Existence and exponential decay estimates for an N-dimensional nonlinear wave equation with a nonlocal boundary condition

Le Thi Phuong Ngoc,Nguyen Anh Triet,Nguyen Thanh Long

doi:10.1186/s13661-016-0527-5

Le Thi Phuong Ngoc, Nguyen Anh Triet + Show 1 more

Open Access

https://doi.org/10.1186/s13661-016-0527-5

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Abstract

Motivated by the recent known results as regards the existence and exponential decay of solutions for wave equations, this paper is devoted to the study of an N-dimensional nonlinear wave equation with a nonlocal boundary condition. We first state two local existence theorems. Next, we give a sufficient condition to guarantee the global existence and exponential decay of weak solutions. The main tools are the Faedo-Galerkin method and the Lyapunov method.

Highlights

In [ ], Munoz-Rivera and Andrade dealt with the global existence and exponential decay of solutions of the nonlinear one-dimensional wave equation with a viscoelastic boundary condition
With less regular initial data, we obtain the following theorem as regards the existence of a weak solution
Under the assumptions of Theorem . , using the Faedo-Galerkin approximation and Lemmas . - . , we find the approximate solution of problem ( . )-( . ) in the form m um(t) = cmj(t)wj, j=

Summary

Introduction

In [ ], Beilin investigated the existence and uniqueness of a generalized solution for the following wave equation with an integral nonlocal condition: ), with f (u) = b|u|p– u, b > has a unique global solution with energy decaying exponentially for any initial data (u , u ) ∈ Ono [ ] studied the global existence and the decay properties of smooth solutions to the Cauchy problem related to In [ ], Munoz-Rivera and Andrade dealt with the global existence and exponential decay of solutions of the nonlinear one-dimensional wave equation with a viscoelastic boundary condition.

Results

Conclusion