Electron. J. Differential Equations, Vol. 2019 (2019), No. 10, pp. 1-27.

Existence and properties of traveling waves for doubly nonlocal Fisher-KPP equations

Dmitri Finkelshtein, Yuri Kondratiev, Pasha Tkachov

We consider a reaction-diffusion equation with nonlocal anisotropic diffusion and a linear combination of local and nonlocal monostable-type reactions in a space of bounded functions on R^d. Using the properties of the corresponding semiflow, we prove the existence of monotone traveling waves along those directions where the diffusion kernel is exponentially integrable. Among other properties, we prove continuity, strict monotonicity and exponential integrability of the traveling wave profiles.

Submitted July 2, 2018. Published Janaury 22, 2019.
Math Subject Classifications: 35C07, 35K57, 45G10.
Key Words: Nonlocal diffusion; reaction-diffusion equation; Fisher-KPP equation; traveling waves; nonlocal nonlinearity; anisotropic kernels; integral equation.

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Dmitri Finkelshtein
Department of Mathematics
Swansea University, Bay Campus
Fabian Way, Swansea SA1 8EN, UK
email: d.l.finkelshtein@swansea.ac.uk
Yuri Kondratiev
Fakultät für Mathematik
Universität Bielefeld, Postfach 110 131
33501 Bielefeld, Germany
email: kondrat@math.uni-bielefeld.de
Pasha Tkachov
Gran Sasso Science Institute
Viale Francesco Crispi, 7
67100 L'Aquila AQ, Italy
email: pasha.tkachov@gssi.it

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