Transmission problems with nonlocal boundary conditions and rough dynamic interfaces

TL;DR

This paper studies complex transmission problems involving semilinear parabolic equations with fractal-like interfaces and nonlinear dynamic boundary conditions, providing a unified framework for solution existence, attractors, and blow-up phenomena.

Contribution

It introduces a comprehensive framework addressing existence, attractors, and blow-up for transmission problems with nonlocal boundary conditions and rough interfaces.

Findings

Established conditions for strong solution existence

Proved existence of finite-dimensional attractors

Analyzed blow-up phenomena under various nonlinearities

Abstract

We consider a transmission problem consisting of a semilinear parabolic equation in a general non-smooth setting with emphasis on rough interfaces which bear a fractal-like geometry and nonlinear dynamic (possibly, nonlocal)\ boundary conditions along the interface. We give a unified framework for existence of strong solutions, existence of finite dimensional attractors and blow-up phenomena for solutions under general conditions on the bulk and interfacial nonlinearities with competing behavior at infinity.