Refined asymptotic expansions for nonlocal diffusion equations

Abstract.

We study the asymptotic behavior for solutions to nonlocal diffusion models of the form u _t = J * u – u in the whole ${\mathbb{R}}^d$ with an initial condition u(x, 0) = u ₀(x). Under suitable hypotheses on J (involving its Fourier transform) and u ₀, it is proved an expansion of the form

$$\left\| {u(u) - \sum\limits_{\left| \alpha \right| \leq k} {\frac{{( - 1)^{\left| \alpha \right|} }}{{\alpha !}}\left( {\int {u_0 (x)x^\alpha\, dx} } \right)\partial ^\alpha K_t } } \right\| _{L^q ({\mathbb{R}}^d )} \leq Ct^{ - A}$$

, where K _t is the regular part of the fundamental solution and the exponent A depends on J, q, k and the dimension d.

Moreover, we can obtain bounds for the difference between the terms in this expansion and the corresponding ones for the expansion of the evolution given by fractional powers of the Laplacian, $\nu_t (x, t) = -(-\Delta)^{\frac{s}{2}} \nu (x, t)$.