Periodic solution of a quasilinear parabolic equation with nonlocal terms and Neumann boundary conditions

  • Raad Awad Hameed1, 2,

    Affiliated with

    • Boying Wu1 and

      Affiliated with

      • Jiebao Sun1Email author

        Affiliated with

        Boundary Value Problems20132013:34

        DOI: 10.1186/1687-2770-2013-34

        Received: 26 November 2012

        Accepted: 2 February 2013

        Published: 21 February 2013

        Abstract

        In this article, we study the periodic solution of a quasilinear parabolic equation with nonlocal terms and Neumann boundary conditions. By using the theory of Leray-Schauder degree, we obtain the existence of a nontrivial nonnegative time periodic solution.

        1 Introduction

        The aim of this work is to consider the following periodic problem for a quasilinear parabolic equation:
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_Equ1_HTML.gif
        (1.1)
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_Equ2_HTML.gif
        (1.2)
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_Equ3_HTML.gif
        (1.3)

        where Ω is a bounded domain in R n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_IEq1_HTML.gif with smooth boundary Ω, n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_IEq2_HTML.gif denotes the outward normal derivative on Ω, Q T = Ω × ( 0 , T ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_IEq3_HTML.gif, a i j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_IEq4_HTML.gif satisfies some suitable smoothness and structure conditions. This model can be used to describe the models for some interesting phenomena in mathematical biology, fisheries and wildlife management. The function u ( x , t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_IEq5_HTML.gif gives the number of individuals (per unit area) of the species at position x and time t, where x represents the spatial variable and t represents the time. The term D i ( a i j ( x , t , u ) D j u ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_IEq6_HTML.gif models a tendency to avoid high density in the habitat, m Φ [ u ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_IEq7_HTML.gif describes the ways in which a given population grows and shrinks over time, as controlled by birth, death, emigration or immigration, and the Neumann boundary condition models the trend of the biology population who survive on the boundary.

        In last decades, linear parabolic equations with nonlocal terms have been investigated by numerous researchers [14]. A typical model was submitted by Allegretto and Nistri [1] and they proposed the following equation:
        u t Δ u = f ( x , t , m , Φ [ u ] , u ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_Equa_HTML.gif
        with the Dirichlet boundary conditions. Also, according to the actual needs, many authors divert attention to nonlinear diffusion equations with nonlocal terms such as the porous equation [5, 6] with a typical form
        u t = Δ u m + ( m Φ [ u ] ) u , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_Equ4_HTML.gif
        (1.4)
        and the p-Laplacian equation [7] with a typical form
        u t = div ( | u | p 2 u ) + ( m Φ [ u ] ) u . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_Equ5_HTML.gif
        (1.5)
        Equation (1.4) is degenerate if m > 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_IEq8_HTML.gif and singular if 0 < m < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_IEq9_HTML.gif. Equation (1.5) is degenerate if p > 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_IEq10_HTML.gif and singular if 1 < p < 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_IEq11_HTML.gif. Only the cases m > 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_IEq8_HTML.gif and p > 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_IEq10_HTML.gif are considered with a few exceptions. All these equations are considered with the Dirichlet boundary condition which describes that the boundary is lethal to the species. Moreover, the methods in these papers are all based on the theory of Leray-Schauder degree. However, results on the quasilinear periodic parabolic equations with nonlocal terms and Neumann boundary conditions are few. In a recent paper [8], Wang and Yin considered the following periodic Neumann boundary value problem:
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_Equb_HTML.gif

        where m > 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_IEq12_HTML.gif. By the parabolic regularized method and the theory of Leray-Schauder degree, they established the existence of nontrivial nonnegative periodic solutions.

        Inspired by the work of [8], we consider the periodic solutions of the Neumann boundary value problem of a quasilinear parabolic equation with nonlocal terms. Compared with the Dirichlet boundary condition, the Neumann boundary condition causes an additional difficulty in establishing some a priori estimates. On the other hand, different from the cases of the Dirichlet boundary condition, an auxiliary problem for (1.1)-(1.3) is considered for using the theory of Leray-Schauder degree. We prove that this problem (1.1)-(1.3) admits a nontrivial nonnegative periodic solution, that is, the following theorem.

