Stability Results for Bounded Stationary Solutions of Reaction-Diffusion-ODE Systems

2022/01/30 by Chris Kowall, Kowall, Chris, Anna Marciniak‐Czochra +3
Computer Science · Engineering · Mathematics · #35B35 (primary) 35B36 #35K57 #35P05 #47D06 (secondary) #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Equations and Numerical Methods #FOS: Mathematics #Spectral Theory (math.SP) #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.2201.12748

openalex publication_date 2022/01/30 · openalex created_date 2022/05/05 · openalex updated_date 2026/07/28

Abstract

Reaction-diffusion equations coupled to ordinary differential equations (ODEs) may exhibit spatially low-regular stationary solutions. This work provides a comprehensive theory of asymptotic stability of bounded, discontinuous or continuous, stationary solutions of reaction-diffusion-ODE systems. We characterize the spectrum of the linearized operator and relate its spectral properties to the corresponding semigroup properties. Considering the function spaces L^∞(Ω)^m+k, L^∞(Ω)^m × C(Ω)^k and C(Ω)^m+k, we establish a sign condition on the spectral bound of the linearized operator, which implies nonlinear stability or instability of the stationary pattern.

Related

• Stable discontinuous stationary solutions to reaction-diffusion-ODE systems
• Instability of all regular stationary solutions to reaction-diffusion-ODE systems
• Stability and instability of constant stationary solutions to some integro-differential equations
• Stability of stationary reaction diffusion-degenerate Nagumo fronts I: spectral analysis
• On the instability of threshold solutions of reaction-diffusion equations, and applications to optimization problems