Fundamental solution to the heat equation with a dynamical boundary condition

TL;DR

This paper derives an explicit fundamental solution for the heat equation with a dynamical boundary condition on a half-space, providing decay estimates and identifying the Fujita exponent for related semilinear problems.

Contribution

It presents the first explicit fundamental solution for the heat equation with dynamical boundary conditions on a half-space and applies it to determine decay rates and critical exponents.

Findings

Explicit fundamental solution derived

Sharp decay estimates established

Fujita exponent identified for semilinear problem

Abstract

We give an explicit representation of the fundamental solution to the heat equation on a half-space of ${\mathbb R}^N$ with the homogeneous dynamical boundary condition, and obtain upper and lower estimates of the fundamental solution. These enable us to obtain sharp decay estimates of solutions to the heat equation with the homogeneous dynamical boundary condition. Furthermore, as an application of our decay estimates, we identify the so-called Fujita exponent for a semilinear heat equation on the half-space of ${\mathbb R}^N$ with the homogeneous dynamical boundary condition.