Convergence to traveling waves in the Fisher-Kolmogorov equation with a non-Lipschitzian reaction term

Abstract

We consider the semi linear Fisher-Kolmogorov-Petrovski-Piscounov equation for the advance of an advantageous gene in biology. Its non-smooth reaction function $f(u)$ allows for the introduction of travelling waves with a new profile. We study existence, uniqueness, and long-time asymptotic behavior of the solutions $u(x,t)$, $(x,t)\in \mathbb{R}\times \mathbb{R}_+$. 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 $x\in \mathbb{R}$) of every solution $u(x,t)$ of the Cauchy problem to a single traveling wave $U(x-ct + ζ)$ as $t\to \infty$. The speed $c$ and the travelling wave $U$ are determined uniquely by $f$, whereas the shift $ζ$ is determined by the initial data.