Solvability of the heat equation on a half-space with a dynamical boundary condition and unbounded initial data

TL;DR

This paper investigates the solvability of the heat equation on a half-space with a dynamic boundary condition, extending previous results to include initial data in weighted Lebesgue spaces beyond bounded functions.

Contribution

It introduces conditions on initial data in weighted Lebesgue spaces that ensure the existence of solutions for the heat equation with dynamical boundary conditions.

Findings

Solvability is guaranteed for initial data in weighted Lebesgue spaces.

Extension of known results from bounded to unbounded initial data.

Identification of appropriate function spaces for solution existence.

Abstract

We study the linear heat equation on a halfspace with a linear dynamical boundary condition. We are interested in an appropriate choice of the function space of initial functions such that the problem possesses a solution. It was known before that bounded initial data guarantee solvability. Here we extend that result by showing that data from a weighted Lebesgue space will also do so.