Study On The Initial-Boundary Value Problem Of A Class Of Wave Equations With Hartree-Type Source Terms

In this paper,we study the local existence,global existence,and longtime behavior(exponential decay and blow-up)of solutions for the following initial boundary value problems of wave equations with damping and Hartree-type source terms:where Ω?Rn(n≥3)is a bounded convex domain with smooth boundary ?Ω,α∈(0,n),n+α/n≤p≤n+α/n-2,and 1/|x|n-α*|u|p=∫Ω|u(y)|p/|x-y|n-αdy.By the standard Faedo-Galerkin method,we obtain the existence result of the local solution.Then,with the help of potential well theory,we prove the existence of the global solution,and by using the perturbed energy method,we derive the exponential decay estimation.Furthermore,by the convexity method,we give the sufficient conditions of blow-up of the solution at finite time in both cases of positive and negative initial energy.