Decay of mass for nonlinear equation with fractional Laplacian

The large time behavior of nonnegative solutions to the reaction-diffusion equation ${\mathrm{\partial }}_{t}u=-(-\mathrm{\Delta }{)}^{\alpha \mathrm{/}2}u-u^p,$ $(\alpha \in (0,2],p>1)$ 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 $p>1+\alpha \mathrm{/}N,$ while nonlinear effects win if $p\le 1+\alpha \mathrm{/}N\mathrm{.}$