Doubly nonlocal Fisher–KPP equation: front propagation

ABSTRACT

We study propagation over ${\mathbb{R}}^d$ of the solution to a doubly nonlocal reaction–diffusion equation of the Fisher–KPP-type with anisotropic kernels. We present both necessary and sufficient conditions which ensure linear in time propagation of the solution in a direction. For kernels with a finite exponential moment over ${\mathbb{R}}^d$ we prove front propagation in all directions for a general class of initial conditions decaying in all directions faster than any exponential function (that includes, for the first time in the literature about the considered type of equations, compactly supported initial conditions).

Keywords:

• Nonlocal diffusion
• Fisher–KPP equation
• nonlocal nonlinearity
• long-time behavior
• front propagation
• anisotropic kernels
• integral equation

2010 Mathematics Subject Classifications:

• 35K55
• 35K57
• 35B40

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

Dmitri Finkelshtein  http://orcid.org/0000-0001-7136-9399

Pasha Tkachov  http://orcid.org/0000-0002-6773-4506

Additional information

Funding

Authors gratefully acknowledge the financial support by the Deutsche Forschungsgemeinschaft (German Research Foundation) through CRC 701 “Stochastic Dynamics: Mathematical Theory and Applications” (DF, YK, PT), the European Commission under the project STREVCOMS PIRSES-2013-612669 (DF, YK), and the “Bielefeld Young Researchers” Fund through the Funding Line Postdocs: “Career Bridge Doctorate – Postdoc” (PT).