General decay for a system of nonlinear viscoelastic wave equations with weak damping

  • Baowei Feng1Email author,

    Affiliated with

    • Yuming Qin2 and

      Affiliated with

      • Ming Zhang1

        Affiliated with

        Boundary Value Problems20122012:146

        DOI: 10.1186/1687-2770-2012-146

        Received: 19 August 2012

        Accepted: 26 November 2012

        Published: 13 December 2012

        Abstract

        In this paper, we are concerned with a system of nonlinear viscoelastic wave equations with initial and Dirichlet boundary conditions in R n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_IEq1_HTML.gif ( n = 1 , 2 , 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_IEq2_HTML.gif). Under suitable assumptions, we establish a general decay result by multiplier techniques, which extends some existing results for a single equation to the case of a coupled system.

        MSC:35L05, 35L55, 35L70.

        Keywords

        viscoelastic system general decay weak damping

        1 Introduction

        In this paper, we are concerned with a coupled system of nonlinear viscoelastic wave equations with weak damping
        { u t t Δ u + 0 t g 1 ( t τ ) Δ u ( τ ) d τ + u t = f 1 ( u , v ) , in  Ω × ( 0 , + ) , v t t Δ v + 0 t g 2 ( t τ ) Δ v ( τ ) d τ + v t = f 2 ( u , v ) , in  Ω × ( 0 , + ) , u = v = 0 , on  Ω × ( 0 , + ) , u ( , 0 ) = u 0 , u t ( , 0 ) = u 1 , v ( , 0 ) = v 0 , v t ( , 0 ) = v 1 , in  Ω , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_Equ1_HTML.gif
        (1.1)

        where Ω R n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_IEq3_HTML.gif ( n = 1 , 2 , 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_IEq2_HTML.gif) is a bounded domain with smooth boundary Ω, u and v represent the transverse displacements of waves. The functions g 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_IEq4_HTML.gif and g 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_IEq5_HTML.gif denote the kernel of a memory, f 1 ( u , v ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_IEq6_HTML.gif and f 2 ( u , v ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_IEq7_HTML.gif are the nonlinearities.

        In recent years, many mathematicians have paid their attention to the energy decay and dynamic systems of the nonlinear wave equations, hyperbolic systems and viscoelastic equations.

        Firstly, we recall some results concerning single viscoelastic wave equation. Kafini and Tatar [1] considered the following Cauchy problem:
        { u t t Δ u + 0 t g ( t s ) Δ u ( x , s ) d s = 0 , x R n , t > 0 , u ( x , 0 ) = u 0 ( x ) , u t ( x , 0 ) = u 1 ( x ) , x R n . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_Equ2_HTML.gif
        (1.2)
        They established the polynomial decay of the first-order energy of solutions for compactly supported initial data and for a not necessarily decreasing relaxation function. Later Tatar [2] studied the problem (1.2) with the Dirichlet boundary condition and showed that the decay of solutions was an arbitrary decay not necessarily at exponential or polynomial rate. Cavalcanti et al. [3] studied the following equation with Dirichlet boundary condition:
        | u t | ρ u t t Δ u Δ u t t + g Δ u γ Δ u t = 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_Equa_HTML.gif

        The authors established a global existence result for γ 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_IEq8_HTML.gif and an exponential decay of energy for γ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_IEq9_HTML.gif. They studied the interaction within the | u t | ρ u t t http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_IEq10_HTML.gif and the memory term g Δ u http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_IEq11_HTML.gif. Later on, several other results were published based on [46]. For more results on a single viscoelastic equation, we can refer to [714].

        For a coupled system, Agre and Rammaha [15] investigated the following system:
        { u t t Δ u + | u t | m 1 u t = f 1 ( u , v ) , in  Ω × ( 0 , T ) , v t t Δ v + | v t | r 1 v t = f 2 ( u , v ) , in  Ω × ( 0 , T ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_Equb_HTML.gif

        where Ω R n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_IEq3_HTML.gif ( n = 1 , 2 , 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_IEq2_HTML.gif) is a bounded domain with smooth boundary. They considered the following assumptions on f i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_IEq12_HTML.gif ( i = 1 , 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_IEq13_HTML.gif):

        (A1) Let
        F ( u , v ) = a | u + v | p + 1 + 2 b | u v | p + 1 2 , f 1 ( u , v ) = F u , f 2 ( u , v ) = F v http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_Equc_HTML.gif

        with a , b > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_IEq14_HTML.gif, p 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_IEq15_HTML.gif if n = 1 , 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_IEq16_HTML.gif and p = 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_IEq17_HTML.gif if n = 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_IEq18_HTML.gif; m , r 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_IEq19_HTML.gif.

