝜕𝜕Δ𝑡𝑢=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 one-dimensional 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 𝑝₀=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 𝑝₀ and 𝑝_𝑐 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 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 set𝑄𝑆==Ωinf_𝑄×(0,𝑇),𝑄_𝜔=Ω×(𝜏,𝜏+𝜔),𝛼(𝑥,𝑡),𝐿=sup_𝑄𝛼(𝑥,𝑡).(2.1) In order to prove the existence of periodic solutions, we only need to consider the following problem:𝜕𝑢𝜕𝑡−𝑘𝜕𝜕Δ𝑡𝑢=Δ𝑢+𝛼(𝑥,𝑡)|𝑢|^𝑝𝑢(𝑥,𝑢𝑡)(𝑥=,𝜏0),=𝑥𝑢∈(𝑥𝜕,Ω𝜏,+𝑡𝜔∈),(𝜏,𝑥𝜏,∈+(Ω𝑥𝜔.,)𝑡,)∈𝑄_𝜔,(((222...234)))

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^𝑝+1(Ω);𝜕𝑢𝐸₀𝜕𝑡∈𝐿²𝑄_𝜔,𝜕𝜕∇𝑡𝑢∈𝐿²𝑄_𝜔,∇𝑢∈𝐿²𝑄_𝜔,={𝑢∈𝐸;𝑢(𝑥,𝑡)=0forany𝑥∈𝜕Ω}.(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∈𝐶¹⁰(𝑄_𝜔), there holds _𝑄𝜔𝜕𝑢𝜕𝑡𝜑d𝑥d𝑡+_𝑄𝜔𝑘𝜕∇𝑢𝜕𝑡∇𝜑d𝑥d𝑡≥−_𝑄𝜔+∇𝑢∇𝜑d𝑥d𝑡_𝑄𝜔𝑢𝑢((𝑥𝑥,,𝜏𝑡))≥≥0𝑢,(𝑥,𝜏𝑥𝛼+∈(𝜔𝑥𝜕,)Ω𝑡,),|𝑢𝑡|^𝑝𝑥∈𝜑∈d(𝜏𝑥Ω,d𝜏.𝑡,+𝜔()𝑥,,𝑡)∈𝑄_𝜔,(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 𝐿^∞(𝑄_𝜔), then there holds 𝑢∈𝐶^{2+𝛼,1+𝛼/2}(𝑄_𝜔) and 𝜕𝑢/𝜕𝑡∈𝐶^{2+𝛼,𝛼/2}(𝑄_𝜔), namely, 𝑢 is just the classical solution. Furthermore, 𝑢 is nonnegative.

