An extension of Wright’s 3/2-theorem for the KPP-Fisher delayed equation
HTML articles powered by AMS MathViewer
- by Karel Hasik and Sergei Trofimchuk PDF
- Proc. Amer. Math. Soc. 143 (2015), 3019-3027 Request permission
Abstract:
We present a short proof of the following natural extension of Wright’s famous $3/2$-stability theorem: the conditions $\tau \leq 3/2, \ c \geq 2$ imply the presence of the positive traveling fronts (not necessarily monotone) $u = \phi (x\cdot \nu +ct), \ |\nu | =1,$ in the delayed KPP-Fisher equation $u_t(t,x) = \Delta u(t,x) + u(t,x)(1-u(t-\tau ,x)),$ $u \geq 0,$ $x \in \mathbb {R}^m.$References
- P. Ashwin, M. V. Bartuccelli, T. J. Bridges, and S. A. Gourley, Travelling fronts for the KPP equation with spatio-temporal delay, Z. Angew. Math. Phys. 53 (2002), no. 1, 103–122. MR 1889183, DOI 10.1007/s00033-002-8145-8
- Henri Berestycki, Grégoire Nadin, Benoit Perthame, and Lenya Ryzhik, The non-local Fisher-KPP equation: travelling waves and steady states, Nonlinearity 22 (2009), no. 12, 2813–2844. MR 2557449, DOI 10.1088/0951-7715/22/12/002
- Arnaud Ducrot and Grégoire Nadin, Asymptotic behaviour of travelling waves for the delayed Fisher-KPP equation, J. Differential Equations 256 (2014), no. 9, 3115–3140. MR 3171769, DOI 10.1016/j.jde.2014.01.033
- Jian Fang and Jianhong Wu, Monotone traveling waves for delayed Lotka-Volterra competition systems, Discrete Contin. Dyn. Syst. 32 (2012), no. 9, 3043–3058. MR 2912070, DOI 10.3934/dcds.2012.32.3043
- Jian Fang and Xiao-Qiang Zhao, Monotone wavefronts of the nonlocal Fisher-KPP equation, Nonlinearity 24 (2011), no. 11, 3043–3054. MR 2844826, DOI 10.1088/0951-7715/24/11/002
- Teresa Faria, Wenzhang Huang, and Jianhong Wu, Travelling waves for delayed reaction-diffusion equations with global response, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 462 (2006), no. 2065, 229–261. MR 2189262, DOI 10.1098/rspa.2005.1554
- Teresa Faria and Sergei Trofimchuk, Positive travelling fronts for reaction-diffusion systems with distributed delay, Nonlinearity 23 (2010), no. 10, 2457–2481. MR 2683776, DOI 10.1088/0951-7715/23/10/006
- Adrian Gomez and Sergei Trofimchuk, Monotone traveling wavefronts of the KPP-Fisher delayed equation, J. Differential Equations 250 (2011), no. 4, 1767–1787. MR 2763555, DOI 10.1016/j.jde.2010.11.011
- Jack K. Hale, Asymptotic behavior of dissipative systems, Mathematical Surveys and Monographs, vol. 25, American Mathematical Society, Providence, RI, 1988. MR 941371, DOI 10.1090/surv/025
- Karel Hasik and Sergei Trofimchuk, Slowly oscillating wavefronts of the KPP-Fisher delayed equation, Discrete Contin. Dyn. Syst. 34 (2014), no. 9, 3511–3533. MR 3190991, DOI 10.3934/dcds.2014.34.3511
- Balázs Bánhelyi, Tibor Csendes, Tibor Krisztin, and Arnold Neumaier, Global attractivity of the zero solution for Wright’s equation, SIAM J. Appl. Dyn. Syst. 13 (2014), no. 1, 537–563. MR 3183042, DOI 10.1137/120904226
- Man Kam Kwong and Chunhua Ou, Existence and nonexistence of monotone traveling waves for the delayed Fisher equation, J. Differential Equations 249 (2010), no. 3, 728–745. MR 2646048, DOI 10.1016/j.jde.2010.04.017
- Eduardo Liz, Manuel Pinto, Gonzalo Robledo, Sergei Trofimchuk, and Victor Tkachenko, Wright type delay differential equations with negative Schwarzian, Discrete Contin. Dyn. Syst. 9 (2003), no. 2, 309–321. MR 1952376, DOI 10.3934/dcds.2003.9.309
- John Mallet-Paret and George R. Sell, Systems of differential delay equations: Floquet multipliers and discrete Lyapunov functions, J. Differential Equations 125 (1996), no. 2, 385–440. MR 1378762, DOI 10.1006/jdeq.1996.0036
- John Mallet-Paret and George R. Sell, The Poincaré-Bendixson theorem for monotone cyclic feedback systems with delay, J. Differential Equations 125 (1996), no. 2, 441–489. MR 1378763, DOI 10.1006/jdeq.1996.0037
- Grégoire Nadin, Benoît Perthame, and Min Tang, Can a traveling wave connect two unstable states? The case of the nonlocal Fisher equation, C. R. Math. Acad. Sci. Paris 349 (2011), no. 9-10, 553–557 (English, with English and French summaries). MR 2802923, DOI 10.1016/j.crma.2011.03.008
- E. M. Wright, A non-linear difference-differential equation, J. Reine Angew. Math. 194 (1955), 66–87. MR 72363, DOI 10.1515/crll.1955.194.66
- Jianhong Wu and Xingfu Zou, Traveling wave fronts of reaction-diffusion systems with delay, J. Dynam. Differential Equations 13 (2001), no. 3, 651–687. MR 1845097, DOI 10.1023/A:1016690424892
Additional Information
- Karel Hasik
- Affiliation: Mathematical Institute, Silesian University, 746 01 Opava, Czech Republic
- Email: Karel.Hasik@math.slu.cz
- Sergei Trofimchuk
- Affiliation: Instituto de Matemática y Física, Universidad de Talca, Casilla 747, Talca, Chile
- MR Author ID: 211398
- Email: trofimch@inst-mat.utalca.cl
- Received by editor(s): February 5, 2013
- Received by editor(s) in revised form: March 7, 2014
- Published electronically: February 13, 2015
- Additional Notes: This research was realized within the framework of the OPVK program, project CZ.1.07/2.300/20.0002
The second author was also partially supported by FONDECYT (Chile), project 1110309, and by CONICYT (Chile) through PBCT program ACT-56. - Communicated by: Yingfei Yi
- © Copyright 2015 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 143 (2015), 3019-3027
- MSC (2010): Primary 34K10, 35K57; Secondary 92D25
- DOI: https://doi.org/10.1090/S0002-9939-2015-12496-3
- MathSciNet review: 3336626