Existence of an attractor and determining modes for structurally damped nonlinear wave equations

Abstract

The paper is devoted to the study of asymptotic behavior as t→+∞ of solutions of initial boundary value problem for structurally damped semi-linear wave equation ∂t2u(x,t)−Δu(x,t)+γ(−Δ)θ∂tu(x,t)+f(u)=g(x),θ∈(0,1),x∈Ω,t>0 under homogeneous Dirichlet’s boundary condition in a bounded domain Ω⊂R3. We proved that the asymptotic behavior as t→∞ of solutions of this problem is completely determined by dynamics of the first N Fourier modes, when N is large enough. We also proved that the semigroup generated by this problem when θ∈(12,1) possesses an exponential attractor.