  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.
