Convergence to travelling waves in Fisher's population genetics model with a non-Lipschitzian reaction term.

Abstract

We consider a one-dimensional population genetics model for the advance of an advantageous gene. The model is described by the semilinear Fisher equation with unbalanced bistable non-Lipschitzian nonlinearity f(u). The "nonsmoothness" of f allows for the appearance of travelling waves with a new, more realistic profile. We study existence, uniqueness, and long-time asymptotic behavior of the solutions u(x,t), [Formula: see text]. We prove also the existence and uniqueness (up to a spatial shift) of a travelling wave U. Our main result is the uniform convergence (for [Formula: see text]) of every solution u(x,t) of the Cauchy problem to a single travelling wave [Formula: see text] as [Formula: see text]. The speed c and the travelling wave U are determined uniquely by f, whereas the shift [Formula: see text] is determined by the initial data.