Asymptotic expansions for nonlocal diffusion equations in -norms for

We study the asymptotic behavior for solutions to nonlocal diffusion models of the form u t ( x , t ) = J ∗ u ( x , t ) − u ( x , t ) = ∫ R d J ( x − y ) u ( y , t ) d y − u ( x , t ) in the whole 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 ‖ u ( x , t ) − ∑ | α | ⩽ k ( − 1 ) | α | α ! ( ∫ u 0 ( x ) x α d x ) ∂ α K t ‖ L q ( R d ) ⩽ C t − 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 v t ( x , t ) = − ( − Δ ) s 2 v ( x , t ) . Here we deal with the case 1 ⩽ q ⩽ 2 . The case 2 ⩽ q ⩽ ∞ was treated previously, by other methods, in L.I. Ignat and J.D. Rossi (2008) [11].