On the instability of threshold solutions of reaction-diffusion equations, and applications to optimization problems

2023/11/13 by Grégoire Nadin, Nadin, Grégoire · 1 citation
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Equations and Numerical Methods #FOS: Mathematics #Nonlinear Partial Differential Equations #Optimization and Control (math.OC)

paper · doi:10.48550/arxiv.2311.07154

openalex publication_date 2023/11/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The first part of this paper is devoted to the derivation of a technical result, related to the stability of the solution of a reaction-diffusion equation u_t-Δu = f(x,u) on (0,∞)× ℝ^N, where the initial datum u(0,x)=u₀(x) is such that lim_{t→ +∞} u(t,x)=W(x) for all x, with W a steady state in H¹(ℝ^N). We characterize the perturbations h such that, if u^h is the solution associated with the initial datum u₀+h, then, if h is small enough in a sense, one has u^h(t,x)>W(x) (resp. u(t,x)0 for all x.

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• Existence of two thresholds in a bistable equation with nonlocal competition

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