Inaccessible States in Time-Dependent Reaction Diffusion

Using asymptotic methods we show that the long‐time dynamic behavior in certain systems of nonlinear parabolic differential equations is described by a time‐dependent, spatially inhomogeneous nonlinear evolution equation. For problems with multiple stable states, the solution develops sharp fronts separating slowly varying regions. By studying the basins of attraction of Abel's nonlinear differential equation, we demonstrate that the presence of explicit time dependence in the asymptotic evolution equation creates "forbidden regions" where the existence of interfaces is excluded. Consequently, certain configurations of stable states in the nonlinear system become inaccessible and cannot be achieved from any set of real initial conditions.