Decay of mass for nonlinear equation with fractional Laplacian

Ahmad Fino (1) , Grzegorz Karch (2)

1 MIA - Mathématiques, Image et Applications - EA 3165
2 Instytut Matematyczny

The large time behavior of nonnegative solutions to the reaction-diffusion equation $\partial_t u=-(-\Delta)^{\alpha/2}u + \lambda u^p,$ $(\alpha\in(0,2], \;p\geq1,\; \lambda\in\{-1,1\})$ posed on $\mathbb{R}^N$ and supplemented with an integrable initial condition is studied. The solutions for the equation with absorbing term $(\lambda=-1)$ are global-in-time. In this case, we show that the anomalous diffusion term determines the large time asymptotics for $p>1+{\alpha}/{N},$ while nonlinear effects win if $p\leq1+{\alpha}/{N}.$ It is known that some solutions to the equation with $\lambda=1$ blow up in a finite time. By the method used in this article, we show the blow-up of all positive solutions in the critical case $p=1+\alpha/N.$

Mots clés: critical exponent; Large time behavior of solutions; fractional Laplacian; blow-up of solutions; critical exponent.

Domaines: Equations aux dérivées partielles [math.AP]

Ahmad Fino, Grzegorz Karch. Decay of mass for nonlinear equation with fractional Laplacian. 2008. ⟨hal-00333433v2⟩