Proof. We lift the regularity of the weak solution 𝑢 step by step, via using the following abstract setting of pseudo-parabolic equation: 𝜕𝑢𝜕𝑡+𝑘1𝑢=(𝐼−𝑘Δ)⁻¹𝑘1𝑢+𝛼(𝑥,𝑡)𝑢^𝑝,(2.7) which can be derived by Fourier transform [9] or by reducing the pseudo-parabolic equation to a system of second-order equations [30], namely, 𝜕𝑣𝑣=(𝐼−𝑘Δ)𝑢,𝜕𝑡+𝑘1𝑣=𝑘1𝑢+𝛼(𝑥,𝑡)𝑢^𝑝.(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 𝐸₀ is a weak solution, then, after a small deformation, it also satisfies _𝑄𝜔𝜕𝜕𝑢𝑡𝜑+𝑘𝜕𝜕∇𝑡𝑢∇𝜑d𝑥d𝑡=−𝑘1_𝑄𝜔+_𝑄𝜔(𝑢𝜑+𝑘∇𝑢∇𝜑)d𝑥d𝑡𝑘1𝑢+𝛼(𝑥,𝑡)|𝑢|^𝑝𝜑d𝑥d𝑡.(2.9) 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 𝑀₀, with domain 𝐷(𝑀₀) dense in 𝑊^∘21,0(𝑄_𝜔), satisfying the identity _𝑄𝜔(𝜙𝜑+𝑘∇𝜙∇𝜑)d𝑥d𝑡=𝑀₀𝜙,𝜑_{𝐿2(𝑄𝜔)},(2.10) whenever 𝜙∈𝐷(𝑀₀) and 𝜑∈𝑊^∘21,0(𝑄_𝜔). The range of 𝑀₀ is all of 𝐿²(𝑄_𝜔), and 𝑀₀ has an inverse which is a bounded mapping of 𝐿²(𝑄_𝜔) into 𝑊^∘21,0(𝑄_𝜔). Then the weak solution 𝑢 in 𝐸₀ is just the weak solution of the following equation: 𝜕𝑢𝜕𝑡=−𝑘1𝑢+𝑀⁰⁻¹𝑘1𝑢+𝛼(𝑥,𝑡)|𝑢|^𝑝.(2.11) We can also relate the extended operator 𝑀₀ to the operator 𝑀₁ which are just the extension of 𝐼−𝑘Δ to the domain 𝑊^∘21,0(𝑄_𝜔)∩𝑊^22,0(𝑄_𝜔) in the sense of generalized derivatives. 𝑀₁ has a continuous inverse operator from 𝐿²(𝑄_𝜔) to 𝑊^∘21,0(𝑄_𝜔)∩𝑊^22,0(𝑄_𝜔). Thus, from 𝜕𝑢𝜕𝑡=−𝑘1𝑢+𝑀¹⁻¹𝑘1𝑢+𝛼(𝑥,𝑡)|𝑢|^𝑝(2.12) and 𝑢∈𝐿^∞(𝑄_𝜔), there hold 𝜕𝑢𝜕𝑡∈𝐿²𝑄_𝜔,𝜕𝜕𝑢𝑡+𝑘1𝑢∈^∘𝑊^21,0𝑄_𝜔∩𝑊^22,0𝑄_𝜔.(2.13) Multiplying 𝑒^𝑡/𝑘 on both sides of (2.12), we get 𝜕𝜕𝑡𝑒^𝑡/𝑘𝑢=𝑒^𝑡/𝑘𝑀¹⁻¹1𝑘𝑢+𝛼(𝑥,𝑡)|𝑢|^𝑝.(2.14) For any 𝑡∈[0,𝜔], integrating the above equation in [𝑡,𝑡+𝜔] and using the periodicity of 𝑢 yield 𝑢(𝑥,𝑡)=𝑒^{(𝑡+𝜔)/𝑘}−𝑒^𝑡/𝑘⁻¹^{𝑡𝑡+𝜔}𝑒^s/𝑘𝑀¹⁻¹1𝑘𝑢(𝑥,𝑠)+𝛼(𝑥,𝑡)|𝑢|^𝑝(𝑥,𝑠)d𝑠,(2.15) which with (2.13) imply that 𝑢∈𝑊^∘21,0𝑄_𝜔∩𝑊^22,1𝑄_𝜔,𝜕𝑢𝜕𝑡∈^∘𝑊^21,0𝑄_𝜔∩𝑊^22,0𝑄_𝜔.(2.16) 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 𝑀_𝑟∶𝑊^𝑟2,0(𝑄_𝜔)∩{𝜙(𝑥,𝑡)=0forany𝑥∈Ω}→𝐿^𝑟(𝑄_𝜔) by 𝑀_𝑟𝜙∶=(𝐼−𝑘Δ)𝜙, whose inverse 𝑀^𝑟−1∶𝐿^𝑟(𝑄_𝜔)→𝑊^𝑟2,0(𝑄_𝜔)∩{𝜙(𝑥,𝑡)=0forany𝑥∈Ω} is continuous. Then, we have 𝜕𝑢𝜕𝑡=−𝑘1𝑢+𝑀^𝑟−1𝑘1𝑢+𝛼(𝑥,𝑡)|𝑢|^𝑝.(2.17) Similar to the above discussion, we can deduce that 𝑢∈𝑊^𝑟2,1𝑄_𝜔,𝜕𝑢𝜕𝑡∈𝑊^𝑟2,0𝑄_𝜔,forany1≤𝑟<∞.(2.18) From the Isotropic Embedding Theorem [34], we know that 𝑢∈𝐶^𝛼,𝛼/2𝑄_𝜔,0<𝛼<2−𝑁+2𝑟with𝑟>𝑁2+2.(2.19) As in [35, 36], (𝐼−𝑘Δ)⁻¹ is bounded from 𝐶^𝛼,𝛼/2(𝑄_𝜔) to 𝐶^{2+𝛼,𝛼/2}(𝑄_𝜔), then 𝑢 satisfies 𝜕𝑢𝜕𝑡=−𝑘1𝑢+(𝐼−𝑘Δ)⁻¹𝑘1𝑢+𝛼(𝑥,𝑡)|𝑢|^𝑝.(2.20) In the same way, we have 𝑢∈𝐶^{2+𝛼,1+𝛼/2}𝑄_𝜔,𝜕𝑢𝜕𝑡∈𝐶^{2+𝛼,𝛼/2}𝑄_𝜔,(2.21) 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 (𝑥₀,𝑡₀)∈Ω×(0,𝜔) such that 𝑢𝑥₀,𝑡₀<0.(2.22) Since 𝑢 is continuous, then there exists a domain Ω₀ such that 𝑢(𝑥,𝑡₀)<0 in Ω₀ and 𝑢(𝑥,𝑡₀)=0 on 𝜕Ω₀. Multiplying (2.2) by 𝜑, which is the principle eigenfunction of −Δ in Ω₀ with homogeneous Dirichlet boundary condition, and integrating on Ω₀, we get 1+𝑘𝜆₀_Ω0𝜕𝑢𝜕𝑡𝜑d𝑥+𝜆⁰_Ω0𝑢𝜑d𝑥=_Ω0𝛼(𝑥,𝑡)|𝑢|^𝑝𝜑d𝑥,(2.23) where 𝜆₀ is the first eigenvalue. Integrating the above inequality from 0 to 𝜔 and using the periodicity of 𝑢, we have 𝜆₀^𝜔0_Ω0𝑢𝜑d𝑥d𝑡>0.(2.24) By the mean value theorem, there exists a point 𝑡^∗∈(0,𝜔) such that _Ω0𝑢𝑥,𝑡^∗𝜑d𝑥>0.(2.25) Actually (2.23) is equivalent to _Ω0𝜕𝑒^{𝑡𝜆0/(1+𝑘𝜆0)}𝑢𝜕𝑡𝜑d𝑥=1+1𝑘𝜆₀_Ω0𝑒^{𝑡𝜆0/(1+𝑘𝜆0)}𝛼(𝑥,𝑡)|𝑢|^𝑝𝜑d𝑥.(2.26) Integrating the above inequality from 𝑡^∗ to 𝜔 implies that _Ω0𝑒^{𝜔𝜆0/(1+𝑘𝜆0)}𝑢(𝑥,𝜔)𝜑d𝑥>0.(2.27) Recalling the periodicity of 𝑢, we see that _Ω0𝑢(𝑥,0)𝜑d𝑥>0.(2.28) Then integrating (2.26) over (0,𝑡) implies that _Ω0𝑒^{𝑡𝜆0/(1+𝑘𝜆0)}𝑢(𝑥,𝑡)𝜑d𝑥>0,𝑡∈(0,𝜔)(2.29) which contradicts with 𝑢(𝑥,𝑡₀)<0 in Ω₀.

3. 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.

