Multidimensional transition fronts for Fisher-KPP reactions

Abstract

We study entire solutions to homogeneous reaction-diffusion equations in several dimensions with Fisher-KPP reactions. Any entire solution $0<u<1$ is known to satisfy \[ \lim_{t\to -\infty} \sup_{|x|\le c|t|} u(t,x) = 0 \qquad \text{for each $c<2\sqrt{f^{\prime }(0)}$,} \] and we consider here those satisfying \[ \lim_{t\to -\infty} \sup_{|x|\le c|t|} u(t,x) = 0 \qquad \text{for some $c>2\sqrt{f^{\prime }(0)}$.} \] When $f$ is $C^2$ and concave, our main result provides an almost complete characterization of transition fronts as well as transition solutions with bounded width within this class of solutions.

17 pages

References in corpus (2)

• Transition fronts for inhomogeneous monostable reaction-diffusion equations via linearization at zero
• Transition Fronts in Time Heterogeneous and Random Media of Ignition Type