Existence and stability for a system of nonlinear damped wave equations

Abstract

—————————————————————————————————— The present thesis is devoted to study the existence, uniqueness and asymptotic behaviour in time of solution for damped systems. This work consists of four chapters. In chapter 1, we recall some fundamental inequalities. In chapter 2, we consider a very important problem from the point of view of application in science and engineering. A system of three wave equations having a different damping effects in an unbounded domain with strong external forces. Using the Faedo-Galerkin method and some energy estimations, we will prove the existence of global solution in Rn owing to the weighted function. By imposing a new appropriate conditions, which are not used in the literature, with the help of some special estimations and generalized Poincar´e’s inequality, we obtain an unusual decay rate for the energy function. In chapter 3, we will be concerned with a problem for m-nonlinear viscoelastic wave equations, under suitable conditions we show the effect of weak and strong damping terms on decay rate for systems of nonlinear mwave equations in viscoelasticity. In chapter 4, we consider Petrovsky-Petrovsky coupled system with nonlinear strong damping. We prove, under some appropriate assumptions, that this system is stable. Furthermore, we use the multiplier method and some general weighted integral inequalities to obtain decay properties of solution.