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. 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 𝑅 to be appropriately large such that Ω⊂𝐵_𝑅/2. Let 𝜆₁, 𝜆̃₁ 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 principle eigenfunctions normalized by ‖𝜙‖_𝐿^∞_(Ω)=1 and ‖𝜓‖_𝐿^∞_{(𝐵𝑅)}=1. Precisely speaking, 𝜙 and 𝜓 satisfy −𝜙Δ(𝜙𝑥)(||𝑥_𝜕Ω)=𝜆₁𝜙,𝑥∈Ω,−Δ𝜓(𝑥)==0𝜆,̃₁𝜙(𝑥)||_{𝜕𝐵𝑅}𝜓,𝑥∈𝐵_𝑅,=0.(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 Φ=𝜅₁𝜙(𝑥),Ψ=𝜅₂𝜓(𝑥),(3.2) where 𝜅₁ and 𝜅₂ are constants which are to be determined later.
Actually, if we choose 𝜅₁=(𝑆/𝜆₁)^1/(1−𝑝), then a simple calculation yields that −ΔΦ(𝑥)=−𝜅₁Δ𝜙=𝜅₁𝜆₁𝜙≤𝑆𝜆₁^1/(1−𝑝)𝜆₁𝜙=𝑆𝑆𝜆₁^{𝑝/(1−𝑝)}𝜙^𝑝≤𝛼(𝑥,𝑡)Φ^𝑝.(3.3) 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 𝜆̃₁Ψ^1−𝑝≥𝛼(x,𝑡),(3.4) which is ensured by 𝜅₂≥max1𝛾𝜆𝐿̃₁^1/(1−𝑝),𝜅¹𝛾.(3.5) Clearly we also have Ψ(𝑥)≥Φ(𝑥).
Set 𝑢̂=Φ and 𝑢̃=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 𝜕𝑢_𝑚𝜕𝑡−𝑘𝜕Δ𝜕𝑡𝑢^𝑚=Δ𝑢^𝑚𝑢+𝛼(𝑥,𝑡)||𝑢_𝑚−1||^𝑝,(𝑥,𝑡)∈𝑄_𝜔,(3.6)_𝑚𝑢(𝑥,𝑡)=0,𝑥∈𝜕Ω,𝑡∈(𝜏,𝜏+𝜔),(3.7)_𝑚(𝑥,𝜏)=𝑢_𝑚−1(𝑥,𝜏+𝜔),𝑥∈Ω(3.8) for 𝑚=1,2,…, where 𝑢₀=𝑢̂. 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, 𝑢̂=𝑢₀≤𝑢₁≤⋯≤𝑢_𝑚≤𝑢_𝑚+1≤⋯≤𝑢.̃(3.9) Since 𝑢₀=𝑢̂ is the lower solution, we get 𝜕𝑢₁−𝑢₀𝜕𝑡−𝑘𝜕Δ𝑢¹−𝑢₀𝜕𝑡≥Δ𝑢¹𝑢₁−𝑢₀,(𝑥,𝑡)∈𝑄_𝜔,−𝑢₀𝑢(𝑥,𝑡)=0,𝑥∈𝜕Ω,𝑡∈(𝜏,𝜏+𝜔),₁−𝑢₀(𝑥,𝜏)≥0,𝑥∈Ω.(3.10) By using the comparison principle of pseudo-parabolic equation [35, 39], we have that 𝑢₁≥𝑢₀, and 𝜕𝑢₁𝜕𝑡−𝑘𝜕Δ𝜕𝑡𝑢¹=Δ𝑢¹𝑢₁+𝛼||𝑢₀||^𝑝≤Δ𝑢₁+𝛼||𝑢₁||^𝑝,(𝑥,𝑡)∈𝑄_𝜔,𝑢(𝑥,𝑡)=0,𝑥∈𝜕Ω,𝑡∈(𝜏,𝜏+𝜔),₁(𝑥,𝜏)=𝑢₀(𝑥,𝜏+𝜔)≤𝑢₁(𝑥,𝜏+𝜔),𝑥∈Ω,(3.11) which means that 𝑢₁ is a lower periodic solution. Furthermore, for 𝑢̃−𝑢₁, there also holds 𝜕𝑢̃−𝑢₁𝜕𝑡−𝑘𝜕Δ𝑢̃−𝑢¹𝜕𝑡≥Δ𝑢̃−𝑢¹,(𝑥,𝑡)∈𝑄_𝜔𝑢̃−𝑢₁,𝑢̃−𝑢₁(𝑥,𝑡)≥0,𝑥∈𝜕Ω,𝑡∈(𝜏,𝜏+𝜔),(𝑥,𝜏)≥𝑢̃(𝑥,𝜏+𝜔)−𝑢̂(𝑥,𝜏+𝜔)≥0,𝑥∈Ω,(3.12) which indicates that 𝑢̃≥𝑢₁ 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 _𝑄𝜔||∇𝑢_𝑚||²d𝑥d𝑡=_𝑄𝜔𝛼(𝑥,𝑡)𝑢_𝑚||𝑢_𝑚−1||^𝑝d𝑥d𝑡≤𝜔||Ω||𝐿‖𝑢‖̃^∞𝑝+1.(3.13) Next, multiplying both sides of (3.7) by 𝜕𝑢_𝑚/𝜕𝑡, we have _𝑄𝜔||||𝜕𝑢_𝑚𝜕𝑡||||²d𝑥d𝑡+𝑘_𝑄𝜔||||𝜕∇𝑢_𝑚𝜕𝑡||||²d𝑥d𝑡=_𝑄𝜔𝛼(𝑥,𝑡)𝜕𝑢^𝑚𝜕𝑡||𝑢^𝑚−1||^𝑝d𝑥d𝑡.(3.14) Using the Young inequality to the above equality yields _𝑄𝜔||||𝜕𝑢_𝑚𝜕𝑡||||²d𝑥d𝑡+_𝑄𝜔||||𝜕∇𝑢_𝑚𝜕𝑡||||²d𝑥d𝑡≤𝜔||Ω||𝐿²‖𝑢‖̃^∞2𝑝.(3.15) Hence, when 𝑛→∞, it follows that, for any 𝑟>0, ∇𝑢𝑢_𝑚𝑚⟶𝑢,in𝐿^𝑟𝑄_𝜔,𝜕𝑢_𝑚⇀∇𝑢,in𝐿²𝑄_𝜔,𝜕∇𝜕𝑢𝑡_𝑚⇀𝜕𝜕𝑢𝑡,in𝐿²𝑄_𝜔,𝜕𝑡⇀𝜕𝜕∇𝑡𝑢,in𝐿²𝑄_𝜔,(3.16) which imply that 𝑢∈𝐸₀ 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 0<Φ(𝑥)≤𝑢≤Ψ(𝑥)≤𝜅₂,𝑥∈Ω,(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).

4. 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].

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 satisfies(a)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 problem 𝑢=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 ℰ∩({0}×𝐸)={(0,0)},(4.2) then ℰ is unbounded in ℝ⁺×𝐸.

Define an operator 𝐺 by𝐺ℝ⁺,𝐿^∞𝜔(𝜏,𝜏+𝜔),𝐿𝑝𝐺̃(Ω(𝑙),𝑣)=⟶𝑢,𝐿^∞𝜔(𝜏,𝜏+𝜔),𝐿𝑝̃(Ω),(4.3) where (2𝑁/(𝑁+2))𝑝<1+𝑝≤𝑝̃<2𝑁/(𝑁−2) is a constant. Let 𝑢 be a solution of the following problem𝜕𝑢𝜕𝑡−𝑘𝜕𝜕Δ𝑡𝑢=Δu+𝛼(𝑥,𝑡)(|𝑣|+𝑙)^𝑝𝑢(𝑥𝑢,𝑡()𝑥=,𝜏0),=𝑢(𝑥(,𝑥𝑡,)𝜏∈+𝜕𝜔Ω)×,(𝑥𝜏,,∈𝜏+Ω(𝑥.𝜔,𝑡)),∈𝑄_𝜔,(((444...456)))

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.

Lemma 4.2. When 1<𝑝<(𝑁+2)/(𝑁−2) with 𝑁>2 or 1<𝑝<∞ with 𝑁≤2, the operator 𝐺 is completely continuous.

