Stability and decay of solutions for some wave equations with delay term

Abstract

This thesis aims to investigate the stability and decay of solutions for wave equations involving a time-delay term. It begins with a presentation of fundamental mathematical concepts, such as metric and normed spaces, followed by a detailed analysis of strong and weak stability, particularly in the presence of delayed dynamic feedback. The second focuses on a wave equation with a delayed term in the dynamical control. It is shown that the delayed system exhibits the same asymptotic decay rate as the corresponding system without delay, indicating that the delay does not alter the long-term decay behaviour. But the last addresses a semi linear wave equation in a bounded domain with boundary damping of memory type. A general decay result is established, which encompasses both exponential and polynomial decay as special cases. This result does not require specific assumptions on the relaxation function, thereby extending and refining previous results in the literature.