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

Abstract

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.

Keywords:

• Fast diffusion equation
• global existence
• memory boundary condition

AMS Subject Classifications:

• 35A01
• 35B44
• 35B51
• 35K59

Notes

No potential conflict of interest was reported by the author.