Structural stability and selection of propagating fronts in semilinear parabolic partial differential equations

Abstract

An alternative viewpoint for the selection problem in propagating front systems is presented. We propose that the selected solution can be identified through analysis of the structural stability of the solutions in question. A solution to a given equation is considered structurally stable if it suffers only an infinitesimal change when the equation (not the solution) is perturbed infinitesimally. Applying the structural stability condition, we identify selected solutions for semilinear parabolic partial differential equations, single-mode equations of the Fisher type, and multiple-mode equations that assume something of a generalized Fisher form. The structural stability condition is confirmed for the subclass of single-mode equations to which the Aronson-Weinberger theorem applies [in Partial Differential Equations and Related Topics, edited by J. A. Goldstein (Springer, Heidelberg, 1975)]. For other single-mode equations and multiple-mode equations, the structural stability condition is confirmed numerically. Equations possessing multiple physically realizable solutions are also studied, and several causes for such behavior are identified. We describe what we believe to be the fundamental feature distinguishing physically realizable and physically unrealizable solutions.