Isothermalization for a Non-local Heat Equation

n this paper we study the asymptotic behavior for a nonlocal heat equation in an inhomogenous medium: $ρ(x)u_t=J\ast u-u \text{in}\mathbb{R}^N\times (0,\infty)\,,$ where $ρ$ 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 $ρ$, we prove various isothermalisation results, i.e. $u(t)$ converges to a constant state in the whole space.

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