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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
first_indexed 2017-09-04T18:57:30Z
last_indexed 2019-02-11T11:35:03Z
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spelling 2019-02-07T20:21:51.9302387 v2 35092 2017-09-04 Global stability in a nonlocal reaction-diffusion equation 4dc251ebcd7a89a15b71c846cd0ddaaf 0000-0001-7136-9399 Dmitri Finkelshtein Dmitri Finkelshtein true false 2017-09-04 MACS 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. Journal Article Stochastics and Dynamics 18 05 1850037 World Scientific 0219-4937 1793-6799 Nonlocal diffusion; Feynman--Kac formula; L\&apos;{e}vy processes; Reaction-diffusion equation; Semilinear parabolic equation; Monostable equation; Nonlocal nonlinearity 28 9 2018 2018-09-28 10.1142/S0219493718500375 http://www.worldscientific.com/worldscinet/sd COLLEGE NANME Mathematics and Computer Science School COLLEGE CODE MACS Swansea University 2019-02-07T20:21:51.9302387 2017-09-04T15:13:35.8891618 Faculty of Science and Engineering School of Mathematics and Computer Science - Mathematics Dmitri Finkelshtein 0000-0001-7136-9399 1 Yuri Kondratiev 2 Stanislav Molchanov 3 Pasha Tkachov 4 0035092-04092017152239.pdf FKMT-ArXiv-v2.pdf 2017-09-04T15:22:39.8970000 Output 446968 application/pdf Accepted Manuscript true 2017-09-05T00:00:00.0000000 true eng
title Global stability in a nonlocal reaction-diffusion equation
spellingShingle Global stability in a nonlocal reaction-diffusion equation
Dmitri Finkelshtein
title_short Global stability in a nonlocal reaction-diffusion equation
title_full Global stability in a nonlocal reaction-diffusion equation
title_fullStr Global stability in a nonlocal reaction-diffusion equation
title_full_unstemmed Global stability in a nonlocal reaction-diffusion equation
title_sort Global stability in a nonlocal reaction-diffusion equation
author_id_str_mv 4dc251ebcd7a89a15b71c846cd0ddaaf
author_id_fullname_str_mv 4dc251ebcd7a89a15b71c846cd0ddaaf_***_Dmitri Finkelshtein
author Dmitri Finkelshtein
author2 Dmitri Finkelshtein
Yuri Kondratiev
Stanislav Molchanov
Pasha Tkachov
format Journal article
container_title Stochastics and Dynamics
container_volume 18
container_issue 05
container_start_page 1850037
publishDate 2018
institution Swansea University
issn 0219-4937
1793-6799
doi_str_mv 10.1142/S0219493718500375
college_str Faculty of Science and Engineering
hierarchytype
hierarchy_top_id facultyofscienceandengineering
hierarchy_top_title Faculty of Science and Engineering
hierarchy_parent_id facultyofscienceandengineering
hierarchy_parent_title Faculty of Science and Engineering
department_str School of Mathematics and Computer Science - Mathematics{{{_:::_}}}Faculty of Science and Engineering{{{_:::_}}}School of Mathematics and Computer Science - Mathematics
url http://www.worldscientific.com/worldscinet/sd
document_store_str 1
active_str 0
description 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.
published_date 2018-09-28T19:17:47Z
_version_ 1821524851042549760
score 11.047674

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