Forced waves of the Fisher–KPP equation in a shifting environment

Abstract

This paper concerns the equation(0.1)ut=uxx+f(x−ct,u),x∈R, where c≥0 is a forcing speed and f:(s,u)∈R×R+→R is asymptotically of KPP type as s→−∞. We are interested in the questions of whether such a forced moving KPP nonlinearity from behind can give rise to traveling waves with the same speed and how they attract solutions of initial value problems when they exist. Under a sublinearity condition on f(s,u), we obtain the complete existence and multiplicity of forced traveling waves as well as their attractivity except for some critical cases. In these cases, we provide examples to show that there is no definite answer unless one imposes further conditions depending on the heterogeneity of f in s∈R.