## Involve: A Journal of Mathematics

- Involve
- Volume 12, Number 6 (2019), 1015-1034.

### Nonstandard existence proofs for reaction diffusion equations

Connor Olson, Marshall Mueller, and Sigurd B. Angenent

#### 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${}^{\ast}$-compactness of the unit ball in the dual of a Banach space.

#### Article information

**Source**

Involve, Volume 12, Number 6 (2019), 1015-1034.

**Dates**

Received: 19 September 2018

Revised: 28 March 2019

Accepted: 2 April 2019

First available in Project Euclid: 13 August 2019

**Permanent link to this document**

https://projecteuclid.org/euclid.involve/1565661770

**Digital Object Identifier**

doi:10.2140/involve.2019.12.1015

**Mathematical Reviews number (MathSciNet)**

MR3990795

**Zentralblatt MATH identifier**

07116067

**Subjects**

Primary: 26E35: Nonstandard analysis [See also 03H05, 28E05, 54J05] 35K57: Reaction-diffusion equations

**Keywords**

nonstandard analysis partial differential equations reaction diffusion equations

#### Citation

Olson, Connor; Mueller, Marshall; Angenent, Sigurd B. Nonstandard existence proofs for reaction diffusion equations. Involve 12 (2019), no. 6, 1015--1034. doi:10.2140/involve.2019.12.1015. https://projecteuclid.org/euclid.involve/1565661770