Existence, blow-up and exponential decay of solutions for a system of nonlinear wave equations with damping and source terms

Le Thi Phuong Ngoc, Vo Anh Khoa, Nguyen Thanh Long

Published 2015-05-23Version 1

This paper deals with a system of wave equations in one-dimensional consisting of nonlinear boundary / interior damping and nonlinear boundary / interior sources. In particular, our interest lies in the theoretical understanding of the existence, finite time blow-up of solutions and their exponential decay. First, two local existence theorems of weak solutions are established by applying the Faedo-Galerkin method and standard arguments of density corresponding to the regularity of initial conditions. The uniqueness is also obtained in some specific cases. Second, it is proved that any weak solutions possessing negative initial energy has the potential to blow up in finite time. Finally, exponential decay estimates for the global solution is achieved through the construction of a suitable Lyapunov functional. In order to corroborate our theoretical decay, a numerical example is provided here.