A mathematical framework for determining the stability of steady states of reaction–diffusion equations with periodic source terms

Abstract

We develop a mathematical framework for determining the stability of steady states of generic nonlinear reaction-diffusion equations with periodic source terms, in one spatial dimension. We formulate an a priori condition for the stability of such steady states, which relies only on the properties of the steady state itself. The mathematical framework is based on Bloch's theorem and Poincaré's inequality for mean-zero periodic functions. Our framework can be used for stability analysis to determine the regions in an appropriate parameter space for which steady-state solutions are stable.

Travelling-wave spatially periodic forcing of asymmetric binary mixtures

“…As the full temporally-evolving Cahn-Hilliard equation with sinusoidal forcing preserves the mean concentration ⟨ψ⟩, it follows that ⟨δC⟩ = 0 for all time. Therefore the boundary conditions on δC(x, t) are either (i) periodic, with δC(η+L, t) = δC(η, t), or (ii) bounded, with δC → 0 as η → ∞ (and the same for the η-derivatives of δC) or In this paper, we deal with Case (i) only, for the following reasons: this case is simple, and it can be used to shed light on the numerical simulations below in Section V. Also, the analysis developed in Case (i) may be combined with the theory developed in Reference [21], such that Case (ii) may be considered an extension of Case (i). As such, we focus in the rest of this section on periodic perturbations, with mean zero, specifically,…”