$L^p$ estimate of the heat equation on a bounded domain

TL;DR

This paper establishes $L^p$ estimates for derivatives of solutions to the heat equation on bounded domains, including endpoint cases, providing a comprehensive proof for these bounds in Lebesgue spaces.

Contribution

It offers a self-contained proof of $L^p$ derivative estimates for the heat equation, covering all relevant endpoint cases on bounded domains.

Findings

Derives $L^p$ estimates for derivatives up to second order

Includes endpoint cases $p=1$ and $p= $

Provides a comprehensive, self-contained proof

Abstract

We consider the linear heat equation on a bounded domain. We study estimates of the derivatives, up to the second order, of the solution locally in time in the Lebesgue spaces. We give a self-contained proof of the estimates in the end-point cases $p = 1, \infty$.