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

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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.

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


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