        Theorem 1 If assumptions (A1), (A2), (A3) hold, then problem (1.1)-(3.3) admits a nontrivial nonnegative periodic solution u L 2 ( 0 , T ; H 1 ( Ω ) ) C T ( Q T ¯ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_IEq13_HTML.gif.

        The article is organized in the following way. In Section 2, we give some necessary preliminaries including the auxiliary problem. In Section 3, we establish some necessary a priori estimations of the solutions of the auxiliary problem. Then we show the proof of the main result of this paper.

        2 Preliminaries

        In the paper, we assume that

        (A1) a i j ( , , u ) = a j i ( , , u ) C T ( Q ¯ T ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_IEq14_HTML.gif and there exist two constants 0 < λ γ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_IEq15_HTML.gif such that
        λ | ξ | 2 a i j ( x , t , u ) ξ i ξ j γ | ξ | 2 for all  ( x , t ) Q T , u R +  and  ξ R n , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_Equc_HTML.gif

        where http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_IEq16_HTML.gif is the class of functions which are continuous in Ω ¯ × R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_IEq17_HTML.gif and T-periodic with respect to t. Furthermore, a i j ( , , u ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_IEq18_HTML.gif is continuous with respect to u.

        (A2) Φ [ ] : L + 2 ( Ω ) R + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_IEq19_HTML.gif is a bounded continuous functional satisfying
        Φ [ u ] C u L 2 ( Ω ) 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_Equd_HTML.gif

        where C is a positive constant independent of u, R + = [ 0 , + ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_IEq20_HTML.gif, L + 2 ( Ω ) = { u L 2 ( Ω ) | u 0 , a.e. in  Ω } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_IEq21_HTML.gif.

        (A3) m ( x , t ) C T ( Q ¯ T ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_IEq22_HTML.gif and satisfies that
        essinf x Ω 1 T 0 T m ( x , t ) d t > γ λ 1 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_Eque_HTML.gif

        where λ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_IEq23_HTML.gif is the first eigenvalue of the Laplacian equation on ω with zero boundary and ϕ 1 ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_IEq24_HTML.gif is the corresponding eigenfunction.

        Since the regularity follows from a quite standard approach, we focus on the discussion of weak solutions in the following sense.

        Definition 1 A function u is said to be a weak solution of problem (1.1)-(1.3), if u L 2 ( 0 , T ; H 1 ( T ) ) C T ( Q ¯ T ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_IEq25_HTML.gif and satisfies
        Q T ( u φ t + a i j ( x , t , u ) D i u D j φ ( m Φ [ u ] ) u φ ) d x d t = 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_Equ6_HTML.gif
        (2.1)

        for any φ C 1 ( Q ¯ T ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_IEq26_HTML.gif with φ ( x , 0 ) = φ ( x , T ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_IEq27_HTML.gif.

        In order to use the theory of Leray-Schauder degree, we introduce a map by considering the following auxiliary problem:
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_Equ7_HTML.gif
        (2.2)
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_Equ8_HTML.gif
        (2.3)
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_Equ9_HTML.gif
        (2.4)

        where ε is a sufficiently small positive constant, τ [ 0 , 1 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_IEq28_HTML.gif is a parameter and f C T ( Q T ¯ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_IEq29_HTML.gif. Then we can define a map u ε = G ( τ , f ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_IEq30_HTML.gif with G : [ 0 , 1 ] × C T ( Q ¯ T ) C T ( Q ¯ T ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_IEq31_HTML.gif. Applying classical estimates (see [9]), we can see that u ε L ( Q T ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_IEq32_HTML.gif is bounded by f L ( Q T ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_IEq33_HTML.gif, and u ε http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_IEq34_HTML.gif is Hölder continuous in Q T http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_IEq35_HTML.gif. Then, by the Arzela-Ascoli theorem, the map G is compact. So, the map G is a compact continuous map. Let f ( u ε ) = ( m Φ [ u ε ] ) u ε + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_IEq36_HTML.gif, where u ε + = max { u ε , 0 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_IEq37_HTML.gif, we can see that the nonnegative solution u ε http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_IEq34_HTML.gif of problem (2.2)-(2.4) is also a nonnegative fixed point of the map u ε = G ( 1 , ( m Φ [ u ε ] ) u ε + ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_IEq38_HTML.gif. So, we will study the existence of nonnegative fixed points of the map u = G ( 1 , ( m Φ [ u ε ] ) u ε + ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_IEq39_HTML.gif instead of a nonnegative solution of problem (2.2)-(2.4). And the desired solution u of (1.1)-(1.3) would be obtained as a limit point of u ε http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_IEq34_HTML.gif.

