The diffusion phenomenon for dissipative wave equations in metric measure spaces

Abstract

We study the long-time behavior of the solution to a type of dissipative wave equation, where the operator in the equation is time-dependent and the solution is defined on a metric measure space (X,m) satisfying appropriate conditions. The operator is assumed to be self-adjoint and is related to a time-dependent Dirichlet form. We link hyperbolic PDEs with the firmly established theories for parabolic PDEs in metric measure spaces and Dirichlet forms, subsequently deriving the asymptotic behavior of the solution to the dissipative wave equation. We present several nontrivial examples of dissipative wave equations in metric measure spaces where our theory works.