Nonstandard existence proofs for reaction diffusion equations

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.