Blow up and global solvability for an absorptive porous medium equation with memory at the boundary

Abstract

Abstract We study the characterization of global solvability versus blow up in finite time for a porous medium model including a balance of internal absorption with memory driven flux through the boundary. Such a boundary condition was previously investigated as part of a model for the transmission of tumour-released growth factor from the site of a pre-vascularized tumour up to and across the wall of a nearby capillary, initiating the process of new capillary growth known as angiogenesis. In previous studies of the model without absorption, we have established the characterization of global solvability in a manner that exactly parallels known results for the corresponding model with localized boundary flux conditions. To include models accounting for internal uptake of growth factor, this analysis has recently been extended to a heat equation with absorption, and herein, we consider the case of a porous medium equation with absorption. Conditions for global solvability emerge naturally out of integral estimates and again provide close parallels with results for localized boundary flux models. It is noted that the results provide a complete characterization in a wide range of models considered.