Partial Differential Equations and Optimization

Abstract

In this thesis, we consider some viscoelastic problems for a strongly elliptic operator of second order with variable coefficients in bounded domains. A review of the recent studies on the generalized thermoelasticity theories and their associated modified models is also presented. In this regard, we investigate several coupled systems with mixed Dirichlet-Neumann boundary conditions and boundary interaction feedback. The coupling is via the acoustic boundary conditions on a portion of the boundary. Using some interesting methods of mathematical analysis as, semigroup theory, Schauder fixed point, Faedo-Galerkin approach, and compactness estimates, to show the local and global existence of energy-associated solutions. In addition, taking into account the Gearhart-Prüss theorem, the exponential stability of the corresponding semigroup is concluded. Moreover, under a very wider class of generality of relaxation functions, we establish several general decay results using the energy methods