Diffusion phenomenon for abstract wave equations with decaying dissipation

Abstract

We consider the initial value problem of the abstract wave equation with dissipation whose coefficient tends to 0 as $t \to \infty$. In the case that the coefficient of the dissipation is a positive constant, Ikehata–Nishihara and Chill–Haraux obtained the decay estimate of the difference between the solution of this equation and the solution of the corresponding abstract heat equation. In the case that $H = L^2 (\mathbb{R}^n)$ and $A$ is the Laplace operator and $b(t)$ is a positive valued monotone $C^2$ function satisfying the sufficient assumption, Wirth obtained the decay estimate of the difference between the solution of the dissipative wave equation and the solution of the corresponding heat equation. The purpose of this paper is to show the decay estimate of the difference between the solution of the abstract wave equation with decaying dissipative term and the solution of the corresponding abstract parabolic equation.