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Global stability in a nonlocal reaction-diffusion equation

Dmitri Finkelshtein Orcid Logo, Yuri Kondratiev, Stanislav Molchanov, Pasha Tkachov

Stochastics and Dynamics, Volume: 18, Issue: 05, Start page: 1850037

Swansea University Author: Dmitri Finkelshtein Orcid Logo

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DOI (Published version): 10.1142/S0219493718500375

Abstract

We study stability of stationary solutions for a class of nonlocal semi-linear parabolic equations. To this end, we prove a Feynman-Kac type formula for a Lévy processes with time-dependent potentials and arbitrary initial condition. We propose sufficient conditions for asymptotic stability of the z...

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Published in: Stochastics and Dynamics
ISSN: 0219-4937 1793-6799
Published: World Scientific 2018
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URI: https://cronfa.swan.ac.uk/Record/cronfa35092
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Abstract: We study stability of stationary solutions for a class of nonlocal semi-linear parabolic equations. To this end, we prove a Feynman-Kac type formula for a Lévy processes with time-dependent potentials and arbitrary initial condition. We propose sufficient conditions for asymptotic stability of the zero solution, and use them to the study of the spatial logistic equation arising in population ecology. For this equation, we find conditions which imply that its positive stationary solution is asymptotically stable. We consider also the case when the initial condition is given by a random field.
Keywords: Nonlocal diffusion; Feynman--Kac formula; L\'{e}vy processes; Reaction-diffusion equation; Semilinear parabolic equation; Monostable equation; Nonlocal nonlinearity
College: Faculty of Science and Engineering
Issue: 05
Start Page: 1850037

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