        3 The proof of the main result

        First, by the same method as in [4], we can obtain the nonnegativity of the solutions of problem (2.2)-(2.4).

        Lemma 1 If a nontrivial function u ε C T ( Q ¯ T ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_IEq40_HTML.gif solves u ε = G ( 1 , ( m Φ [ u ε ] ) u ε + ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_IEq38_HTML.gif, then
        u ε ( x , t ) > 0 for all ( x , t ) Q ¯ T . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_Equf_HTML.gif

        In the following, we will show some a priori estimates for the upper bound of a nonnegative periodic solution of problem (2.2)-(2.4). Here and below, we denote by p http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_IEq41_HTML.gif ( 1 p http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_IEq42_HTML.gif) the L p ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_IEq43_HTML.gif norm.

        Lemma 2 For λ [ 0 , 1 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_IEq44_HTML.gif, let u ( x , t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_IEq45_HTML.gif be a nonnegative periodic solution which solves u ε = G ( 1 , λ ( m Φ [ u ε ] ) u ε + ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_IEq46_HTML.gif, then there exists a constant K independent of λ, ε such that
        u ( t ) < K , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_Equ10_HTML.gif
        (3.1)

        where u ( t ) = u ( , t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_IEq47_HTML.gif.

        Proof Multiplying (2.2) by u m + 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_IEq48_HTML.gif ( m 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_IEq49_HTML.gif) and integrating over Ω, we have
        1 m + 2 d d t u ( t ) m + 2 m + 2 + 4 ( m + 1 ) ( m + 2 ) 2 ( | u ( t ) | m 2 u ( t ) ) 2 2 m ( x , t ) L ( Ω × ( 0 , T ) ) u ( t ) m + 2 m + 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_Equg_HTML.gif
        and hence
        d d t u ( t ) m + 2 m + 2 + C 1 ( | u ( t ) | m 2 u ( t ) ) 2 2 C 2 ( m + 2 ) u ( t ) m + 2 m + 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_Equ11_HTML.gif
        (3.2)
        where C i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_IEq50_HTML.gif ( i = 1 , 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_IEq51_HTML.gif) are positive constants independent of u and m. Assume that u ( t ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_IEq52_HTML.gif and set
        u k ( t ) = | u ( t ) | m k 2 u ( t ) , m k = 2 k 2 ( k = 1 , 2 , ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_Equh_HTML.gif
        then m k = 2 m k 1 + 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_IEq53_HTML.gif. For convenience, we denote by C a positive constant independent of k and m, which may take different values. From (3.2), we obtain
        d d t u k ( t ) 2 2 + C u k ( t ) 2 2 C ( m + 2 ) u k ( t ) 2 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_Equ12_HTML.gif
        (3.3)
        By using the Gagliardo-Nirenberg inequality, we have
        u k ( t ) 2 C u k ( t ) 2 θ u k ( t ) 1 1 θ , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_Equ13_HTML.gif
        (3.4)
        with
        θ = N N + 2 ( 0 , 1 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_Equi_HTML.gif
        By inequalities (3.3), (3.4) and the fact that u k ( t ) 1 = u k 1 ( t ) 2 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_IEq54_HTML.gif, we obtain the following differential inequality:
        d d t u k ( t ) 2 2 C u k ( t ) 2 2 θ u k ( t ) 1 2 ( θ 1 ) θ + C ( m k + 2 ) u k ( t ) 2 2 C u k ( t ) 2 2 θ u k 1 ( t ) 2 4 ( θ 1 ) θ + C ( m k + 2 ) u k ( t ) 2 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_Equj_HTML.gif
        Let
        λ k = max { 1 , sup t u k ( t ) 2 } , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_Equk_HTML.gif
        we have
        d d t u k ( t ) 2 2 u k ( t ) 2 2 ( m k + 1 ) m k + 2 { C u k ( t ) 2 2 θ 2 ( m k + 1 ) m k + 2 λ k 1 4 ( θ 1 ) θ + C ( m k + 2 ) u k ( t ) 2 2 m k + 2 } . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_Equ14_HTML.gif
        (3.5)
        By Young’s inequality,
        a b ϵ a p + ϵ q p b q , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_Equl_HTML.gif
        where p > 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_IEq55_HTML.gif, q > 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_IEq56_HTML.gif, a > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_IEq57_HTML.gif, b > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_IEq58_HTML.gif, ϵ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_IEq59_HTML.gif and 1 p + 1 q = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_IEq60_HTML.gif. Set
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_Equm_HTML.gif
        then we obtain
