The heat equation with the $L^p$ primitive integral

TL;DR

This paper introduces a new Banach space of distributions derived from $L^p$ functions, solves the heat equation within this space, and establishes existence, smoothness, and uniqueness of solutions with initial data as derivatives of $L^p$ functions.

Contribution

It defines a novel space of distributions based on $L^p$ primitives, solves the heat equation in this space, and proves key properties like smoothness, estimates, and uniqueness.

Findings

Solutions are smooth functions.

Sharp estimates of solutions are obtained.

Uniqueness of solutions is proved.

Abstract

For each $1\leq p<\infty$ a Banach space of integrable Schwartz distributions is defined by taking the distributional derivative of all functions in $L^p({\mathbb R})$. Such distributions can be integrated when multiplied by a function that is the integral of a function in $L^q({\mathbb R})$, where $q$ is the conjugate exponent of $p$. The heat equation on the real line is solved in this space of distributions. The initial data is taken to be the distributional derivative of an $L^p({\mathbb R})$ function. The solutions are shown to be smooth functions. Initial conditions are taken on in norm. Sharp estimates of solutions are obtained and a uniqueness theorem is proved.