Non-autonomous approximations governed by the fractional powers of damped wave operators

Abstract

In this article we study non-autonomous approximations governed by the fractional powers of damped wave operators of order $\alpha \in (0,1)$ subject to Dirichlet boundary conditions in an $n$-dimensional bounded domain with smooth boundary. We give explicitly expressions for the fractional powers of the wave operator, we compute their resolvent operators and their eigenvalues. Moreover, we study the convergence as $\alpha\nearrow 1$ with rate $1-\alpha$.

Keywords

• Non-autonomous damped wave equations
• fractional powers
• rate of convergence
• eigenvalues