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Finite time blow-up of parabolic systems with nonlocal terms

In this paper, we study the blow-up phenomena for a class of parabolic systems with nonlocal terms, called shadow systems, which are often used to approximate reaction-diffusion systems when one of the diffusion rates is large. Existence of finite blow-up solutions is characterized based on the parameters in the shadow systems. Two different approaches are employed to overcome the difficulties caused by the appearance of nonlocal terms and the lack of comparison principles. One is based on integral estimates, while the other relies on the Schauder Fixed-Point Theorem. This paper continues the work begun in [13]. In particular, we improve the earlier results concerning blow-up solutions to the optimal case.