        (A2) There exist two positive constants c 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_IEq20_HTML.gif, c 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_IEq21_HTML.gif such that for all u , v R 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_IEq22_HTML.gif, F ( u , v ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_IEq23_HTML.gif satisfies
        c 0 ( | u | p + 1 + | v | p + 1 ) F ( u , v ) c 1 ( | u | p + 1 + | v | p + 1 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_Equd_HTML.gif
        Under the assumptions (A1)-(A2), they established the global existence of weak solutions and the global existence of small weak solutions with initial and Dirichlet boundary conditions. Moreover, they also obtained the blow up of weak solutions. Mustafa [16] studied the following system:
        { u t t Δ u + 0 t g 1 ( t τ ) Δ u ( τ ) d τ + f 1 ( u , v ) = 0 , v t t Δ v + 0 t g 2 ( t τ ) Δ v ( τ ) d τ + f 2 ( u , v ) = 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_Equ3_HTML.gif
        (1.3)

        in Ω × ( 0 , + ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_IEq24_HTML.gif with initial and Dirichlet boundary conditions, proved the existence and uniqueness to the system by using the classical Faedo-Galerkin method and established a stability result by multiplier techniques. But the author considered the following different assumptions on f i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_IEq12_HTML.gif ( i = 1 , 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_IEq13_HTML.gif) from (A1)-(A2):

        ( A 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_IEq25_HTML.gif) f i : R 2 R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_IEq26_HTML.gif ( i = 1 , 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_IEq13_HTML.gif) are C 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_IEq27_HTML.gif functions and there exists a function F such that
        f 1 ( x , y ) = F x , f 2 ( x , y ) = F y , F 0 , x f 1 ( x , y ) + y f 2 ( x , y ) F ( x , y ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_Eque_HTML.gif
        ( A 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_IEq28_HTML.gif)
        | f i x ( x , y ) | + | f i y ( x , y ) | d ( 1 + | x | β i 1 1 + | y | β i 2 1 ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_Equf_HTML.gif

        for all ( x , y ) R 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_IEq29_HTML.gif, where the constant d > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_IEq30_HTML.gif and β i j 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_IEq31_HTML.gif, ( n 2 ) β i j n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_IEq32_HTML.gif for i , j = 1 , 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_IEq33_HTML.gif.

        Han and Wang [17] considered the following coupled nonlinear viscoelastic wave equations with weak damping:
        { u t t Δ u + 0 t g 1 ( t τ ) Δ u ( τ ) d τ + | u t | m 1 u t = f 1 ( u , v ) , in  Ω × ( 0 , T ) , v t t Δ v + 0 t g 2 ( t τ ) Δ v ( τ ) d τ + | v t | r 1 v t = f 2 ( u , v ) , in  Ω × ( 0 , T ) , u = v = 0 , on  Ω × ( 0 , T ) , u ( , 0 ) = u 0 , u t ( , 0 ) = u 1 , v ( , 0 ) = v 0 , v t ( , 0 ) = v 1 , in  Ω , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_Equ4_HTML.gif
        (1.4)
        where Ω R n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_IEq3_HTML.gif is a bounded domain with smooth boundary Ω. Under the assumptions (A1)-(A2) on f i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_IEq12_HTML.gif ( i = 1 , 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_IEq13_HTML.gif), the initial data and the parameters in the equations, they established the local existence, global existence uniqueness and finite time blow up properties. When the weak damping terms | u t | m 1 u t http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_IEq34_HTML.gif, | v t | r 1 v t http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_IEq35_HTML.gif were replaced by the strong damping terms Δ u t http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_IEq36_HTML.gif, Δ v t http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_IEq37_HTML.gif, Liang and Gao [18] showed that under certain assumption on initial data in the stable set, the decay rate of the solution energy is exponential when they take
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_Equg_HTML.gif

        a , b > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_IEq14_HTML.gif and p > 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_IEq38_HTML.gif if n = 1 , 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_IEq16_HTML.gif, 1 < p 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_IEq39_HTML.gif if n = 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_IEq18_HTML.gif. Moreover, they obtained that the solutions with positive initial energy blow up in a finite time for certain initial data in the unstable set. For more results on coupled viscoelastic equations, we can refer to [1921].

