Global theorems for species distributions governed by reaction-diffusion equations

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Abstract
For an essentially nonlinear reaction‐diffusion equation of the form ∂c/∂t = D▿2c + Q(c) supplemented with a Robin type boundary condition over the surface of a closed bounded three‐dimensional region R, it is shown that any solution for the concentration distribution c=c(x, t) in R is restricted to a prescribed Hilbert space neighborhood of any steady state solution c̄ = c̄(x) for a generic reaction rate expression Q(c). Under certain derived conditions, the Hilbert space neighborhood shrinks to zero with increasing time, implying that there is only one unique steady state solution, while under more general circumstances a Hilbert space proximity condition is obtained for any pair of steady state solutions. By introducing a special functional that is constant with time for a steady state solution and monotone‐decreasing otherwise, necessary conditions for a dynamical transition between a pair of steady state solutions are deduced. From the latter results follow sufficient conditions which guarantee the uniqueness of the steady state solution and a lower bound on the transition time for dynamical evolution from one (perturbed unstable) steady state solution to another in cases of multiplicity.