Acta mathematica scientia, Series B >
STABILITY OF TIME-PERIODIC TRAVELING FRONTS IN BISTABLE REACTION-ADVECTION-DIFFUSION EQUATIONS
Received date: 2015-03-13
Revised date: 2015-05-28
Online published: 2016-06-25
Supported by
This work was supported by National Natural Science Foundation of China (11401134), China Postdoctoral Science Foundation Funded Project (2012M520716), the Fundamental Research Funds for the Central Universities (HIT.NSRIF.2014063)
This paper is concerned with the global exponential stability of time periodic traveling fronts of reaction-advection-diffusion equations with time periodic bistable nonlinearity in infinite cylinders. It is well known that such traveling fronts exist and are asymptotically stable. In this paper, we further show that such fronts are globally exponentially stable. The main difficulty is to construct appropriate supersolutions and subsolutions.
Weijie SHENG . STABILITY OF TIME-PERIODIC TRAVELING FRONTS IN BISTABLE REACTION-ADVECTION-DIFFUSION EQUATIONS[J]. Acta mathematica scientia, Series B, 2016 , 36(3) : 802 -814 . DOI: 10.1016/S0252-9602(16)30041-8
[1] Alikakos N, Bates P W, Chen X. Periodic traveling waves and locating oscillating patterns in multidimensional domains. Trans Amer Math Soc, 1999, 351: 2777-2805
[2] Zhao G Y. Multidimensional periodic traveling waves in infinite cylinders. Discrete Contnu Dyn Syst, 2009, 24: 1025-1045
[3] Aronson D G, Weinberger H F. Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation//Goldstein J A, ed. Partial differential Equations and Related Topics. Lecture Notes in Math, 446. Springer, 1975: 5-49
[4] Aronson D G, Weinberger H F. Multidimensional nonlinear diffusions arising in population genetics. Adv Math, 1978, 30: 33-76
[5] Berestycki H, Nirenberg L. Traveling fronts in cylinders. Ann Inst H Poincaré Anal Non Linéaire, 1992, 9: 497-572
[6] Berestycki H, Hamel F. Front propagation in periodic excitable media. Comm Pure Appl Math, 2002, 55949-1032
[7] Hamel F. Qualitative properties of monostable pulsating fronts: exponential decay and monotonicity. J Math Pures Appl, 2008, 89: 355-399
[8] Nadin G. Traveling fronts in space-time periodic media. J Math Pures Appl, 2009, 92: 232-262
[9] Berestycki H, Hamel F, Matano H. Bistable travelling waves around an obstacle. Comm Pure Appl Math, 2009, 62: 729-788
[10] Berestycki H, Rossi L. Reaction-diffusion equations for population dynamics with forced speed I-The case of the whole space. Discrete Contin Dyn Syst, 2008, 21: 41-67
[11] Berestycki H, Rossi L. Reaction-diffusion equations for population dynamics with forced speed II-cylindricaltype domains. Discrete Contin Dyn Syst, 2009, 25: 19-61
[12] Hamel F, Roques L. Uniqueness and stability properties of monostable pulsating fronts. J Eur Math Soc, 2011, 13: 345-390
[13] Vega J M. Travelling wavefronts of reaction-diffusion equations in cylindrical domains. Comm Partial Differential Equations, 1993, 18: 505-531
[14] Vega, J M. The asymptotic behavior of the solutions of some semilinear elliptic equations in cylindrical domains. J Differential Equations, 1993, 102: 119-152
[15] Hamel F, Nadirashvili N. Entire solutions of the KPP equation. Comm Pure Appl Math, 1999, 52: 1255-1276
[16] Hamel F, Nadirashvili N. Traveling fronts and entire solutions of the Fisher-KPP equation in RN. Arch Ration Mech Anal, 2001, 157: 91-163
[17] Chen X, Guo J S. Existence and uniqueness of entire solutions for a reaction-diffusion equation. J Differential Equations, 2005, 212: 62-84
[18] Li W T, Liu N W, Wang Z C. Entire solutions in reaction-advection-diffusion equations in cylinders. J Math Pures Appl, 2008, 90: 492-504
