Qualitative studies of some dissipative systems for wave equations

Abstract

The present thesis is devoted to the study of well-posedness and asymptotic behaviour in time of solution for damped systems. This work consists of four chapters. In chapter 1, we recall of some fundamental inequalities. In chapter 2, we consider a very important problem from the point of view of application in sciences and engineering. A system of three wave equations having a different damping effects in an unbounded domain with strong external forces. Using the FaedoGalerkin method and some energy estimates, we will prove the existence of global solution in Rn owing to to the weighted function. By imposing a new appropriate conditions, which are not used in the literature, with the help of some special estimates and generalized Poincar´e’s inequality, we obtain an unusual decay rate for the energy function. In chapter 3, we will concerned with a problem for coupled nonlinear viscoelastic wave equation with distributed delay and strong damping and source terms, under suitable conditions we prove a blow up/growth results of solutions. In chapter 4, we consider one-dimensional porous-elastic system with nonlinear damping, infinite memory and distributed delay terms. We show the well posedness of solution by the semigroup theory and that the solution energy has an explicit and optimal decay, for the cases of equal and nonequal speeds of wave propagation.