Isothermalization for a Non-local Heat Equation

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.