Journal of Evolution Equations

, Volume 8, Issue 4, pp 617–629

Refined asymptotic expansions for nonlocal diffusion equations

Article

Abstract.

We study the asymptotic behavior for solutions to nonlocal diffusion models of the form utJ * uu in the whole \({\mathbb{R}}^d\) with an initial condition u(x, 0) = u0(x). Under suitable hypotheses on J (involving its Fourier transform) and u0, 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 Kt 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, \(\nu_t (x, t) = -(-\Delta)^{\frac{s}{2}} \nu (x, t)\).

Keywords:

Nonlocal diffusion asymptotic behavior fractional Laplacian 

Mathematics Subject Classifications (2000):

35B40 45A05 45M05 

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Copyright information

© Birkhaueser 2008

Authors and Affiliations

  1. 1.Institute of Mathematics of the Romanian AcademyBucharestRomania
  2. 2.IMDEA MatematicasMadridSpain
  3. 3.Departamento de MatemáticaFCEyN UBABuenos AiresArgentina

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