Global Attractors For Strongly Damped Wave Equations With Periodic Boundary Condition

In this paper,we investigate the asymptotic behavior of solutions for the following strongly damped wave equations with periodic boundary condition:u_tt+ω（-Δ）^θu_t-Δu+Φ（u）=f,x∈Ω,t>0.where Ω is a bounded domain in R3.u（x,t）:Ω×R+ → R,θ∈(0,1],the strongly damped coefficient ω is a positive constant,nonlinear term Φ:R→R satisfy some growth conditions.f:Ω→R is the external force.The wave equation is supplemented with the initial conditions:u（x,0）=u₀（x）,ut（x,0）=u₁（x）,x∈Ω.The dynamical properties of the partial differential equations,such as asymptotical be-haviors of solutions and global attractors,are important for the study of diffusion systems,which influence the stability of nonlinear diffusion phenomena and provides the mathe-matical foundation for the study of nonlinear dynamical system.This paper we prove the existence of the global attractor for the above equations with a quite general strongly damped term ω（-Δ）θut,θ∈(0,1].When the nonlinearity is subcritical case,we prove the existence of an exponential attractor of optimal regularity and the bound of the Hausdorff dimension and fractal dimension of the global attractor.