        ( m k + 2 ) u k ( t ) 2 2 m k + 2 1 2 u k ( t ) 2 2 θ 2 ( m k + 1 ) m k + 2 λ k 1 4 ( θ 1 ) θ + C ( m k + 2 ) l k l k 1 λ k 1 4 1 θ θ 1 l k 1 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_Equ15_HTML.gif
        (3.6)
        Here we have used the fact that p = l k > r > 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_IEq61_HTML.gif for some r independent of k. In fact, it is easy to verify that
        lim k l k = + . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_Equn_HTML.gif
        Denoting
        a k = l k l k 1 , b k = 1 θ θ 4 l k 1 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_Equo_HTML.gif
        and combining (3.5) with (3.6), we have
        d d t u k ( t ) 2 2 u k ( t ) 2 2 ( m k + 1 ) m k + 2 { C u k ( t ) 2 2 θ 2 ( m k + 1 ) m k + 2 λ k 1 4 ( θ 1 ) θ + C ( m k + 2 ) a k λ k 1 b k } . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_Equ16_HTML.gif
        (3.7)
        Then
        ( m k + 2 ) d d t u k ( t ) 2 2 m k + 2 C u k ( t ) 2 2 θ 2 ( m k + 1 ) m k + 2 λ k 1 4 ( θ 1 ) θ + C ( m k + 2 ) a k λ k 1 b k . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_Equ17_HTML.gif
        (3.8)
        From the periodicity of u k ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_IEq62_HTML.gif, we know that there exists t 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_IEq63_HTML.gif at which u k ( t ) 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_IEq64_HTML.gif reaches its maximum and thus the left-hand side of (3.8) vanishes. Then we obtain
        u k ( t ) 2 { C [ ( m k + 2 ) a k λ k 1 b k + 4 ( 1 θ ) θ ] } 1 α k , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_Equp_HTML.gif
        where
        α k = 2 θ 2 ( m k + 1 ) m k + 2 = 2 l k m k + 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_Equq_HTML.gif
        Therefore, we conclude that
        u k ( t ) 2 { C ( m k + 2 ) a k λ k 1 b k + 4 ( 1 θ ) θ } 1 α k = { C ( m k + 2 ) a k } m k + 2 2 l k λ k 1 2 ( 1 θ ) ( m k + 2 ) ( l k 1 ) θ . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_Equr_HTML.gif
        Since m k + 2 ( l k 1 ) θ = 1 1 θ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_IEq65_HTML.gif and m k + 2 2 l k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_IEq66_HTML.gif and α k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_IEq67_HTML.gif are bounded, we get
        u k ( t ) 2 C A k λ k 1 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_Equs_HTML.gif
        where A > 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_IEq68_HTML.gif is a positive constant independent of k. Then we have
        ln u k ( t ) 2 ln λ k ln C + k ln A + 2 ln λ k 1 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_Equt_HTML.gif
        thus
        ln u k ( t ) 2 ln C i = 0 k 2 2 i + 2 k 1 ln λ 1 + ln A ( j = 0 k 2 ( k j ) 2 j ) ( 2 k 1 1 ) ln C + 2 k 1 ln λ 1 + f ( k ) ln A , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_Equu_HTML.gif
        or
        u k ( t ) m k + 2 { C 2 k 1 1 λ 1 2 k 1 A f ( k ) } 2 m k + 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_Equv_HTML.gif
        where
        f ( k ) = k 2 ( k + 1 ) 2 k 1 + 2 k + 1 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_Equw_HTML.gif
        Letting k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_IEq69_HTML.gif, we obtain
        u ( t ) C λ 1 C ( max { 1 , sup t u ( t ) 2 } ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_Equ18_HTML.gif
        (3.9)
        Now, we just need to show the estimate of u ( t ) 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_IEq70_HTML.gif. Multiplying (2.2) by u and integrating by parts over Q T http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_IEq35_HTML.gif, by the periodicity of u, we have
        Q T λ 1 | u | 2 + ε u 2 d t d x Q T λ u 2 ( m ϕ [ u ] ) d t d x , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_Equx_HTML.gif
        which implies that
        Q T u 2 ( m ϕ [ u ] ) d t d x 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_Equy_HTML.gif
        Let M = max ( x , t ) Q ¯ T m ( x , t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_IEq71_HTML.gif, by assumption (A2), we have
        0 Q T u 2 ( m ϕ [ u ] ) d t d x M Q T u 2 d t d x Q T u 2 ϕ [ u ] d t d x M Q T u 2 d t d x C 0 T u 2 2 d t , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_Equz_HTML.gif
        that is,
        0 T u 2 4 d t C 0 T u 2 2 d t , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_Equaa_HTML.gif
        where C is a positive independent of λ. By Young’s inequality, we have
        0 T u 2 2 d t 0 T 1 4 ε 2 + ε 2 u 2 4 d t . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_Equab_HTML.gif
        Combining with the above inequality, we have
        u k ( t ) 2 C , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_Equac_HTML.gif

