Stationary points of the “reaction-diffusion” equation and transitions to stable states

Of concern is a stationary “reaction-diffusion” equation with cubic non-linearity is Neumann boundary conditions and fixed average value of the desired bifurcating solutions. A method of approximate calculation of bifurca-ting solutions for small and finite values of supercritical parameter increment are presented. Computing is based on the Lyapunov–Schmidt reducing procedure and is leaning on key functions Ritz' approximation of the set of eigenfunctions (modes) of main linear part of gradient energy functional. A technique of evaluating of a functional space size, where Lyapunov–Schmidt reduction can be applied is performed. In case of local reduction the main part of the key function has been found and asymptotic presentation of bifurcating solutions for small supercritical increment of bifurcation parameter is calculated. The relation between solutions search procedures for “reaction-diffusion” equations and Cahn–Hilliard equation (with extended Neumann boundary conditions) is also performed. Graphs are presented.