        If we take m = r = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_IEq40_HTML.gif in (1.4), the system will be transformed into (1.1). To the best of our knowledge, there is no result on general energy decay for the viscoelastic problem (1.1). Motivated by [16, 17], in this paper, we shall establish the general energy decay for the problem (1.1) by multiplier techniques, which extends some existing results for a single equation to the case of a coupled system. The rest of our paper is organized as follows. In Section 2, we give some preparations for our consideration and our main result. The statement and the proof of our main result will be given in Section 3.

        For the reader’s convenience, we denote the norm and the scalar product in L 2 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_IEq41_HTML.gif by http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_IEq42_HTML.gif and ( , ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_IEq43_HTML.gif, respectively. C 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_IEq44_HTML.gif denotes a general constant, which may be different in different estimates.

        2 Preliminaries and main result

        To state our main result, in addition to (A1)-(A2), we need the following assumption.

        (A3) g i : R + R + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_IEq45_HTML.gif, i = 1 , 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_IEq13_HTML.gif, are differentiable functions such that
        g i ( 0 ) > 0 , 1 0 + g i ( s ) d s = l i > 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_Equh_HTML.gif
        and there exist nonincreasing functions ξ 1 , ξ 2 : R + R + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_IEq46_HTML.gif satisfying
        g i ( t ) ξ i ( t ) g i ( t ) , t 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_Equi_HTML.gif
        Now, we define the energy functional
        E ( t ) = 1 2 Ω ( u t 2 + ( 1 0 t g 1 ( s ) d s ) | u | 2 ) d x + 1 2 ( g 1 u ) ( t ) + 1 2 ( g 2 v ) ( t ) + 1 2 Ω ( v t 2 + ( 1 0 t g 2 ( s ) d s ) | v | 2 ) d x Ω F ( u , v ) d x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_Equ5_HTML.gif
        (2.1)
        and the functional
        D ( t ) = ( 1 0 t g 1 ( s ) d s ) u ( t ) 2 + ( 1 0 t g 2 ( s ) d s ) v ( t ) 2 + 2 [ ( g 1 u ) ( t ) + ( g 2 v ) ( t ) ] 4 Ω F ( u ( t ) , v ( t ) ) d x , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_Equ6_HTML.gif
        (2.2)
        where
        ( g y ) ( t ) = 0 t g ( t s ) y ( t ) y ( s ) 2 d s . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_Equj_HTML.gif

        The existence of a global solution to the system (1.1) is established in [17] as follows.

        Proposition [17]

        Let (A1)-(A3) hold. Assume that D ( 0 ) = u 0 2 + v 0 2 4 Ω F ( u 0 , v 0 ) d x > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_IEq47_HTML.gif, 2 p C 0 l ( E ( 0 ) l ) p 1 2 < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_IEq48_HTML.gif and that ( u 0 , u 1 ) H 0 1 ( Ω ) × L 2 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_IEq49_HTML.gif, ( v 0 , v 1 ) H 0 1 ( Ω ) × L 2 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_IEq50_HTML.gif, where C 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_IEq51_HTML.gif is a computable constant and l = min { l 1 , l 2 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_IEq52_HTML.gif. Then the problem (1.1) has a unique global solution ( u ( t ) , v ( t ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_IEq53_HTML.gif satisfying
        ( u ( t ) , u t ( t ) ) C ( R + ; H 0 1 ( Ω ) × L 2 ( Ω ) ) , ( v ( t ) , v t ( t ) ) C ( R + ; H 0 1 ( Ω ) × L 2 ( Ω ) ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_Equk_HTML.gif

        We are now ready to state our main result.

        Theorem 2.1 Let (A1)-(A3) hold. Assume that D ( 0 ) = u 0 2 + v 0 2 4 Ω F ( u 0 , v 0 ) d x > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_IEq47_HTML.gif, 2 p C 0 l ( E ( 0 ) l ) p 1 2 < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_IEq48_HTML.gif and that ( u 0 , u 1 ) H 0 1 ( Ω ) × L 2 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_IEq49_HTML.gif, ( v 0 , v 1 ) H 0 1 ( Ω ) × L 2 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_IEq50_HTML.gif, where C 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_IEq51_HTML.gif is a computable constant and l = min { l 1 , l 2 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_IEq52_HTML.gif. Then there exist constants C , η > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_IEq54_HTML.gif such that, for t large, the solution of (1.1) satisfies
        E ( t ) C e η 0 t ξ ( s ) d s , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_Equ7_HTML.gif
        (2.3)
        where
        ξ ( t ) = min { ξ 1 ( t ) , ξ 2 ( t ) } , t 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_Equ8_HTML.gif
        (2.4)

        3 Proof of Theorem 2.1

        In this section, we carry out the proof of Theorem 2.1. Firstly, we will estimate several lemmas.

