Jeffrey R. Anderson

Local Existence And Uniqueness of Solutions of Degenerate Parabolic Equations

We consider a class of degenerate parabolic equaitons on a bounded domain with mixed boundary conditions. These problems arise, for example, in the study of flow through porous media. Under appropriate hypotheses, we establish the existence of a nonegative solution which is obtainable as a monotone limit of solutions of quasilinear parabolic equations. This construction is used establish uniqueness, cinparison, and L1 continuous dependence theorems, as well some results on blow up of solutions in finite time

Global solvability for the heat equation with boundary flux governed by nonlinear memory

We introduce the study of global existence and blowup in finite time for the heat equation with flux at the boundary governed by a nonlinear memory term. Via a simple transformation, the model may be written in a form which has been 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. Additionally, we provide a brief summary of related studies regarding heat equation models where memory terms are incorporated within reaction or diffusion.

A fast diffusion model with memory at the boundary: global solvability in the critical case

Necessary and sufficient conditions for the global solvability of a slow diffusion model with boundary flux governed by memory have been previously shown to be the same as those for a corresponding model with localized nonlinear flux at the boundary. Recent investigations of a similar fast diffusion model with memory have also successfully replicated conditions in parallel with the corresponding localized problem, except for the critical case separating global solvability from blow up in finite time. We provide a suitable modification of an estimate, typically applied to the case of slow diffusion, which also applies to the fast diffusion model and subsequently establishes global solvability in the critical case. Memory terms appearing in the model are of the type which have been introduced in studies of tumor-induced angiogenesis.