The Diffusion Phenomenon for Dissipative Wave Equations with Time-Dependent Operators

In this dissertation, we study the long-time behavior of the solution to a type of dissipative wave equation, where the operator in this equation is time-dependent and self-adjoint, and the solution to this equation is defined in a metric measure space satisfying appropriate conditions. We first consider the solution to an important particular dissipative wave equation in Euclidean space, where the operator is uniformly elliptic and in divergence form for each fixed time. We derive the asymptotic behavior of the solution to this equation. Furthermore, the work done for this particular problem serves as a stepping stone, allowing us to study the solution to the general type of dissipative wave equation. When we study the general problem, the operator in the dissipative wave is assumed to correspond to a Dirichlet form. We link hyperbolic PDEs with the firmly established theories for parabolic PDEs and Dirichlet forms, subsequently deriving the asymptotic behavior of the solution to the general dissipative wave equation.