Vol. 12, No. 6, 2019

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Nonstandard existence proofs for reaction diffusion equations

Connor Olson, Marshall Mueller and Sigurd B. Angenent

Vol. 12 (2019), No. 6, 1015–1034
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.

\vrule height 2.5pt depth -2pt width 26pt To the memory of Terry Millar \vrule height 2.5pt depth -2pt width 26pt

Keywords
nonstandard analysis, partial differential equations, reaction diffusion equations
Mathematical Subject Classification 2010
Primary: 26E35, 35K57
Milestones
Received: 19 September 2018
Revised: 28 March 2019
Accepted: 2 April 2019
Published: 3 August 2019

Communicated by Suzanne Lenhart
Authors
Connor Olson
Department of Mathematics
University of Wisconsin
Madison, WI
United States
Marshall Mueller
Department of Mathematics
Tufts University
Medford, MA
United States
Sigurd B. Angenent
Department of Mathematics
University of Wisconsin
Madison, WI
United States