Weighted energy estimates for wave equation with space-dependent damping term for slowly decaying initial data

Abstract

This paper is concerned with weighted energy estimates for solutions to wave equation ${\partial }_t^{2}u-\Delta u+a(x){\partial }_{t}u=0$ with space-dependent damping term $a(x)=|x{|}^{-\alpha }$ $(\alpha \in [0,1))$ in an exterior domain $\Omega$ having a smooth boundary. The main result asserts that the weighted energy estimates with weight function like polymonials are given and these decay rate are almost sharp, even when the initial data do not have compact support in $\Omega$. The crucial idea is to use special solution of ${\partial }_{t}u=|x{|}^{\alpha }\Delta u$ including Kummer's confluent hypergeometric functions.

26 pages

References in corpus (2)

• Diffusion phenomena for the wave equation with space-dependent damping in an exterior domain
• Remarks on an elliptic problem arising in weighted energy estimates for wave equations with space-dependent damping term in an exterior domain

Cited by in corpus (3)

• Supersolutions for parabolic equations with unbounded diffusion and its applications to some classes of parabolic and hyperbolic equations
• Decay property of solutions to the wave equation with space-dependent damping, absorbing nonlinearity, and polynomially decaying data
• Asymptotic expansion of solutions to the wave equation with space-dependent damping