The large diffusion limit for the heat equation with a dynamical boundary condition

TL;DR

This paper investigates the behavior of solutions to the heat equation with a dynamical boundary condition in a half-space as the diffusion coefficient becomes very large, showing convergence to a Laplace equation solution.

Contribution

It establishes the limit behavior of the heat equation with dynamical boundary conditions as the diffusion coefficient approaches infinity.

Findings

Solutions converge to Laplace equation solutions as diffusion coefficient increases

Provides a rigorous mathematical framework for the diffusion limit

Extends understanding of boundary conditions in PDEs

Abstract

We study the heat equation on a half-space with a linear dynamical boundary condition. Our main aim is to show that, if the diffusion coefficient tends to infinity, then the solutions converge (in a suitable sense) to solutions of the Laplace equation with the same dynamical boundary condition.