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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Uniqueness of travelling waves for nonlocal monostable equations


Authors: Jack Carr and Adam Chmaj
Journal: Proc. Amer. Math. Soc. 132 (2004), 2433-2439
MSC (2000): Primary 92D15, 39B99, 45G10
Published electronically: March 4, 2004
MathSciNet review: 2052422
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Abstract | References | Similar Articles | Additional Information

Abstract: We consider a nonlocal analogue of the Fisher-KPP equation

\begin{displaymath}u_t =J*u-u+f(u),~x\in R,~f(0)=f(1)=0,~f>0 ~{\rm on}~(0,1),\end{displaymath}

and its discrete counterpart ${\dot u}_n =(J*u)_n -u_n +f(u_n )$, $n\in Z$, and show that travelling wave solutions of these equations that are bounded between $0$ and $1$ are unique up to translation. Our proof requires finding exact a priori asymptotics of a travelling wave. This we accomplish with the help of Ikehara's Theorem (which is a Tauberian theorem for Laplace transforms).


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Additional Information

Jack Carr
Affiliation: Department of Mathematics, Heriot-Watt University, Riccarton, Edinburgh, EH14 4AS, UK
Email: j.carr@ma.hw.ac.uk

Adam Chmaj
Affiliation: Department of Mathematics, Heriot-Watt University, Riccarton, Edinburgh, EH14 4AS, UK
Address at time of publication: Department of Mathematics, Michigan State University, East Lansing, Michigan 48824
Email: chmaj@math.msu.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-04-07432-5
PII: S 0002-9939(04)07432-5
Received by editor(s): August 6, 2002
Received by editor(s) in revised form: May 7, 2003
Published electronically: March 4, 2004
Additional Notes: This work was supported by a Marie Curie Fellowship of the European Community IHP programme under contract number HPMFCT-2000-00465 and in part by NSF grant DMS-0096182
Communicated by: Mark J. Ablowitz
Article copyright: © Copyright 2004 American Mathematical Society