  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
$p\leq1+{\alpha}/{N}.$
