Solutions to certain classes of linearized reaction–diffusion equations

A systematic study is presented for the linear manifold of solutions to a generic system of reaction–diffusion equations in the neighborhood of a constant uniform (equilibrium) solution. The theory pertains directly to an arbitrary number of reacting and diffusing molecular or biological species in an arbitrary bounded spatial (1-, 2- or 3-dimensional) region with an impermeable boundary, so that the normal gradient of any species concentration function is zero at all boundary points. The stability analysis developed by previous authors is streamlined here for the case of two reacting and diffusing species, worked out completely for the case of three species, and made more amenable to specialized treatment for cases with four or more species. With the use of modern algebraic computational methods, explicit analytical general solutions to the linearized reaction–diffusion equations are derived for certain classes of model theories. These results either apply directly or admit extension to a wide range of practical reaction–diffusion problems in physical chemistry and biology.