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Zbl 1061.45003
Carr, Jack; Chmaj, Adam
Uniqueness of travelling waves for nonlocal monostable equations. (English)
[J] Proc. Am. Math. Soc. 132, No. 8, 2433-2439 (2004). ISSN 0002-9939; ISSN 1088-6826/e

The authors study uniqueness of travelling waves of two nonlocal models. One is the integro-differential equation $$u_t= J* u- u+ f(u),\tag1$$ where $J* u(z)= \int_{\bbfR^1} J(z-y) u(y)\,dy$, with $J\ge 0$, even, compactly supported and $\int_{\bbfR^1} J=1$. The second model is a discrete version of (1), namely, an infinite ordinary differential equation system $$\dot u_n= (J* u)_n- u_n+ f(u_n),\quad n\in\bbfZ,$$ where $(J* u)_n= \sum_{|i|\ge 1} J(n- i)u_i$ and $J$ and $f$ as in (1). To this end the authors use a Tauberian theorem for the Laplace transform.
[Messoud A. Efendiev (Berlin)]

MSC 2000:: *45J05 Integro-ordinary differential equations
45G10 Nonsingular nonlinear integral equations
34A34 Nonlinear ODE and systems, general
45M05 Asymptotic theory of integral equations

Keywords: uniqueness; travelling wave; Tauberian theorem; integro-differential equation; infinite ordinary differential equation system; Laplace transform

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