Global solvability and blow-up for a nonlinear diffusion model with competing local and memory driven boundary flux

We characterize global solvability versus blow-up in finite time for a nonlinear diffusion model in one space dimension with opposing fluxes at the boundary. Memory driven absorption at one side is introduced through the time integral of a power law, while dissipation at the other side is contained in a localized power law. Previously, a complete analysis appeared in the literature on global solvability for such a model incorporating a balance of local power laws according to orderings of the powers. The present study is part of our continuing investigation of impacts on solvability due to introducing memory within corresponding purely local diffusion models. Such memory boundary conditions have been introduced in models for the process of new capillary growth, as initiated by a pre-vascularized tumor, known as angiogenesis. Global solvability established for the model with memory flux emerges out of integral estimates, providing sharp conditions. We find either all solutions of such models are global or all nontrivial solutions blow up in finite time. This is notably in contrast to cases of the local model with both global and blow-up solutions, as well as equilibrium states available for establishing global boundedness.