        which together with (3.9) implies (3.1), and thus the proof is complete. □

        Corollary 1 There exists a positive constant R independent of ε such that
        deg ( I G ( 1 , ( m Φ [ u ε ] ) u ε + ) , B R , 0 ) = 1 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_Equad_HTML.gif

        where B R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_IEq72_HTML.gif is a ball centered at the origin with radius R in L ( Q T ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_IEq73_HTML.gif.

        Proof It follows from Lemma 2 that there exists a positive constant R independent of λ, ε such that
        u G ( 1 , ( m Φ [ u ε ] ) u ε + ) , u B R , λ [ 0 , 1 ] . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_Equae_HTML.gif
        So, the degree is well defined on B R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_IEq72_HTML.gif. From the homotopy invariance of the Leray-Schauder degree and the existence and uniqueness of the solution of G ( 1 , 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_IEq74_HTML.gif, we can see that
        deg ( 1 G ( 1 , ( m Φ [ u ε ] ) u ε + ) , B R , 0 ) = deg ( 1 G ( 1 , λ ( m Φ [ u ε ] ) u ε + ) , B R , 0 ) = deg ( 1 G ( 1 , 0 ) , B R , 0 ) = 1 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_Equaf_HTML.gif

        The proof is completed. □

        Lemma 3 There exist constants r 0 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_IEq75_HTML.gif and ε > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_IEq76_HTML.gif such that for any r < r 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_IEq77_HTML.gif, ε < ε 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_IEq78_HTML.gif, u = G ( τ , ( m Φ [ u ] ) u + + ( 1 τ ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_IEq79_HTML.gif admits no nontrivial solution u satisfying
        0 < u L ( Q T ) r , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_Equag_HTML.gif

        where r is a positive constant independent of ε.

