On Global Solutions for Quasilinear One-Dimensional Parabolic Problems with Dynamical Boundary Conditions

TL;DR

This paper establishes near-optimal conditions ensuring the global existence of classical solutions for one-dimensional quasilinear parabolic problems with dynamical boundary conditions, advancing the understanding of such PDEs.

Contribution

It introduces sufficient and nearly optimal criteria for the global existence of solutions to these specific parabolic problems, filling a gap in the existing literature.

Findings

Derived near-optimal conditions for solution existence

Proved global existence in parabolic H"older spaces

Enhanced understanding of dynamical boundary conditions

Abstract

We provide sufficient and almost optimal conditions for global existence of classical solutions in parabolic H\"older spaces to quasilinear one-dimensional parabolic problems with dynamical boundary conditions.