Existence And General Decay Of Solutions For Two Types Of Coupled Wave Equations

In this paper, we study the existence and general decay of solutions for two types of coupling wave equations. This paper is mainly composed of three chapters.In chapter 1, We introduce the research status of the existence and general decay of solutions for coupled wave equations.In chapter 2, we study the local existence of weak solutions for a class of coupled viscoelastic wave equations with strongly damped. We use the Galerkin method to prove the local existence of weak solutions for the following equation, where Ω is an open bounded domain of Rn with a smooth boundary (?)Ω. This chapter generalizes and improves the earlier condition α= 0, ε=0, and we study the local existence of weak solutions for the wave equations under the condition a> 0,0<ε<1.In chapter 3, we study the existence and general decay of solutions for the wave equation with a memory type in boundary, where Ω is an open bounded domain of Rn with a smooth boundary (?)Ω= Γ0∪Γ1, υ is the unit outward normal vector. The partition Γ1 and Γ2 are disjoint, closed, with meas(Γo)> 0 and satisfying where m(x)=x-xo, for some xo ∈ Rn.In the chapter, we study the general decay of solutions for the wave equations with a bounded,where the memory-type damping controles a part of the boundary. Firstly, we give the existence of solutions, secondly, we can construct Lyapunov function equivalented to the energy function E(t) by using multiplier method.