| Sumario: | We study the semilinear nonlocal equation u t =Ju-u-u p in the whole. First, we prove the global well-posedness for initial conditions. 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. © 2007 - IOS Press and the authors. All rights reserved.
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