, Volume 160, Issue 4, pp 375-384
Date: 30 Jan 2009

Decay of mass for nonlinear equation with fractional Laplacian

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Abstract

The large time behavior of non-negative solutions to the reaction–diffusion equation \({\partial_t u=-(-\Delta)^{\alpha/2}u - u^p}\) , \({(\alpha\in(0,2], \;p > 1)}\) posed on \({\mathbb{R}^N}\) and supplemented with an integrable initial condition is studied. We show that the anomalous diffusion term determines the large time asymptotics for p > 1 + α/N, while nonlinear effects win if p ≤ 1 + α/N.

Communicated by A. Jüngel.
The preparation of this paper was supported in part by the European Commission Marie Curie Host Fellowship for the Transfer of Knowledge “Harmonic Analysis, Nonlinear Analysis and Probability” MTKD-CT-2004-013389. A. Fino gratefully thanks the Mathematical Institute of Wrocław University for the warm hospitality. The preparation of this paper by G. Karch was also partially supported by the MNiSW grant N201 022 32/09 02.