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 12dd𝑡_Ω|𝑢|²d𝑥+𝑘2dd𝑡_Ω||∇𝑢||²d𝑥+_Ω||∇𝑢||²d𝑥≤𝐿_Ω𝑢(|𝑣|+𝑙)^𝑝d𝑥.(4.7) Integrating the above inequality from 𝜏 to 𝜏+𝜔 and combining with the Hölder inequality and the Isotropic Embedding Theorem [34], we conclude that _𝑄𝜔||∇𝑢||²d𝑥d𝑡≤𝐿_𝑄𝜔|𝑢|𝑝̃^/(𝑝̃_−𝑝)d𝑥d𝑡⁽𝑝̃_−𝑝)/𝑝̃_𝑄𝜔(|𝑣|+𝑙)𝑝d̃𝑥d𝑡^𝑝/≤𝐶_𝑄𝜔𝑝̃||∇𝑢||²d𝑥d𝑡^1/2_𝑄𝜔(|𝑣|+𝑙)𝑝d̃𝑥d𝑡^𝑝/𝑝.̃(4.8) Thus, we have _𝑄𝜔|𝑢|²d𝑥d𝑡≤𝐶₁,_𝑄𝜔||∇𝑢||²d𝑥d𝑡≤𝐶₁,(4.9) where 𝐶₁ depends only on 𝑙, 𝑝, 𝑝̃, ‖𝛼‖_𝐿^∞_(𝑄𝜔), 𝑄_𝜔, and sup_{𝑡∈(𝜏,𝜏+𝜔)}‖𝑣‖_𝐿𝑝̃_(Ω). By the mean value theorem, we see that there exists a point 𝑡̂∈(𝜏,𝜏+𝜔) such that _Ω||𝑢𝑥,𝑡̂||²+𝑘||∇𝑢𝑥,𝑡̂||²d𝑥≤𝐶¹𝜔.(4.10) Integrating (4.7) from 𝑡̂ to 𝑡 gives _Ω|𝑢|²+𝑘||∇𝑢||²d𝑥≤𝐶²,𝑡∈𝑡,̂𝜏+𝜔.(4.11) Noticing the periodicity of 𝑢, we arrive at _Ω||𝑢(𝑥,𝜏)||²+𝑘||∇𝑢(𝑥,𝜏)||²d𝑥≤𝐶²,(4.12) from which it is easy to obtain that sup_{[]𝑡∈𝜏,𝜏+𝜔}_Ω|𝑢|²+𝑘||∇𝑢||²(𝑥,𝑡)d𝑥≤𝐶²,(4.13) where 𝐶₂ depends only on 𝑙, 𝑝, 𝑝̃, 𝑘, ‖𝛼‖_𝐿^∞_(𝑄𝜔), 𝑄_𝜔, and sup_{𝑡∈(𝜏,𝜏+𝜔)}‖𝑣‖_𝐿𝑝̃_(Ω). Using the Isotropic Embedding Theorem [34] yields _Ω|𝑢|^𝑟d𝑥≤𝐶₃,(4.14) where 𝑟=11≤≤𝑟𝑟<≤+𝑁2∞𝑁−,2,𝑁𝑁>≤22,,(4.15) and 𝐶₃depends on 𝑙, 𝑝, 𝑝̃, 𝑘, 𝑁, ‖𝛼‖_𝐿^∞_(𝑄𝜔), 𝑄_𝜔, and sup_{𝑡∈(𝜏,𝜏+𝜔)}‖𝑣‖_𝐿𝑝̃_(Ω). Multiplying (4.4) by 𝜕𝑢/𝜕𝑡 and integrating over Ω yield _Ω|||𝜕𝑢𝜕𝑡|||²d𝑥+𝑘_Ω|||𝜕∇𝑢𝜕𝑡|||²d𝑥+12dd𝑡_Ω||∇𝑢||²d𝑥=_Ω𝜕𝑢𝜕𝑡𝛼(𝑥,𝑡)(|𝑣|+𝑙)^𝑝d𝑥.(4.16) Integrating the above equality from 𝜏 to 𝜏+𝜔 and using the Hölder inequality and the Isotropic Embedding Theorem [34], we get _𝑄𝜔|||𝜕𝜕𝑢𝑡|||²+𝑘|||𝜕∇𝑢≤𝐿_𝑄𝜔𝜕𝑡|||²d𝑥d𝑡|||𝜕𝑢𝜕𝑡|||𝑝̃^/(𝑝̃_−𝑝)d𝑥d𝑡⁽𝑝̃_−𝑝)/𝑝̃_𝑄𝜔(|𝑣|+𝑙)𝑝d̃𝑥d𝑡^𝑝/≤𝐶_𝑄𝜔𝑝̃|||𝜕∇𝑢𝜕𝑡|||²d𝑥d𝑡^1/2_𝑄𝜔(|𝑣|+𝑙)𝑝d̃𝑥d𝑡^𝑝/𝑝.̃(4.17) Then one has _𝑄𝜔|||𝜕𝜕𝑢𝑡|||²+𝑘|||𝜕∇𝑢𝜕𝑡|||²d𝑥d𝑡≤𝐶⁴,(4.18) where 𝐶₄ depends on 𝑙, 𝑝, 𝑝̃, ‖𝛼‖_𝐿^∞_(𝑄𝜔), 𝑄_𝜔, and sup_{𝑡∈(𝜏,𝜏+𝜔)}‖𝑣‖_𝐿𝑝̃_(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.

