Asymptotic Behavior of Solutions to a Model System of a Radiating Gas

TL;DR

This paper studies the long-term behavior of solutions to a coupled hyperbolic-elliptic system modeling radiating gas, establishing global existence, decay rates, and asymptotic convergence to a diffusion wave.

Contribution

It introduces a time-weighted energy method to prove global solutions and their decay, and characterizes the asymptotic profile as a diffusion wave.

Findings

Global existence of solutions established

Optimal decay estimates obtained

Solutions asymptotically approach a diffusion wave

Abstract

In this paper we focus on the initial value problem for a hyperbolic-elliptic coupled system of a radiating gas in multi-dimensional space. By using a time-weighted energy method, we obtain the global existence and optimal decay estimates of solutions. Moreover, we show that the solution is asymptotic to the linear diffusion wave which is given in terms of the heat kernel.