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

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 \textit{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 Poincar\'e'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.