Isothermalization for a Non-local Heat Equation

TL;DR

This paper investigates the long-term behavior of solutions to a nonlocal heat equation in an inhomogeneous medium, demonstrating conditions under which solutions stabilize to a constant state over time.

Contribution

It provides new results on the asymptotic convergence of solutions to a nonlocal heat equation with variable density, extending understanding of isothermalization in inhomogeneous media.

Findings

Solutions converge to a constant state under certain integrability conditions.

Different asymptotic behaviors are characterized based on initial data and medium properties.

The paper establishes conditions for global convergence in nonlocal heat equations.

Abstract

n this paper we study the asymptotic behavior for a nonlocal heat equation in an inhomogenous medium: $$\rho(x)u_t=J\ast u-u \text{in}\mathbb{R}^N\times (0,\infty)\,,$$ where $\rho$ is a continous positive function, $u$ is nonnegative and $J$ is a probability measure having finite second-order momentum. Depending on integrability conditions on the initial data $u_0$ and $\rho$, we prove various isothermalisation results, i.e. $u(t)$ converges to a constant state in the whole space.