Fundamental solution and diffusion limits for the heat equation in a half-space with a diffusive dynamical boundary condition

TL;DR

This paper derives explicit fundamental solutions and sharp bounds for the heat equation with a diffusive boundary condition in a half-space, analyzing solution decay and diffusion limits.

Contribution

It provides an explicit fundamental solution and bounds for the heat equation with a diffusive boundary condition, and studies the solution behavior and diffusion limits.

Findings

Explicit fundamental solution derived for the heat equation with boundary condition

Sharp pointwise upper and lower bounds established

Analysis of decay estimates and diffusion limits of solutions

Abstract

We derive an explicit representation of the fundamental solution to the heat equation in a half-space of ${\mathbb R}^N$ with a diffusive dynamical boundary condition, and establish sharp pointwise upper and lower bounds. We also investigate qualitative properties of the associated solutions, including precise decay estimates. Furthermore, we analyze the diffusion limits of solutions to the initial--boundary value problem, and reveal the role of the diffusive dynamical boundary condition in the behavior of solutions.