Long-time behaviour for a non-autonomous Klein-Gordon-Zakharov system

Abstract

The aim of this paper is to study the long-time dynamics of solutions of the evolution system{utt−Δu+u+η(−Δ)12ut+aϵ(t)(−Δ)12vt=f(u),(x,t)∈Ω×(τ,∞),vtt−Δv+η(−Δ)12vt−aϵ(t)(−Δ)12ut=0,(x,t)∈Ω×(τ,∞), subject to boundary conditionsu=v=0,(x,t)∈∂Ω×(τ,∞), where Ω is a bounded smooth domain in Rn, n≥3, with the boundary ∂Ω assumed to be regular enough, η>0 is constant, aϵ is a Hölder continuous function and f is a dissipative nonlinearity. This problem is a non-autonomous version of the well known Klein-Gordon-Zakharov system. Using the uniform sectorial operators theory, we will show the local and global well-posedness of this problem in H01(Ω)×L2(Ω)×H01(Ω)×L2(Ω). Additionally, we prove existence, regularity and upper semicontinuity of pullback attractors.