        Lemma 3.1 Let u ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_IEq55_HTML.gif, v ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_IEq56_HTML.gif be the solution of (1.1). Then the following energy estimate holds for any t 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_IEq57_HTML.gif:
        E ( t ) = ( u t 2 + v t 2 ) + 1 2 [ ( g 1 u ) + ( g 2 v ) ] 1 2 [ g 1 ( t ) u ( t ) 2 + g 2 ( t ) v ( t ) 2 ] 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_Equ9_HTML.gif
        (3.1)

        Proof Multiplying the first equation of (1.1) by u t http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_IEq58_HTML.gif and the second equation by v t http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_IEq59_HTML.gif, respectively, integrating the results over Ω, performing integration by parts and noting that F t ( u , v ) = f 1 ( u , v ) u t + f 2 ( u , v ) v t http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_IEq60_HTML.gif, we can easily get (3.1). The proof is complete. □

        Lemma 3.2 Under the assumption (A3), the following hold:
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_Equ10_HTML.gif
        (3.2)
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_Equ11_HTML.gif
        (3.3)
        Proof Using Hölder’s inequality, we get
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_Equl_HTML.gif

        On the other hand, we repeat the above proof with g http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_IEq61_HTML.gif, instead of g, we can get (3.3). The proof is now complete. □

        Lemma 3.3 Let (A1)-(A3) hold and u ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_IEq55_HTML.gif, v ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_IEq56_HTML.gif be the solution of (1.1). Then the functional I ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_IEq62_HTML.gif defined by
        I ( t ) : = Ω ( u u t + v v t ) d x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_Equm_HTML.gif
        satisfies
        I ( t ) l 1 2 u ( t ) 2 l 2 2 v ( t ) 2 + ( 1 + 1 4 δ ) ( u t 2 + v t 2 ) + C 1 δ ( g 1 u ) + C 1 δ ( g 2 v ) + C 1 Ω F ( u , v ) d x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_Equ12_HTML.gif
        (3.4)

        for all δ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_IEq63_HTML.gif.

        Proof By (1.1), a direct differentiation gives
        I ( t ) = u t 2 u 2 + Ω u 0 t g 1 ( t τ ) u ( τ ) d τ d x Ω u t u d x + Ω f 1 u d x + v t 2 v 2 + Ω v 0 t g 2 ( t τ ) v ( τ ) d τ d x Ω v t v d x + Ω f 2 v d x . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_Equ13_HTML.gif
        (3.5)
        From the assumptions (A1)-(A2), we derive
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_Equn_HTML.gif
        and
        f 1 u + f 2 v = a ( p + 1 ) | u + v | p + 1 + b ( p + 1 ) | u v | p + 1 2 C 1 F ( u , v ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_Equ14_HTML.gif
        (3.6)
        By Young’s inequality and (3.2), we deduce for any δ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_IEq63_HTML.gif
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_Equ15_HTML.gif
        (3.7)
        Similarly, we have
        Ω v 0 t g 2 ( t τ ) v ( τ ) d τ d x v 2 0 t g 2 ( τ ) d τ + δ v 2 + C 1 4 δ ( g 2 v ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_Equ16_HTML.gif
        (3.8)
        Using Young’s inequality and Poincaré’s inequality, we obtain for any δ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_IEq63_HTML.gif
        Ω u u t d x δ u 2 + 1 4 δ u t 2 δ λ 2 u 2 + 1 4 δ u t 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_Equ17_HTML.gif
        (3.9)
        where λ is the first eigenvalue of −Δ with the Dirichlet boundary condition. Similarly,
        Ω v v t d x δ v 2 + 1 4 δ v t 2 δ λ 2 v 2 + 1 4 δ v t 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_Equo_HTML.gif
        which together with (3.5)-(3.9) gives
        I ( t ) ( l 1 δ δ λ 2 ) u 2 ( l 2 δ δ λ 2 ) v 2 + ( 1 + 1 4 δ ) ( u t 2 + v t 2 ) + C 1 4 δ ( g 1 u ) + C 1 4 δ ( g 2 v ) + C 1 Ω F ( u , v ) d x . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_Equ18_HTML.gif
        (3.10)
        Now, we choose δ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_IEq63_HTML.gif so small that
        l 1 δ δ λ 2 l 1 2 , l 2 δ δ λ 2 l 2 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_Equp_HTML.gif

