# Movement of time-delayed hot spots in Euclidean space for special initial states

## TL;DR

This paper investigates the long-term behavior of specific spatial features of solutions to a damped wave equation with special initial conditions, revealing geometric and topological changes over time.

## Contribution

It provides a detailed analysis of the evolution of null, critical, maximum, and minimum sets for solutions with particular initial states, highlighting their asymptotic properties.

## Key findings

- Null set becomes a smooth sphere-like surface over time
- Critical points increase to at least three after large time
- Maximum points move outside the initial support's convex hull

## Abstract

We consider the Cauchy problem for the damped wave equation under the initial state that the sum of an initial position and an initial velocity vanishes. When the initial position is non-zero, non-negative and compactly supported, we study the large time behavior of the spatial null, critical, maximum and minimum sets of the solution. The spatial null set becomes a smooth hyper-surface homeomorphic to a sphere after a large enough time. The spatial critical set has at least three points after a large enough time. The set of spatial maximum points escapes from the convex hull of the support of the initial position. The set of spatial minimum points consists of one point after a large time, and the unique spatial minimum point converges to the centroid of the initial position at time infinity.