New scenario of decay rate for system of three nonlinear wave equations with visco-elasticities

Abstract

A system of three semilinear wave equations with strong external forces in $ \mathbb{R}^n $ is considered. We use weighted phase spaces, where the problem is well defined, to compensate the lack of Poincare's inequality. Using the Faedo-Galerkin method and some energy estimates, we prove the existence of global solution. By imposing a new appropriate conditions, which are not used in the literature, with the help of some special estimates and generalized Poincare's inequality, we obtain an unusual decay rate for the energy function. It is a generalization of similar results in [ 16 , 29 , 31 ]. The work is relevant in the sense that the problem is more complex than what can be found in the literature.