        which together with (3.10) gives (3.4). The proof is complete. □

        Lemma 3.4 Let (A1)-(A3) hold and u ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_IEq55_HTML.gif, v ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_IEq56_HTML.gif be the solution of (1.1). Then the functional J ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_IEq64_HTML.gif defined by
        J ( t ) = J 1 ( t ) + J 2 ( t ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_Equq_HTML.gif
        with
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_Equr_HTML.gif
        satisfies
        J ( t ) ( 0 t g 1 ( τ ) 2 δ ) u t 2 + δ C 1 u 2 + C 1 δ ( g 1 u ) C 1 δ ( g 1 u ) ( 0 t g 2 ( τ ) 2 δ ) v t 2 + δ C 1 v 2 + C 1 δ ( g 2 v ) C 1 δ ( g 2 v ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_Equ19_HTML.gif
        (3.11)
        Proof A direct differentiation for J 1 ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_IEq65_HTML.gif yields
        J 1 ( t ) = Ω u t t 0 t g 1 ( t τ ) ( u ( t ) u ( τ ) ) d τ Ω u t 0 t g 1 ( t τ ) ( u ( t ) u ( τ ) ) d τ d x ( 0 t g 1 ( τ ) d τ ) Ω u t 2 d x . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_Equ20_HTML.gif
        (3.12)
        Using the first equation of (1.1) and integrating by parts, we obtain
        J 1 ( t ) = ( 1 0 t g 1 ( τ ) d τ ) Ω u 0 t g 1 ( t τ ) ( u ( t ) u ( τ ) ) d τ d x + Ω ( 0 t g 1 ( t τ ) | u ( t ) u ( τ ) | d τ ) 2 d x + Ω u t 0 t g 1 ( t τ ) ( u ( t ) u ( τ ) ) d τ d x Ω f 1 ( u , v ) 0 t g 1 ( t τ ) ( u ( t ) u ( τ ) ) d τ d x Ω u t 0 t g 1 ( t τ ) ( u ( t ) u ( τ ) ) d τ d x ( 0 t g 1 ( τ ) d τ ) Ω u t 2 d x . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_Equ21_HTML.gif
        (3.13)
        From Young’s inequality, Poincaré’s inequality and Lemma 3.2, we derive
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_Equ22_HTML.gif
        (3.14)
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_Equ23_HTML.gif
        (3.15)
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_Equ24_HTML.gif
        (3.16)
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_Equ25_HTML.gif
        (3.17)
        Now, we estimate the first term on the right-hand side of (3.17). Using the assumptions (A1)-(A2) and Young’s inequality, we arrive at
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_Equ26_HTML.gif
        (3.18)
        where we used the embedding H 0 1 ( Ω ) L s ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_IEq66_HTML.gif for 2 s 2 n / ( n 2 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_IEq67_HTML.gif if n = 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_IEq18_HTML.gif or s 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_IEq68_HTML.gif if n = 1 , 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_IEq69_HTML.gif and the fact 1 2 ( u t 2 + v t 2 ) + 1 4 l 1 u 2 + 1 4 l 2 v 2 2 E ( 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_IEq70_HTML.gif proved in Lemma 5.1 in [17]. Combining (3.13)-(3.18), we get
        J 1 ( t ) ( 0 t g 1 ( τ ) d τ 2 δ ) u t 2 + δ C 1 u 2 + δ C 1 v 2 + C 1 δ ( g 1 u ) C 1 δ ( g 1 u ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_Equ27_HTML.gif
        (3.19)
        The same estimate to J 2 ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_IEq71_HTML.gif, we can derive
        J 2 ( t ) ( 0 t g 2 ( τ ) d τ 2 δ ) v t 2 + δ C 1 u 2 + δ C 1 v 2 + C 1 δ ( g 2 v ) C 1 δ ( g 2 v ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_Equs_HTML.gif

        which together with (3.19) gives (3.11). The proof is now complete. □

        Proof of Theorem 2.1 For N 1 , N 2 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_IEq72_HTML.gif, we define the functional K http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_IEq73_HTML.gif by
        K : = N 1 E ( t ) + N 2 J ( t ) + I ( t ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_Equt_HTML.gif
        and let
        g 0 = min { 0 t 0 g 1 ( s ) d s , 0 t 0 g 2 ( s ) d s } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_Equu_HTML.gif

        for some fixed t 0 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_IEq74_HTML.gif.

