On a singular heat equation with dynamic boundary conditions

TL;DR

This paper studies a nonlinear singular heat equation with dynamic boundary conditions in 3D, focusing on existence, regularization, long-term behavior, and uniqueness of solutions.

Contribution

It introduces analysis of a singular diffusion heat equation with dynamic boundary conditions, highlighting existence, regularization, and long-term solution properties.

Findings

Existence of weak solutions established

Solutions exhibit instantaneous regularization

Long-time behavior characterized

Abstract

In this paper we analyze a nonlinear parabolic equation characterized by a singular diffusion term describing very fast diffusion effects. The equation is settled in a smooth bounded three-dimensional domain and complemented with a general boundary condition of dynamic type. This type of condition prescribes some kind of mass conservation; hence extinction effects are not expected for solutions that emanate from strictly positive initial data. Our main results regard existence of weak solutions, instantaneous regularization properties, long-time behavior, and, under special conditions, uniqueness.