We introduce the study of global existence and blow up in finite time for nonlinear diffusion equations with flux at the boundary governed by memory. Via a simple transformation, the memory term arises out of a corresponding model introduced in previous studies of tumor-induced angiogenesis. The present study is also in the spirit of extending work on models of the heat equation with local, nonlocal, and delay nonlinearities present in the boundary flux. Specifically, we establish an identical set of necessary and sufficient conditions for blow up in finite time as previously established in the case of local flux conditions at the boundary.
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