Decay of Solution of a Viscoelastic Problem

The aim of this thesis is to investigate the asymptotic behavior of solutions of some viscoelastic problems in bounded domains. In this regard, we study several problems and establish exponential, polynomial and general decay rate results. The decay results are established in the absence, as well as in the presence of a source term. This thesis contains five chapters. In Chapter 1, we discuss the properties and significance of viscoelastic materials and we end the chapter by reviewing some literatures related to our problems. In Chapter 2, we present some principal concepts, some theorems on Sobolev embeddings and some lemmas which are of essential use in the proofs of our results. In Chapter 3, we study the asymptotic behaviors of solution of a viscoelastic problem when the relaxation function is decaying exponentially in the presence of a source term. In this regard, we establish a decay result which depends on the behavior of the external force. The case when the relaxation function is decaying polynomially in the absence of a source term is treated in Chapter 4. A general decay result for the finite history case is studied in the last chapter.