[19] Li W T, Liu N W, Wang Z C. Entire solutions of reaction-advection-diffusion equations with bistable nonlinearity in cylinders. J Differential Equations, 2009, 246: 4249-4267
[20] Li W T, Liu N W, Wang Z C. Pulsating type entire solutions of monostable reaction-advection-diffusion equations in periodic excitable media. Nonlinear Anal, 2012, 75: 1869-1880
[21] Liu N W, Li W T. Entire solutions in reaction-advection-diffusion equations with bistable nonlinearities in heterogeneous media. Sci China Math, 2010, 53: 1775-1786
[22] Sheng W J, Cao M L. Entire solutions of the Fisher-KPP equation in time periodic media. Dyn Partial Differ Equ, 2012, 9: 133-145
[23] Sheng W J, Liu N W. Entire solutions in monostable reaction-advection-diffusion equations in infinite cylinders. Nonlinear Anal, 2011, 74: 3540-3547
[24] Sheng W J, Li W T, Wang Z C. Multidimensional stability of V-shaped traveling fronts in the Allen-Cahn equation. Sci China Math, 2013, 56: 1969-1982
[25] Sheng W J. Time periodic traveling curved fronts of bistable reaction-diffusion equations in RN. Appl Math Lett, 2016, 54: 22-30
[26] Wei D, Wu J Y, Mei M. Remark on critical speed of traveling wavefronts for Nicholson's blowflies equation with diffusion. Acta Math Sci, 2010, 30B(5): 1561-1566
[27] Cao W T, Huang F M. On the convergence rate of a class of reaction hyperbolic systems for axonal transport. Acta Math Sci, 2015, 35B(4): 945-954
[28] Jiang M N, Xiang J L. Asymptotic stability of traveling waves for a dissipative nonlinear evolution system. Acta Math Sci, 2015, 35B(6): 1325-1338
[29] Volpert A I, Volpert V A, Volpert V A. Traveling Wave Solutions of Parabolic Systems. Transl Math Monogr, Vol 140. Providence: Amer Math Soc, 1994
[30] Fife P C, McLeod J B. The approach of solutions of nonlinear diffusion equations to travelling front solutions. Arch Ration Mech Anal, 1977, 65: 335-361
[31] Chen X. Existence, uniqueness, and asymptotic stability of traveling waves in nonlocal evolution equations. Adv Differential Equations, 1997, 2: 125-160
[32] Shen W. Traveling waves in time almost periodic structures governed by bistable nonlinearities, I: Stability and uniqueness. J Differential Equations, 1999, 159: 1-54
[33] Bates P W, Chen F. Periodic traveling waves for a non-local integro-differential model. Electronic J Differential Equations, 1999, 26: 1-19
[34] Berestycki H, Larrouturou B, Roquejoffre J M. Stability of travelling fronts in a model for flame propagation. part I: linear analysis. Arch Ration Mech Anal, 1992, 117: 97-117
[35] Roquejoffre J M. Stability of travelling fronts in a model for flame propagation, part Ⅱ: nonlinear stability. Arch Ration Mech Anal, 1992, 117: 119-153
[36] Roquejoffre J M. Convergence to travelling waves for solutions of a class of semilinear parabolic equations. J Differential Equations, 1994, 108: 262-295
[37] Roquejoffre J M. Eventual monotonicity and convergence to travelling fronts for the solutions of semilinear parabolic equations in cylinders. Ann Inst H Poincaré Anal Non Linéaire, 1997, 14: 499-552
[38] Lunardi A. Analytic Semigroups and Optimal Regularity in Parabolic Problems. Boston: Birkhäuser, 1995
[39] Wang Z C, Wu J. Periodic traveling curved fronts in reaction-diffusion equation with bistable time-periodic nonlinearity. J Differential Equations, 2011, 250: 3196-3229
[40] Sheng W J, Li W T, Wang Z C. Periodic pyramidal traveling fronts of bistable reaction-diffusion equations with time-periodic nonlinearity. J Differential Equations, 2012, 252: 2388-2424
/
| 〈 |
|
〉 |