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Asymptotic behaviour of solutions to one-dimensional reaction diffusion cooperative systems involving infinitesimal generators

  • Miguel Yangari EMAIL logo


The aim of this paper is to study the large-time behaviour of mild solutions to the one-dimensional cooperative systems with anomalous diffusion when at least one entry of the initial condition decays slower than a power. We prove that the solution moves at least exponentially fast as time goes to infinity. Moreover, the exponent of propagation depends on the decay of the initial condition and of the reaction term.


1 D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusions arising in population genetics, Adv. Math. 30 (1978), 33–76. 10.1016/0001-8708(78)90130-5Search in Google Scholar

2 X. Cabre and J. Roquejoffre, The influence of fractional diffusion in Fisher–KPP equation, Comm. Math. Phys. 320 (2013), 679–722. 10.1007/s00220-013-1682-5Search in Google Scholar

3 A.-C. Coulon and M. Yangari, Exponential propagation for fractional reaction-diffusion cooperative systems with fast decaying initial conditions, J. Dynam. Differential Equations (2015), 10.1007/s10884-015-9479-1. 10.1007/s10884-015-9479-1Search in Google Scholar

4 P. Felmer and M. Yangari, Fast propagation for fractional KPP equations with slowly decaying initial conditions, SIAM J. Math. Anal. 45 (2013), 2, 662–678. 10.1137/120879294Search in Google Scholar

5 F. Hamel and L. Roques, Fast propagation for KPP equations with slowly decaying initial conditions, J. Differential Equations 249 (2010), 1726–1745. 10.1016/j.jde.2010.06.025Search in Google Scholar

6 A. N. Kolmogorov, I. G. Petrovsky and N. S. Piskunov, Étude de l'equation de la diffusion avec croissance de la quantité de matiére et son application á un probléme biologique, Bull. Univ. État Moscou Sér. Inter. A 1 (1937), 1–26. Search in Google Scholar

7 M. Lewis, B. Li and H. Weinberger, Spreading speed and linear determinacy for two-species competition models, J. Math. Biol. 45 (2002), 219–233. 10.1007/s002850200144Search in Google Scholar

8 R. Lui, Biological growth and spread modeled by systems of recursions. I. Mathematical theory, Math. Biosci. 93 (1989), 2, 269–295. 10.1016/0025-5564(89)90026-6Search in Google Scholar

9 H. F. Weinberger, M. Lewis and B. Li, Analysis of linear determinacy for spread in cooperative models, J. Math. Biol. 45 (2002), 183–218. 10.1007/s002850200145Search in Google Scholar PubMed

10 H. F. Weinberger, M. Lewis and B. Li, Anomalous spreading speeds of cooperative recursion systems, J. Math. Biol. 55 (2007), 207–222. 10.1007/s00285-007-0078-6Search in Google Scholar PubMed

11 M. Yangari, Existence and uniqueness of global mild solutions for nonlocal Cauchy systems in Banach spaces, Rev. Politécnica 35 (2015), 2, 149–152. Search in Google Scholar

Received: 2015-3-30
Revised: 2015-10-26
Accepted: 2015-11-12
Published Online: 2016-5-5
Published in Print: 2016-6-1

© 2016 by De Gruyter

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