Asymptotic behaviour for a semilinear nonlocal equation

We study the semilinear nonlocal equation u_t=J*u−u−u^p in the whole ${\mathbb{R}}^{N}$ . First, we prove the global well-posedness for initial conditions $u(x,0)=u_{0}(x)\in L^{1}({\mathbb{R}}^{N})\cap L^{\infty}({\mathbb{R}}^{N})$ . Next, we obtain the long time behaviour of the solutions. We show that different behaviours are possible depending on the exponent p and the kernel J: finite time extinction for p<1, faster than exponential decay for the linear case p=1, a weakly nonlinear behaviour for p large enough and a decay governed by the nonlinear term when p is greater than one but not so large.