Movement of time-delayed hot spots in Euclidean space

TL;DR

This paper analyzes the behavior of solutions to the damped wave equation, focusing on the movement of hot spots over time, and provides estimates for the solution's behavior in various function spaces.

Contribution

It introduces a decomposition of the solution into heat and wave parts and studies the evolution of hot spots, offering new insights into the solution's shape and properties.

Findings

Existence and location of spatial maximizers analyzed

Decomposition into heat and wave components developed

L^p-L^q estimates for the solution established

Abstract

We investigate the shape of the solution of the Cauchy problem for the damped wave equation. In particular, we study the existence, location and number of spatial maximizers of the solution. Studying the shape of the solution of the damped wave equation, we prepare a decomposed form of the solution into the heat part and the wave part. Moreover, as its another application, we give $L^p$-$L^q$ estimates of the solution.