Theorem 4.3. Assume 𝛼(𝑥,𝑡)∈𝐶¹(ℝ^𝑁,ℝ). If Ω is a convex domain and 1<𝑝<<+𝑁𝑁∞+−,22,ffoorr𝑁𝑁≤>22,,(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 4.1. 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: 𝜕𝑢𝜕𝑡−𝑘𝜕𝜕Δ𝑡𝑢=Δ𝑢,(𝑥,𝑡)∈𝑄^𝜔𝑢(𝑥𝑢,𝑡()𝑥=,𝜏0),=𝑢(𝑥(,𝑥𝑡,)𝜏∈+𝜕𝜔Ω)×,(𝑥𝜏,∈𝜏+Ω.𝜔,),(((444...222012))) 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 _𝑄𝜔||∇𝑢||²d𝑥d𝑡=0.(4.23) Recalling the Poincaré inequality, we see that _𝑄𝜔𝑢²d𝑥d𝑡≤𝐶_𝑄𝜔||∇𝑢||²d𝑥d𝑡=0,(4.24) 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 sup_𝑡‖𝑢(⋅,𝑡)‖_𝐿𝑝̃_(Ω)<𝑅,(4.25) 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 12dd𝑡_Ω|𝑢|²d𝑥+𝑘2dd𝑡_Ω≤𝐿_Ω||∇𝑢||²d𝑥+_Ω||∇𝑢||²d𝑥|𝑢|^𝑝+1d𝑥≤𝐿||Ω||⁽𝑝̃_{−𝑝−1)/}𝑝̃_Ω|𝑢|𝑝d̃𝑥^(𝑝+1)/𝑝.̃(4.26) By virtue of the Isotropic Embedding Theorem, we get 12dd𝑡_Ω|𝑢|²d𝑥+𝑘2dd𝑡_Ω||∇𝑢||²d𝑥+𝜇_Ω|𝑢|𝑝d̃𝑥^2/𝑝̃≤𝐿||Ω||₍𝑝̃_{−𝑝−1)/}𝑝̃_Ω|𝑢|𝑝d̃𝑥^(𝑝+1)/𝑝,̃(4.27) where 𝜇 is the constant in the Isotropic Embedding Theorem; that is, 12𝑑𝑑𝑡_Ω|𝑢|²d𝑥+𝑘2dd𝑡_Ω≤𝐿||Ω||||∇𝑢||²d𝑥₍𝑝̃_{−𝑝−1)/}𝑝̃−𝜇_Ω|𝑢|𝑝d̃𝑥^(1−𝑝)/𝑝̃_Ω|𝑢|𝑝d̃𝑥^(𝑝+1)/𝑝.̃(4.28) Thus, if sup_𝑡_Ω|𝑢|𝑝̃≤𝑅₀,(4.29) where 𝑅₀=𝜇2𝐿𝑝̃^/(𝑝−1)||Ω||⁻⁽𝑝̃_{−𝑝−1)/(𝑝−1)},(4.30) then we have 12dd𝑡_Ω|𝑢|²d𝑥+𝑘2dd𝑡_Ω||∇𝑢||²d𝑥≤−𝐿||Ω||⁽𝑝̃_{−𝑝−1)/}𝑝̃_Ω|𝑢|^𝑝+1d𝑥,(4.31) which means 𝑢=0 a.e. in 𝑄_𝜔.
Next, we check the condition (b) in Lemma 4.1, namely, there exists an 𝑅<𝑅₀, such that deg(𝐢𝐝−𝐺(0,⋅),𝐵(0,𝑅),0)=1.(4.32) Consider the following problem: 𝜕𝑢𝜕𝑡−𝑘𝜕𝜕Δ𝑡𝑢=Δ𝑢+𝜎𝛼(𝑥,𝑡)|𝑣|^𝑝𝑢(𝑥𝑢,𝑡()𝑥=,𝜏0),=𝑢(𝑥(𝑥,𝑡,)𝜏∈+𝜕𝜔Ω),×(,𝑥𝜏,∈𝜏(𝑥Ω+,𝑡,𝜔)∈),𝑄_𝜔,(4.33) where 𝜎∈[0,1]. Construct a homotopic mapping 𝑇∶[0,1]×𝐿^∞𝜔(𝜏,𝜏+𝜔),𝑇𝐿(𝑝𝜎̃(,Ω𝑣))=⟶𝑢.𝐿^∞𝜔(𝜏,𝜏+𝜔),𝐿𝑝̃(Ω),(4.34) Similar to Lemma 4.2, we see that 𝑇 is completely continuous. Assume that sup_{𝑡∈(𝜏,𝜏+𝜔)}‖𝑣(⋅,𝑡)‖𝑝_𝐿̃𝑝̃_(Ω)≤𝑅,(4.35) where 𝑅≤𝑅₀ is to be determined. Multiplying the first equation of the above problem by 𝑢 and integrating over Ω yield 12dd𝑡_Ω|𝑢|²d𝑥+𝑘2dd𝑡_Ω≤𝐿_Ω||∇𝑢||²d𝑥+_Ω||∇𝑢||²d𝑥|𝑢|𝑝̃^/(𝑝̃_−𝑝)d𝑥⁽𝑝̃_−𝑝)/𝑝̃_Ω|𝑣|𝑝d̃𝑥^𝑝/≤𝐿_Ω𝑝̃||∇𝑢||²d𝑥^1/2_Ω|𝑣|𝑝d̃𝑥^𝑝/≤𝐶_Ω𝑝̃||∇𝑢||²d𝑥^1/2𝑅^𝑝/𝑝.̃(4.36) Integrating from 𝜏 to 𝜏+𝜔 gives _𝑄𝜔||∇𝑢||²d𝑥≤𝐶𝑅^2𝑝/𝑝.̃(4.37) Then, we further have _𝑄𝜔|𝑢|²d𝑥≤𝐶𝑅^2𝑝/𝑝.̃(4.38) By means of the integral mean value theorem, we see that there exists a 𝑡_𝜎∈[𝜏,𝜏+𝜔] such that _Ω||𝑢𝑥,𝑡^𝜎||²+||∇𝑢𝑥,𝑡_𝜎||²d𝑥≤𝐶𝑅^2𝑝/𝑝.̃(4.39) By the periodicity of 𝑢 and a similar process as Lemma 4.2, we obtain that sup_{𝑡∈(𝜏,𝜏+𝜔)}_Ω|𝑢|²+𝑘||∇𝑢||²d𝑥≤𝐶𝑅^2𝑝/𝑝.̃(4.40) Using the Isotropic Embedding Theorem gives sup_{𝑡∈(𝜏,𝜏+𝜔)}_Ω||𝑢(𝑥,𝑡)||𝑝d̃𝑥≤𝐶𝑅^𝑝<𝑅.(4.41) If 𝑅 with 𝑅≤𝑅₀ is appropriately small, such that 𝑝>1, therefore, deg(id−𝑇(1,⋅),𝐵(0,𝑅),0)=deg(id−𝑇(0,⋅),𝐵(0,𝑅),0)=1,(4.42) which means that deg(id−𝐺(0,⋅),𝐵(0,𝑅),0)=1.(4.43)
