Blow-up in nonlocal reaction-diffusion equations

We present new blow-up results for reaction-diffusion equations with nonlocal nonlinearities. The nonlocal source terms we consider are of several types, and are relevant to various models in physics and engineering. 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 localized values of the solution. For each type of problems, we give finite time blow-up results which significantly improve or extend previous results of several authors. In some cases, when the nonlocal source term is in competition with a local dissipative or convective term, optimal conditions on the parameters for finite time blow-up or global existence are obtained. Our proofs rely on comparison techniques and on a variant of the eigenfunction method combined with new properties on systems of differential inequalities. Moreover, a unified local existence theory for general nonlocal semilinear parabolic equations is developed.