Global stability of noncritical traveling front solutions of Fisher-type equations with degenerate nonlinearity

Abstract

In this paper, for degenerate n-degree Fisher-type equations, we discuss the stability of their traveling front solutions with noncritical speeds. In fact, when the initial perturbations around these noncritical traveling front solutions are in some weighted Banach spaces, we have proved that these solutions are globally exponentially stable in the form of (1+t)13e−νt for ν ∈ (0, 1) via L1-energy estimates, L2-energy estimates, and the weighted energy method. Furthermore, by Fourier transform and the weighted energy method, we will prove that traveling front solutions with noncritical speeds are also globally exponentially stable in the form of t−12e−νt for some positive constant ν when the initial perturbations around these solutions are in some weighted Sobolev spaces. Our conclusions extend the local stability of noncritical traveling front solutions into the global case and also give some novel forms of exponential stability of these solutions.