Refined asymptotics for the infinite heat equation with homogeneous Dirichlet boundary conditions

TL;DR

This paper studies the long-term behavior of solutions to the infinite heat equation with Dirichlet boundary conditions, showing convergence to a unique limit and analyzing support expansion using advanced mathematical techniques.

Contribution

It provides a rigorous proof of convergence and support expansion for solutions to the infinite heat equation with homogeneous Dirichlet boundary conditions.

Findings

Solutions converge to a unique limit over time.

Support of solutions expands in a predictable manner.

The proof employs half-relaxed limits and boundary estimates.

Abstract

The nonnegative viscosity solutions to the infinite heat equation with homogeneous Dirichlet boundary conditions are shown to converge as time increases to infinity to a uniquely determined limit after a suitable time rescaling. The proof relies on the half-relaxed limits technique as well as interior positivity estimates and boundary estimates. The expansion of the support is also studied.