Uniqueness of travelling waves for nonlocal monostable equations

Carr, Jack; Chmaj, Adam

doi:10.1090/S0002-9939-04-07432-5

Uniqueness of travelling waves for nonlocal monostable equations
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by Jack Carr and Adam Chmaj

Proc. Amer. Math. Soc. 132 (2004), 2433-2439

DOI: https://doi.org/10.1090/S0002-9939-04-07432-5

Published electronically: March 4, 2004

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Abstract:

We consider a nonlocal analogue of the Fisher-KPP equation \[ u_t =J*u-u+f(u),~x\in R,~f(0)=f(1)=0,~f>0 ~\textrm {on}~(0,1),\] 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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