Asymptotic behavior for the heat equation in nonhomogeneous media with critical density

TL;DR

This paper investigates the long-term behavior of solutions to a heat equation with a critical singular density in nonhomogeneous media, revealing two distinct asymptotic profiles based on initial data properties.

Contribution

It identifies how initial data vanishing at the origin influences asymptotic profiles and highlights the necessity of radial symmetry for certain convergence results.

Findings

Two different asymptotic profiles depending on initial data at the origin.

Radial symmetry is required for convergence in cases where initial data vanishes at zero.

Counterexamples show non-convergence to symmetric profiles in general.

Abstract

We study the asymptotic behavior of solutions to the heat equation in nonhomogeneous media with critical singular density $$ |x|^{-2}\partial_{t}u=\Delta u, \quad \hbox{in} \ \real^N\times(0,\infty). $$ The asymptotic behavior proves to have some interesting and quite striking properties. We show that there are two completely different asymptotic profiles depending on whether the initial data $u_0$ vanishes at $x=0$ or not. Moreover, in the former the results are true only for radially symmetric solutions, and we provide counterexamples to convergence to symmetric profiles in the general case.