Uniqueness of travelling waves for nonlocal monostable equations

Proceedings of the American Mathematical Society - Tập 132 Số 8 - Trang 2433-2439
Jack Carr1, Adam Chmaj1,2
1Department of Mathematics, Heriot-Watt University, Riccarton, Edinburgh, EH14 4AS, UK
2Department of Mathematics, Michigan State University, East Lansing, Michigan, 48824

Tóm tắt

We consider a nonlocal analogue of the Fisher-KPP equation \[ u t = J ∗ u − u + f ( u ) ,   x ∈ R ,   f ( 0 ) = f ( 1 ) = 0 ,   f > 0   on   ( 0 , 1 ) , u_t =J*u-u+f(u),~x\in R,~f(0)=f(1)=0,~f>0 ~\textrm {on}~(0,1), \] and its discrete counterpart u ˙ n = ( J ∗ u ) n − u n + f ( u n ) {\dot u}_n =(J*u)_n -u_n +f(u_n ) , n ∈ Z n\in Z , and show that travelling wave solutions of these equations that are bounded between 0 0 and 1 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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Tài liệu tham khảo

Atkinson, C., 1976, Deterministic epidemic waves, Math. Proc. Cambridge Philos. Soc., 80, 315, 10.1017/S0305004100052944

Bates, Peter W., 1999, A discrete convolution model for phase transitions, Arch. Ration. Mech. Anal., 150, 281, 10.1007/s002050050189

Bramson, Maury, 1983, Convergence of solutions of the Kolmogorov equation to travelling waves, Mem. Amer. Math. Soc., 44, iv+190, 10.1090/memo/0285

Brown, K. J., 1977, Deterministic epidemic waves of critical velocity, Math. Proc. Cambridge Philos. Soc., 81, 431, 10.1017/S0305004100053494

Chen, Xinfu, 2003, Uniqueness and existence of traveling waves for discrete quasilinear monostable dynamics, Math. Ann., 326, 123, 10.1007/s00208-003-0414-0

Diekmann, Odo, 1978, On the bounded solutions of a nonlinear convolution equation, Nonlinear Anal., 2, 721, 10.1016/0362-546X(78)90015-9

Ellison, William, 1985, Prime numbers

Fife, Paul C., 1979, Mathematical aspects of reacting and diffusing systems, 28, 10.1007/978-3-642-93111-6

R. A. Fisher, The advance of advantageous genes, Ann. Eugenics 7 (1937), 355-369.

1988, Dynamics of curved fronts

Schumacher, Konrad, 1980, Travelling-front solutions for integro-differential equations. I, J. Reine Angew. Math., 316, 54, 10.1515/crll.1980.316.54

Weinberger, H. F., 1978, Asymptotic behavior of a model in population genetics, 47

Sispanov, Sergio, 1939, Generalización del teorema de Laguerre, Bol. Mat., 12, 113

Wu, Jianhong, 2001, Traveling wave fronts of reaction-diffusion systems with delay, J. Dynam. Differential Equations, 13, 651, 10.1023/A:1016690424892

Zinner, B., 1993, Traveling wavefronts for the discrete Fisher’s equation, J. Differential Equations, 105, 46, 10.1006/jdeq.1993.1082

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Proceedings of the American Mathematical Society

Tập 132 Số 8

2433-2439

Thông tin tác giả