On a class of degenerate parabolic equations with dynamic boundary conditions

TL;DR

This paper extends the analysis of a class of quasi-linear parabolic equations with dynamic boundary conditions, establishing uniform bounds and conditions for nonlinearities, thus broadening understanding of their long-term behavior.

Contribution

It demonstrates that previous well-posedness results can be generalized to more complex nonlinearities by deriving uniform L-infinity estimates and new balance conditions.

Findings

Established uniform L-infinity bounds for solutions.

Derived new conditions for nonlinearities balancing source terms.

Extended previous results to more general nonlinear cases.

Abstract

We consider a quasi-linear parabolic equation with nonlinear dynamic boundary conditions occurring as a natural generalization of the semilinear reaction-diffusion equation with dynamic boundary conditions. The corresponding class of initial and boundary value problems has already been studied previously, proving well-posedness and the existence of the global attractor. The goal of this note is to show that the previous analysis can be redone for more general nonlinearities by proving an additional (uniform) L\infty-estimate on the solutions. In particular, we derive new conditions which reflect an exact balance between the two nonlinear mechanisms involved, even when both the nonlinear (source) terms contribute in opposite directions.