A Computer-Assisted Stability Proof for a Stationary Solution of Reaction-Diffusion Equations

TL;DR

This paper presents a computer-assisted method to rigorously verify the stability of stationary solutions in reaction-diffusion equations using eigenvalue exclusion techniques and numerical verification.

Contribution

It introduces a novel eigenvalue exclusion approach combined with numerical verification to rigorously establish stability of solutions in reaction-diffusion systems.

Findings

Successfully verified stability of a stationary solution

Developed computable criteria for eigenvalue exclusion

Proved invertibility of relevant operators

Abstract

The main subject of this paper is a computer assisted stability proof for a stationary solution of reaction diffusion equations in one dimensional space. We use Nakao's numerical verification method to enclose a stationary solution of reaction-diffusion equations. Considering the linearized stability of the solution, a method of excluding eigenvalues in a half plane is adopted. We first focus on the eigenvalues for an operator linearized at an approximate solution. An excluding theorem is presented such that we know under some condition, and there is no eigenvalue in some disks. Some computable criteria are constructed to apply the theorem in a computer. And also the invertibility of some operator is proved theoretically in the paper. However, we need the information of the eigenvalues for the operator linearized at the exact solution. This can be obtained by combining with the verification results of the solution. Then we judge the stability of the solution from the domain where the eigenvalues are located. At last there are some verification results.