On a nonlinear wave equation with boundary conditions involved in a Cauchy problem

Abstract

We consider the initial boundary value problem for the nonlinear wave equation utt-∂∂x(μ(x,t)ux)+K|u|p-2u+λ|ut|q-2ut=F(x,t), 0<x<1,0<t<T, where the boundary condition at x=0 is associated with a second-order differential equation P′′(t)+γ1P′(t) +γ2P(t)=β(t)utt(0,t),0<t<T, and the boundary condition at x=1 is of the form μ(1,t)ux(1,t)=λ1|ut(1,t)|α-2ut(1,t), where K,λ,p,q,γ1,γ2,λ1,α,p, and q are given positive constants, F,μ, and β are given functions. First, the existence and uniqueness of a weak solution are proved by using the FaedoGalerkin method. Next, the asymptotic behavior of the solution as λ1→0+ is discussed. Finally, we obtain an asymptotic expansion of the solution up to order N in a small parameter λ1 with error λ1α(N+1)-12(α-1).