Asymptotic behavior for wave equations with space-dependent damping in a weighted energy class

Abstract

In this paper, we discuss diffusion phenomena for the wave equation with space-dependent damping. It is known that this phenomenon occurs in the constant damping case with initial data belonging to a suitable energy class. This paper clarifies that diffusion phenomenon also occurs when the damping is effective and space-dependent, and initial data belong to a certain energy class. The proof relies on an energy method involving Kummer's confluent hypergeometric functions with a modified version of the technique of decomposition of solutions introduced in [20].