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 𝑙_𝑛+sup_{𝑡∈(𝜏,𝜏+𝜔)}‖‖𝑢_𝑛(⋅,𝑡)‖‖_𝐿∞(Ω)⟶∞,(4.44) which implies that 𝑙_𝑛+‖‖𝑢_𝑛‖‖_{𝐿∞(𝑄𝜔)}⟶∞.(4.45) If this were true, then we would have 𝑙_𝑛‖‖𝑢_𝑛‖‖_{𝐿∞(𝑄𝜔)}⟶0.(4.46) Suppose the contrary, and note that if 𝑙_𝑛 is bounded, then ‖𝑢_𝑛‖_𝐿^∞_(𝑄𝜔)→∞, which means (4.46). Thus, without loss of generality, we may assume that 0<𝑙_𝑛→+∞. Making change of variable 𝑣_𝑛=𝑢^𝑛𝑙_𝑛,(4.47) we have 𝜕𝑣_𝑛𝜕𝑡−𝑘𝜕Δ𝜕𝑡𝑣^𝑛−Δ𝑣^𝑛=𝛼(𝑥,𝑡)𝑙^𝑛𝑝−1||𝑣_𝑛||+1^𝑝.(4.48) If ‖𝑣_𝑛‖_𝐿^∞_(𝑄𝜔) are bounded uniformly, that is, there is a constant 𝐶>0 such that ‖𝑣_𝑛‖_𝐿^∞_(𝑄𝜔)<𝐶, then, for any 𝜑∈𝐶^1𝑇(𝑄_𝜔) with 𝜑|_𝜕Ω=0, we have _𝑄𝜔𝜕𝑣_𝑛𝜕𝑡𝜑d𝑥d𝑡+_𝑄𝜔𝑘𝜕∇𝑣^𝑛𝜕𝑡∇𝜑d𝑥d𝑡+_𝑄𝜔=∇𝑣_𝑛∇𝜑d𝑥d𝑡_𝑄𝜔𝛼(𝑥,𝑡)𝑙^𝑛𝑝−1||𝑣_𝑛||+1^𝑝𝜑d𝑥d𝑡.(4.49) Noticing the density of 𝐶^1𝑇(𝑄_𝜔) in 𝐿^2𝑇((𝜏,𝜏+𝜔),𝑊^1,2(Ω)), then it is sound to take 𝜑=𝑣_𝑛 thus, we have _𝑄𝜔||∇𝑣_𝑛||²d𝑥d𝑡≤𝐶𝐿𝑙^𝑛𝑝−1.(4.50) In addition, for any 0≤𝜑(𝑥)∈𝐶¹⁰(Ω), we also have 𝑙^𝑛𝑝−1_𝑄𝜔𝛼(𝑥,𝑡)𝜑d𝑥d𝑡≤𝑙^𝑛𝑝−1_𝑄𝜔≤𝛼(𝑥,𝑡)||𝑣_𝑛||+1^𝑝𝜑d𝑥d𝑡_𝑄𝜔||∇𝑣_𝑛||²d𝑥d𝑡^1/2_𝑄𝜔||∇𝜑||²d𝑥d𝑡^1/2≤_𝑄𝜔||∇𝜑||²d𝑥d𝑡^1/2𝐶𝐿𝑙^𝑛𝑝−1^1/2,(4.51) that is, 𝑙^{𝑛(𝑝−1)/2}_𝑄𝜔𝛼(𝑥,𝑡)𝜑d𝑥d𝑡≤_𝑄𝜔||∇𝜑||²d𝑥d𝑡^1/2.(4.52) Clearly, it is a contradiction since 𝑙_𝑛→∞. Therefore, (4.46) holds, which also implies that ‖𝑢_𝑛‖_𝐿^∞_(𝑄𝜔)→∞. Let 𝜌_𝑛=‖𝑢_𝑛‖_𝐿^∞_(𝑄𝜔)=𝑢_𝑛(𝑥_𝑛,𝑡_𝑛)→∞. By the convexity of Ω, we see that there exists a 𝛿₀>0 such that dist(𝑥_𝑛,𝜕Ω)≥𝛿₀; see, for example, [41, 42]. Then, there exists a subsequence, and for simplicity, we still denote it by 𝑥_𝑛 such that 𝑥_𝑛→𝑥₀, 𝑡_𝑛→𝑡₀ with dis(𝑥_𝑛,𝜕Ω)≥𝛿₀. Let 𝜔_𝑛𝑗(𝑦,𝑠)=𝜌⁻¹_n𝑢_𝑛𝜌^{𝑛−(𝑝−1)/2}𝑦+𝑥₀,𝑡_𝑗+𝑗𝑠,𝛼_𝑛𝑗=𝛼𝜌^{𝑛−(𝑝−1)/2}𝑦+𝑥₀,𝑡_𝑗+𝑗𝑠,(4.53) and let Ω_𝑛=𝑦;𝑦=𝜌^{𝑛(𝑝−1)/2}𝑥−𝑥₀for𝑥∈Ω,𝑄_𝑛𝑗=Ω_𝑛×𝜏−𝑡^𝑗𝑗,𝜏+𝜔𝑗−𝑡^𝑗.(4.54) Then, 𝜔_𝑛𝑗 with ‖𝜔_𝑛𝑗‖_𝐿^∞_{(𝑄𝑛𝜔)}=1 on 𝑄_𝑛𝑗 satisfies 𝜌^𝑛1−𝑝𝜕𝜔_𝑛𝑗𝜕𝑠−𝑘𝜕Δ𝜕𝜔𝑠^𝑛𝑗−𝑗Δ𝜔^𝑛𝑗=𝑗𝛼(𝑦,𝑠)||𝜔_𝑛𝑗||+𝑙_𝑛𝜌^𝑛−1^𝑝.(4.55) Similar to the proof of Theorem 2.2, we can deduce that 𝜔_𝑛𝑗≥0 which admits throwing off the symbol of absolute value of |𝜔_𝑛𝑗|. Therefore, for any 𝜙(𝑦,𝑠)∈𝐶^1𝑇(𝑄_𝑛𝑗) with 𝜙=0 on 𝜕Ω_𝑛, we have _𝑄𝑛𝑗𝜌^𝑛1−𝑝𝜕𝜔_𝑛𝑗𝜕𝑠𝜙d𝑦d𝑠+_𝑄𝑛𝑗𝑘𝜕∇𝜔^𝑛𝑗𝜕𝑠∇𝜙d𝑦d𝑠+𝑗_𝑄𝑛𝑗=𝑗_𝑄𝑛𝑗∇𝜔_𝑛𝑗∇𝜙d𝑦d𝑠𝛼(𝑦,𝑠)𝜔_𝑛𝑗+𝑙_𝑛𝜌^𝑛−1^𝑝𝜙d𝑦d𝑠.(4.56) Taking 𝜙=𝜔_𝑛𝑗, we have 𝑗_𝑄𝑛𝑗||∇𝜔_𝑛𝑗||²d𝑦d𝑠=𝑗_𝑄𝑛𝑗𝛼(𝑦,𝑠)𝜔_𝑛𝑗+𝑙_𝑛𝜌^𝑛−1^𝑝𝜔_𝑛𝑗d𝑦d𝑠≤𝐶||Ω_𝑛||,(4.57) which means that there exists 𝜎_𝑗∈[(𝜏−𝑡_𝑗)/𝑗,(𝜏+𝜔−𝑡_𝑗)/𝑗) such that _Ω𝑛||∇𝜔_𝑛𝑗𝑦,𝜎_𝑗||²d𝑦≤𝐶||Ω_𝑛||.