The Asymptotic Behavior Of A Damped Wave Equation With A Nonlinear Memory Term

LetΩbe an open bounded subset of R n with smooth boundary. A damped wave equation with a nonlinear memory term is considered inΩ. u_tt +αut -Δu -∫₀^tμ(t - s ) | u ( s ) | ^βu ( s ) ds + g (u )=f, ( x , t )∈Ω×R⁺; u ( x,t)=0, ( x , t )∈Γ×R⁺; u ( x,0)= u0 (x), ut ( x,0)= u_t (x), x∈Ω. Whereα, ^βare positive constants,∫₀^tμ(t -s ) | u ( s ) | ^βu ( s )ds is the nonlinear memory term and g (u ) is a nonlinear source term. The equations with a linear memory term (∫₀^tμ(t - s )Δu ( s )ds) have been carried out extensive and in-depth research into by many scholars. However, the case of memory term as nonlinear memory exists widely in nature. In the literature [13, 14], Cavalcanti and Park have proved the existence and uniform decay of solutions to the wave equations with nonlinear boundary damping and nonlinear boundary memory source term. This paper is about the existence of global solutions and the existence of the global attractor for the wave equation with the nonlinear memory term in an open bounded domainΩ. First, Faedo-Galerkin's approximation method is used in order to obtain the existence of global week solutions and strong solutions. Second, the uniqueness and regularity of global solutions are proved. Then the existence of an absorbing set in L² (Ω)×H₀¹(Ω) is obtained. Finally, in L² (Ω)×H₀¹(Ω) the semi-group S (t ) associated with the equation is split into S₁ (t )and S₂(t ). For any bounded set B the uniform decay of S₂(t )B is proved, and S₁ (t )B is asymptotic compactness in L² (Ω)×H₀¹(Ω)is also proved with the method of the differentiation of the equation. Therefore, the existence of global attractor is obtained.