        Proof By contradiction, let u be a nontrivial fixed point of u = G ( τ , ( m Φ [ u ] ) u + + 1 τ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_IEq80_HTML.gif satisfying 0 < u L ( Q T ) r http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_IEq81_HTML.gif. For any given ϕ ( x ) C 0 ( B δ ( x 0 ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_IEq82_HTML.gif, multiplying (2.2) by ϕ 2 u http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_IEq83_HTML.gif and integrating over Q T = B δ ( x 0 ) × ( 0 , T ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_IEq84_HTML.gif, we have
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_Equ19_HTML.gif
        (3.10)
        By the periodicity of u, the first term on the left-hand side is zero. The second term on the left-hand side can be rewritten as
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_Equah_HTML.gif
        The third term of the left-hand side of equation (3.11) can be rewritten as
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_Equai_HTML.gif
        Then from (3.10), we obtain
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_Equaj_HTML.gif
        From assumption (A1), we can see that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_Equak_HTML.gif
        Since τ [ 0 , 1 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_IEq28_HTML.gif, we have
        Q T γ | D ϕ | 2 d t d x Q T ϕ 2 ( m ε Φ [ u ] ) d t d x 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_Equal_HTML.gif
        By an approaching process, we choose ϕ = ϕ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_IEq85_HTML.gif, where ϕ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_IEq86_HTML.gif is the eigenvector of the first eigenvalue λ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_IEq23_HTML.gif in (A3), and then we obtain
        0 Q T γ | D ϕ 1 | 2 d t d x Q T ϕ 1 2 ( m ε Φ [ u ] ) d t d x = Q T γ ϕ 1 ϕ 1 d t d x Q T ϕ 1 2 ( m ε Φ [ u ] ) d t d x = Q T γ λ 1 ϕ 1 2 d t d x Q T ϕ 1 2 ( m ε Φ [ u ] ) d t d x = Ω ϕ 1 2 0 T ( γ λ 1 m + ε + Φ [ u ] ) d t d x . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_Equ20_HTML.gif
        (3.11)
        Thus, there exists x 0 Ω http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_IEq87_HTML.gif such that 0 T ( λ 1 m ( x 0 , t ) + Φ [ u ( x 0 , t ) ] ) d t 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_IEq88_HTML.gif, then
        1 T 0 T m ( x 0 , t ) d t γ λ 1 + ε + 1 T 0 T Φ [ u ( x 0 , t ) ] d t . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_Equam_HTML.gif
        From assumption (A2), we can see that
        1 T 0 T m ( x 0 , t ) d t γ μ 1 + ε + C r 2 | Ω | http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_Equan_HTML.gif

        holds for any sufficiently small r and ε, which is a contradiction to assumption (A3). The proof is complete. □

        Corollary 2 There exists a small positive constant r < R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_IEq89_HTML.gif which is independent of ε, τ such that
        deg ( I G ( 1 , ( m Φ [ u ε ] ) u ε + ) , B r , 0 ) = 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_Equao_HTML.gif

        where B r http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_IEq90_HTML.gif is a ball centered at the origin with radius r in L ( Q T ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_IEq73_HTML.gif.

        Proof Similar to Lemma 3, we can see that there exists a positive constant 0 < r < R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_IEq91_HTML.gif independent of ε such that
        u ε G ( τ , ( m Φ [ u ε ] ) u ε + + 1 τ ) , u B r , λ [ 0 , 1 ] . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_Equap_HTML.gif
        So, the degree is well defined on B r http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_IEq90_HTML.gif. By Lemma 3.3, we can easily see that u = G ( 0 , ( m Φ [ u ] ) u + + 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_IEq92_HTML.gif admits no solution in B r http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_IEq90_HTML.gif. Then, by the homotopy invariance of the Leray-Schauder degree, we have
        deg ( I G ( 1 , ( m Φ [ u ε ] ) u ε + ) , B r , 0 ) = deg ( 1 G ( 0 , ( m Φ [ u ε ] ) u ε + + 1 ) , B r , 0 ) = 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_Equaq_HTML.gif

        The proof is completed. □

        Now, we show the proof of the main result of this paper.

        Proof of Theorem 1

        Using Corollaries 1 and 2, we have
        deg ( 1 G ( f ( ) ) , B R B r , 0 ) = 1 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_Equar_HTML.gif

        where R and r are positive constants and R > r http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_IEq93_HTML.gif. Problem (2.2)-(2.4) admits a nonnegative nontrivial solution u ε http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_IEq34_HTML.gif with r u ε R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_IEq94_HTML.gif. Combining with the regularity results [9] and a similar argument as in [10], we can prove that the limit function of u ε http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-34/MediaObjects/13661_2012_Article_288_IEq34_HTML.gif is a nonnegative nontrivial periodic solution of problem (1.1)-(1.3). □

        Declarations

        Acknowledgements

        This work is partially supported by the National Science Foundation of China (11271100, 11126222), the Fundamental Research Funds for the Central Universities (Grant No. HIT. NSRIF. 2011006), the Natural Sciences Foundation of Heilongjiang Province (QC2011C020) and also by the 985 project of Harbin Institute of Technology.

        Authors’ Affiliations

        (1)
        Department of Mathematics, Harbin Institute of Technology
        (2)
        Department of Mathematics, College of Education, University of Tikrit

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        Copyright

        © Hameed et al.; licensee Springer. 2013

        This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://​creativecommons.​org/​licenses/​by/​2.​0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.