Decay of mass for nonlinear equation with fractional Laplacian

TL;DR

This paper investigates the long-term decay behavior of solutions to a reaction-diffusion equation with a fractional Laplacian, revealing how diffusion or nonlinear effects dominate depending on the parameters.

Contribution

It provides a detailed analysis of the asymptotic behavior of solutions, distinguishing regimes where either anomalous diffusion or nonlinear effects are dominant.

Findings

Diffusion dominates for p > 1 + α/N.

Nonlinear effects dominate for p ≤ 1 + α/N.

Large time asymptotics are characterized based on parameter regimes.

Abstract

The large time behavior of nonnegative 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 $p>1+{\alpha}/{N},$ while nonlinear effects win if $p\leq1+{\alpha}/{N}.$