Summary: 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 \(|x|^{\alpha}u_0(x)\rightarrow A>0\) as \(|x|\rightarrow\infty\). 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 \(u_0\).