Isothermalization for a Non-local Heat Equation
Résumé
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.
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