(4.58) For any 𝑠>𝜎_𝑗, taking 𝜙=𝜒(𝜎_𝑗,𝑠)(𝜕𝜔_𝑛𝑗/𝜕𝑠) in (4.56) yields 12_Ω𝑛||∇𝜔_𝑛𝑗(𝑦,𝑠)||²d𝑦≤12_Ω𝑛+1||∇𝜔_𝑛𝑗𝑦,𝜎_𝑗||²d𝑦𝑝+1_Ω𝑛−1𝛼_𝑛𝑗𝑦,𝜎_𝑗𝜔_𝑛𝑗𝑦,𝜎_𝑗+𝑙_𝑛𝜌^𝑛−1^𝑝+1d𝑦𝑝+1_Ω𝑛−𝑗𝛼_𝑛𝑗(𝑦,𝑠)𝜔_𝑛𝑗(𝑦,𝑠)+𝑙_𝑛𝜌^𝑛−1^𝑝+1d𝑦𝑝+1^𝑠𝜎_𝑗_Ω𝑛𝜕𝛼_𝑛𝑗𝜕𝑠(𝑦,𝑠)𝜔^𝑛𝑗≤𝐶₅+𝑙_𝑛𝜌^𝑛−1^𝑝+1d𝑦d𝑠||Ω_𝑛||,(4.59) where 𝐶₅ is a constant independent of 𝑗, 𝑛, Ω_𝑛. By the periodicity of 𝜔_𝑛𝑗, we further have _Ω𝑛||||∇𝜔_𝑛𝑗𝑦,𝜏−𝑗𝑡^𝑗||||²d𝑦≤𝐶₅||Ω_𝑛||.(4.60) Repeating the process above, we finally obtain that for any 𝑠∈((𝜏−𝑡_𝑗)/𝑗,𝜎_𝑗), _Ω𝑛||∇𝜔_𝑛𝑗(𝑦,𝑠)||²d𝑦≤𝐶₆||Ω_𝑛||.(4.61) Summing up, we finally obtain that sup_𝑠_Ω𝑛||∇𝜔_𝑛𝑗(𝑦,𝑠)||²d𝑦≤𝐶||Ω_𝑛||.(4.62) In addition, we note that, for any 𝜑∈𝐶¹⁰(Ω_𝑛), we have 𝑗_𝑄𝑛𝑗∇𝜔_𝑛𝑗∇𝜑d𝑦d𝑠=𝑗_𝑄𝑛𝑗𝛼_𝑛𝑗(𝑦,𝑠)𝜔_𝑛𝑗+𝑙_𝑛𝜌^𝑛−1^𝑝𝜑d𝑦d𝑠.(4.63) By Lebesgue differential theorem, there exists 𝑠_𝑗∈((𝜏−𝑡_𝑗)/𝑗,(𝜏+𝜔−𝑡_𝑗)/𝑗) such that _Ω𝑛∇𝜔_𝑛𝑗𝑦,𝑠_𝑗∇𝜑d𝑦=_Ω𝑛𝛼_𝑛𝑗𝑦,𝑠_𝑗𝜔_𝑛𝑗𝑦,𝑠_𝑗+𝑙_𝑛𝜌^𝑛−1^𝑝𝜑d𝑦.(4.64) Then, there exists a function 𝜔_𝑛∈𝑊^1,2(Ω_𝑛) with ‖𝜔_𝑛‖_𝐿^∞=1 such that as 𝑗→∞ (passing to a subsequence if necessary) ∇𝜔_𝑛𝑗⇀∇𝜔_𝑛in𝐿²Ω_𝑛;𝜔_𝑛𝑗⟶𝜔_𝑛in𝐿^𝑟Ω_𝑛for1≤𝑟≤2𝑁𝛼_𝑛𝑗𝑁−2,𝑦,𝑠_𝑗⟶𝛼_𝑛(𝑦)uniformly,𝛼_𝑛(𝑦)∈𝐶^𝛽Ω_𝑛forsome0<𝛽<1,(4.65) we obtain that _Ω𝑛∇𝜔_𝑛∇𝜑d𝑦=_Ω𝑛𝛼(𝑦)𝜔_𝑛+𝑙_𝑛𝜌^𝑛−1^𝑝d𝑦.(4.66) Take 𝜑=𝜔_𝑛𝜂⁴(𝑥), where 𝜂=1,𝑥∈𝐵^𝑅0,𝑥∈𝐵_2𝑅(0),(0),(4.67) with 0≤𝜂≤1 is sufficiently smooth and |∇𝜂|≤𝐶/𝑅. Then, for sufficiently large 𝑛, we have 𝐵_2𝑅⊂Ω_𝑛 and _𝐵2𝑅𝜂⁴||∇𝜔_𝑛||²d𝑦=≤−1_𝐵2𝑅4𝜂³𝜔_𝑛∇𝜔_𝑛∇𝜂d𝑦+_𝐵2𝑅𝛼_𝑛(𝑦)𝜔_𝑛+𝑙_𝑛𝜌^𝑛−1^𝑝𝜔𝜂⁴d𝑦2_𝐵2𝑅𝜂⁴||∇𝜔_𝑛||²d𝑦+𝐶𝑅²_𝐵2𝑅≤1𝜂²𝜔^2𝑛d𝑦+𝐿_𝐵2𝑅𝜔_𝑛+𝑙_𝑛𝜌^𝑛−1^𝑝𝜔𝜂⁴d𝑦2_𝐵2𝑅𝜂⁴||∇𝜔_𝑛||²d𝑦+𝐶𝑅^𝑁−2+𝐶𝑅^𝑁.(4.68) Then, for sufficiently large 𝑅>0, we have _𝐵𝑅||∇𝜔_𝑛||²d𝑦≤𝐶𝑅^𝑁,(4.69) 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→∞𝛼_𝑛(𝑦)⟶𝛼(𝑦)uniformly,∇𝜔_𝑛⇀∇𝜔in𝐿²𝐵_𝑅,𝜔fo_𝑛r⟶1≤𝜔𝑟≤in2𝐿𝑁^𝑟𝐵_𝑅𝑁−2.(4.70) Then, we have _𝐵𝑅‖‖𝜔‖‖∇𝜔∇_{𝐿∞(𝐵𝑅)}𝜑d𝑦=_𝐵𝑅𝛼(𝑦)𝜔^𝑝𝜑d𝑦,forany𝜑∈𝐶¹⁰𝐵_𝑅,=1,𝜔≥0,for𝑥∈Ω_𝑛.(4.71) Moreover, since 𝜔≢0, we have 𝜔(𝑥)>0 for all 𝑥∈𝐵_𝑅 by the strong maximum principle [43]. Taking balls larger and larger and repeating the argument for the subsequence 𝜔_𝑘 obtained at the previous step, we get a Cantor diagonal subsequence, and for simplicity, we still denote it by 𝜔_𝑘 which converges in 𝑊^1,2_l𝑜𝑐(ℝ^𝑁) to a function 𝜔∈𝑊^1,2_l𝑜𝑐(ℝ^𝑁); namely, _ℝ𝑁‖∇𝜔𝜔‖∇𝜑d𝑦_{𝐿∞(ℝ𝑁)}=_ℝ𝑁𝛼(𝑦)𝜔^𝑝𝜑d𝑦,forany𝜑∈𝐶¹⁰ℝ^𝑁,=1,𝜔>0,for𝑥∈ℝ^𝑁,(4.72) which means that (4.72), is a contradiction. Indeed, for the case 1<𝑝<(𝑁+2)/(𝑁−2) with 𝑁>2 and the case 𝑝>1 with 𝑁≤2, 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 𝑙_𝑛+‖𝑢_𝑛‖_∞ 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, 𝑢 is the nontrivial nonnegative classical periodic solution.

