Stability Analysis of the Wave Equation with Fractional Damping

Abstract

This thesis investigates the stability of a damped wave equation featuring a combination of classical viscous damping and a tempered fractional damping term that incorporates memory effects. The inclusion of this nonlocal term introduces significant analytical complexities due to its convolution structure. We consider the wave equation under homogeneous Dirichlet boundary conditions. The

primary objective is to establish the well-posedness of the system and analyze the long- term behavior of its solutions. To handle the nonlocal fractional operator, we employ an

augmented formulation, transforming the problem into an equivalent evolution system within a suitable Hilbert space. By applying the theory of C0-semigroups, we demonstrate that the associated linear operator is maximal dissipative, which ensures the existence, uniqueness, and continuous dependence of the solutions on the initial data. Furthermore, by constructing an appropriate Lyapunov functional that accounts for both damping mechanisms, we prove the exponential stability of the system, demonstrating that the energy decays to zero at an exponential rate. These theoretical results are also validated numerically.