BLOW-UP IN NONLOCAL REACTION-DIFFUSION EQUATIONS

We present new blow-up results for reaction-diffusion equations with n onlocal nonlinearities. The nonlocal source terms we consider are of s everal types, and are relevant to various models in physics and engine ering. They may involve an integral of the unknown function, either in space, in time, or both in space and time, or they may depend on loca lized values of the solution. For each type of problems, we give finit e time blow-up results which significantly improve or extend previous results of several authors. In some cases, when the nonlocal source te rm is in competition with a local dissipative or convective term, opti mal conditions on the parameters for finite time blow-up or global exi stence are obtained. Our proofs rely on comparison techniques and on a variant of the eigenfunction method combined with new properties on s ystems of differential inequalities. Moreover, a unified local existen ce theory for general nonlocal semilinear parabolic equations is devel oped.