A rigorous reduction of the L2-stability of the solutions to a nonlinear binary reaction–diffusion system of PDE's to the stability of the solutions to a linear binary system of ODE's

A basic peculiar Lyapunov functional V is introduced for the dynamical systems generated by a pair of nonlinear reaction–diffusion PDE's, with nonconstant coefficients. The sign of V and of its derivative along the solutions is linked—through an immediate simple relation—to the eigenvalues. By using V and the L2-norm, the non-linear L2-stability (instability) is rigorously reduced to the stability (instability) of the solutions to a linear binary system of ODE's.