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Acta Mathematica Vietnamica

ON A NONLINEAR WAVE EQUATION WITH A NONLOCAL BOUNDARY CONDITION

LE THI PHUONG NGOC, TRAN MINH THUYET, PHAM THANH SON, NGUYEN THANH LONG

Abstract

Consider the initial-boundary value problem for the nonlinear wave equation {uttμ(t)uxx+K|u|p2u+λ|ut|q2ut=F(x,t),0<x<1,0<t<T,μ(t)ux(0,t)=K0u(0,t)+0tk(ts)u(0,s)ds+g(t),μ(t)ux(1,t)=K1u(1,t)+λ1|ut(1,t)|α2ut(1,t),u(x,0)=u~0(x),ut(x,0)=u~1(x), where p,q,α2; K0,K1,K0; λ,λ1>0 are given constants and μ,F,g,k,u~0,u~1 are given functions. First, the existence and uniqueness of a weak solution are proved by using the Galerkin method. Next, with α=2, we obtain an asymptotic expansion of the solution up to order N in two small parameters λ,λ1 with error (λ2+λ12)N+12.