Weighted L∞-estimates for solutions of the damped wave equation in three space dimensions and its application

Abstract

In this paper, we derive a weighted $L^{\infty }$-estimate of the solution to the damped wave equation in three space dimensions. Our proof uses a concrete representation formula of the solution to the damped wave equation that does not rely on the Fourier transform or the energy method. Moreover, by applying our weighted estimate, we consider the global existence of solutions to nonlinear damped wave equations for small data and obtain a new pointwise decay estimate.