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

Theorem 4.4. Assume 𝛼(𝑥)∈𝐶¹(ℝ^𝑛) and 𝑁>2. If 𝑝≥(𝑁+2)/(𝑁−2) and Ω is star shaped, then there is no nontrivial and nonnegative periodic solution.

Proof. If 𝛼(𝑥,𝑡) is independent of 𝑡, 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 𝑢_𝑡 on both sides and integrating over 𝑄_𝜔 yield 12_𝑄𝜔||𝑢_𝑡||²d𝑥+𝑘2_𝑄𝜔||∇𝑢_𝑡||²d𝑥=0,(4.73) which means that 𝑢 is a steady state and satisfies the steady-state equation −Δ𝑢=𝛼(𝑥)𝑢^𝑝.(4.74) However, by [46, 47], if 𝑁>2, 𝑝≥(𝑁+2)/(𝑁−2), and Ω 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 𝑝≥(𝑁+2)/(𝑁−2), if the domain Ω is an annulus domain, then there may exist nontrivial and nonnegative periodic solution. In 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.

5. The Singular Case 𝑝=1

In this section, we consider the case 𝑝=1, in which the problem is written as𝜕𝑢𝜕𝑡−𝑘𝜕𝜕Δ𝑡𝑢=Δ𝑢+𝛼(𝑥,𝑡)𝑢,𝑥∈Ω×ℝ,(5.1) subject to𝑢(𝑥,𝑡)||_𝜕Ω=0.(5.2)

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 condition𝑢(𝑥,0)=𝑢₀(𝑥),(5.3) where 𝑢₀(𝑥)≥0 for 𝑥∈Ω and satisfies some compatibility conditions.

We have the following theorem.

Theorem 5.1. Assume that 𝑝=1. Let 𝜆₁ be the first eigenvalue of −Δ with homogeneous boundary condition on Ω.(1)If 𝐿=sup𝛼(𝑥,𝑡)<𝜆₁, 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=inf𝛼(𝑥,𝑡)>𝜆₁, 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)≥𝜆₁, then there exists a nontrivial and nonnegative periodic solution for (1.1) and (1.2). Moreover, if 𝛼≡𝜆₁, then there is a positive periodic solution for (1.1) and (1.2).

Proof. Firstly, we consider the case 𝐿<𝜆₁. We note that there exist 𝜆̃ with 𝐿<𝜆̃<𝜆₁ and domain Ω with Ω⊂Ω such that 𝜆̃ 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 𝐾𝜓 is an upper solution of the problem (5.1)–(5.3) for appropriately large 𝐾>0. Then, we have 𝑢≤𝐾𝜓. Let 𝑤(𝑥,𝑡) be the solution of the following problem: 𝜕𝑢𝜕𝑡−𝑘𝜕𝜕Δ𝑡𝑢=Δ𝑢+𝛼(𝑥,𝑡)𝑢,(𝑥,𝑡)∈Ω×ℝ⁺𝑢(𝑥,𝑡)=𝐾𝜓𝑒_−𝑡,𝑢(𝑥,0)=𝐾𝜓(𝑥),,𝑥𝑥∈∈Ω𝜕.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)<𝜆₁, which means that 𝑢(𝑥,𝑡) goes to 0 uniformly as 𝑡→∞.
Next, let us consider the case 𝑆>𝜆₁. Take Ω⊂Ω such that the first eigenvalue 𝜆̂ of −Δ with homogeneous boundary condition on Ω satisfies 𝜆₁<𝜆̂<𝑆. Let 𝜙 be the first eigenfunction of Ω. Set 𝑤=𝑔(𝑡)𝜙,(5.6) where 𝑔(𝑡) satisfies 𝑔′(𝑡)=𝑆−𝜆̂𝑔𝑔((𝑡0))>=001,.+𝑡𝑘>𝜆̂𝑔0,(𝑡),𝑡>0,(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, 𝑢=𝑤0,,𝑥𝑥∈∈ΩΩ/,Ω(5.8) 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→∞.
However if 𝛼(𝑥,𝑡) is independent of 𝑡 and 𝛼(𝑥)≥𝜆₁, then there exists a domain Ω′⊂Ω, such that one can find a function 𝜙>0 on Ω′ which satisfies −Δ𝜙=𝜙𝛼||(𝑥)𝜙,𝑥∈Ω,_𝜕Ω′=0.(5.9) Then, 𝜙 is the periodic solution. Specially, when 𝛼(𝑥)=𝜆₁, 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: 𝑢𝑢′((𝑡𝑡))==𝑢𝛼((𝑡𝑡)+𝑢𝜔(𝑡)),,(5.10) is a periodic solution to our problem. However, a simple calculation indicates that the above problem has periodic solutions provided that ∫^𝜔0𝛼(𝑠)d𝑠=0, which does not coincide with our assumption 𝛼(𝑥,𝑡)>0.

Acknowledgments

This work is partially supported by the National Natural Science Foundation of China, partially supported by China Postdoctoral Science Foundation, partially supported by the Specialized Research Fund for the Doctoral Program of High Education of China, and partially supported by the Fundamental Research Funds for the Central Universities. The authors would also like to express their sincere thanks to the referees for their value suggestions for the revision of the paper which contributed greatly to this work.