  We study the large time behavior of nonnegative solutions of the Cauchy
problem $u_t=\int J(x-y)(u(y,t)-u(x,t))\,dy-u^p$, $u(x,0)=u_0(x)\in L^\infty$,
where $|x|^{\alpha}u_0(x)\to A>0$ as $|x|\to\infty$. One of our main goals is
the study of the critical case $p=1+2/\alpha$ for $0<\alpha<N$, left open in
previous articles, for which we prove that $t^{\alpha/2}|u(x,t)-U(x,t)|\to 0$
where $U$ is the solution of the heat equation with absorption with initial
datum $U(x,0)=C_{A,N}|x|^{-\alpha}$. Our proof, involving sequences of
rescalings of the solution, allows us to establish also the large time behavior
of solutions having more general nonintegrable initial data $u_0$ in the
supercritical case and also in the critical case ($p=1+2/N$) for bounded and
integrable $u_0$.