        Using Lemma 3.1 and Lemmas 3.3-3.4, a direct differentiation gives
        K ( t ) ( l 2 N 2 δ C 1 ) ( u 2 + v 2 ) + ( C 1 δ + N 2 C 1 δ ) [ ( g 1 u ) + ( g 2 v ) ] ( N 1 + N 2 2 δ 1 1 4 δ ) ( u t 2 + v t 2 ) + C 1 Ω F ( u , v ) d x + ( N 1 2 N 2 C 1 δ ) [ ( g 1 u ) + ( g 2 v ) ] , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_Equ28_HTML.gif
        (3.20)

        where l = min { l 1 , l 2 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_IEq52_HTML.gif.

        Now, we choose δ = 1 4 C 1 N 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_IEq75_HTML.gif and N 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_IEq76_HTML.gif, N 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_IEq77_HTML.gif large enough so that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_Equ29_HTML.gif
        (3.21)
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_Equ30_HTML.gif
        (3.22)
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_Equ31_HTML.gif
        (3.23)
        Inserting (3.21)-(3.23) into (3.20), we have
        K ( t ) c 1 ( u 2 + v 2 ) c 2 ( u t 2 + v t 2 ) + c 3 [ ( g 1 u ) + ( g 2 v ) ] + ( 4 C 1 2 N 2 l + 4 C 1 2 N 2 l ) [ ( g 1 u ) + ( g 2 v ) ] + C 1 Ω F ( u , v ) d x . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_Equ32_HTML.gif
        (3.24)
        Therefore, for two positive constants ω and C, we obtain
        K ( t ) ω E ( t ) + C [ ( g 1 u ) + ( g 2 v ) ] , for all  t t 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_Equ33_HTML.gif
        (3.25)
        On the other hand, we choose N 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_IEq76_HTML.gif even larger so that K ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_IEq78_HTML.gif is equivalent to E ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_IEq79_HTML.gif, i.e.,
        K ( t ) E ( t ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_Equ34_HTML.gif
        (3.26)
        Multiplying (3.25) by ξ ( t ) = min { ξ 1 ( t ) , ξ 2 ( t ) } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_IEq80_HTML.gif and using (A3), we get
        ξ ( t ) K ( t ) ω ξ ( t ) E ( t ) + C Ω 0 t ξ 1 ( t τ ) g 1 ( t τ ) | u ( t ) u ( τ ) | 2 d τ d x + C Ω 0 t ξ 2 ( t τ ) g 2 ( t τ ) | v ( t ) v ( τ ) | 2 d τ d x ω ξ ( t ) E ( t ) C Ω 0 t g 1 ( t τ ) | u ( t ) u ( τ ) | 2 d τ d x C Ω 0 t g 2 ( t τ ) | v ( t ) v ( τ ) | 2 d τ d x ω ξ ( t ) E ( t ) C E ( t ) , for all  t t 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_Equ35_HTML.gif
        (3.27)
        By virtue of (A3) and ξ ( t ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_IEq81_HTML.gif, we have
        d d t ( ξ ( t ) K ( t ) + C E ( t ) ) ω ξ ( t ) E ( t ) , for all  t t 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_Equ36_HTML.gif
        (3.28)
        Using (3.26), we can easily get
        L ( t ) : = ξ ( t ) K ( t ) + C E ( t ) E ( t ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_Equ37_HTML.gif
        (3.29)
        which together with (3.28) yields, for some positive constant η,
        L ( t ) η ξ ( t ) L ( t ) , for all  t t 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_Equ38_HTML.gif
        (3.30)
        Integrating (3.30) over ( t 0 , t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_IEq82_HTML.gif, we arrive at
        L ( t ) L ( t 0 ) e η t t 0 ξ ( τ ) d τ C e η t t 0 ξ ( τ ) d τ , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-146/MediaObjects/13661_2012_Article_238_Equv_HTML.gif

        which together with (3.29) and the boundedness of E and ξ yields (2.3). The proof is now complete. □

        Declarations

        Acknowledgements

        Baowei Feng was supported by the Doctoral Innovational Fund of Donghua University with contract number BC201138, and Yuming Qin was supported by NNSF of China with contract numbers 11031003 and 11271066 and the grant of Shanghai Education Commission (No. 13ZZ048).

        Authors’ Affiliations

        (1)
        College of Information Science and Technology, Donghua University
        (2)
        Department of Applied Mathematics, Donghua University

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