A Computer-Assisted Stability Proof for a Stationary Solution of Reaction-Diffusion Equations
This work provides a rigorous computational framework for proving stability of stationary solutions in reaction-diffusion equations, which is important for applied mathematicians and scientists studying pattern formation.
The paper presents a computer-assisted proof of stability for a stationary solution of one-dimensional reaction-diffusion equations by combining Nakao's numerical verification method with eigenvalue exclusion techniques. The method successfully verifies stability for a specific solution, demonstrating the feasibility of rigorous computer-assisted proofs for such problems.
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.