Nonstandard existence proofs for reaction diffusion equations

TL;DR

This paper presents a nonstandard analysis approach to proving the existence of distribution solutions for reaction diffusion equations, highlighting differences and similarities with standard methods.

Contribution

It introduces a novel nonstandard proof technique that replaces traditional compactness theorems in reaction diffusion equations.

Findings

Nonstandard methods can effectively establish existence of solutions.

Standard part operation replaces classical compactness theorems.

Approach simplifies certain aspects of existence proofs.

Abstract

We give an existence proof for distribution solutions to a scalar reaction diffusion equation, with the aim of illustrating both the differences and the common ingredients of the nonstandard and standard approaches. In particular, our proof shows how the operation of taking the standard part of a nonstandard real number can replace several different compactness theorems, such as Ascoli's theorem and the Banach--Alaoglu theorem on weak$^*$-compactness of the unit ball in the dual of a Banach space.