Curved fronts of monostable reaction-advection-diffusion equations in
space-time periodic media

Zhen-Hui  Bu; Zhi-Cheng Wang

doi:10.3934/cpaa.2016.15.139

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January 2016, 15(1): 139-160. doi: 10.3934/cpaa.2016.15.139

Curved fronts of monostable reaction-advection-diffusion equations in space-time periodic media

Zhen-Hui Bu ^1, and Zhi-Cheng Wang ^2,

1.	School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, China
2.	School of Mathematics and Statistics, Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou University, Lanzhou, Gansu 730000

Received July 2015 Revised September 2015 Published December 2015

Abstract
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This paper is to study traveling fronts of reaction-diffusion equations with space-time periodic advection and nonlinearity in $\mathbb{R}^N$ with $N\geq3$. We are interested in curved fronts satisfying some ``pyramidal" conditions at infinity. In $\Bbb{R}^3$, we first show that there is a minimal speed $c^{*}$ such that curved fronts with speed $c$ exist if and only if $c\geq c^{*}$, and then we prove that such curved fronts are decreasing in the direction of propagation. Furthermore, we give a generalization of our results in $\mathbb{R}^N$ with $N\geq4$.

Keywords: reaction-advection-diffusion equations, Curved fronts, space-time periodic., minimal speed.

Mathematics Subject Classification: 35K57, 35C07, 35B35, 35B4.

Citation: Zhen-Hui Bu, Zhi-Cheng Wang. Curved fronts of monostable reaction-advection-diffusion equations in space-time periodic media. Communications on Pure & Applied Analysis, 2016, 15 (1) : 139-160. doi: 10.3934/cpaa.2016.15.139

References:

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[8]	J. Földes and P. Poláčik, On cooperative parabolic systems: Harnack inequalities and asymptotic symmmetry, Discrete Contin. Dyn. Syst., 25 (2009), 133-157. doi: 10.3934/dcds.2009.25.133. Google Scholar
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[11]	F. Hamel, R. Monneau and J.-M. Roquejoffre, Stability of travelling waves in a model for conical flames in two space dimensions, Ann. Sci. Ecole Norm. Sup., 37 (2004), 469-506. doi: 10.1016/j.ansens.2004.03.001. Google Scholar
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[15]	F. Hamel and L. Roques, Uniqueness and stability properties of monostable pulsating fronts, J. Eur. Math. Soc., 13 (2011), 345-390. doi: 10.4171/JEMS/256. Google Scholar
[16]	R. Huang, Stability of travelling fronts of the Fisher-KPP equation in $R^N$, NoDEA Nonlinear Differential Equations Appl., 15 (2008), 599-622. doi: 10.1007/s00030-008-7041-0. Google Scholar
[17]	N. V. Krylov and M. V. Safonov, An estimate for the probability of a diffusion process hitting a set of positive measure, Dokl. Akad. Nauk SSSR, 245 (1979), 18-20. Google Scholar
[18]	Y. Kurokawa and M. Taniguchi, Multi-dimensional pyramidal travelling fronts in the Allen-Cahn equations, Proc. Roy. Soc. Edinburgh Sect. A, 141 (2011), 1031-1054. doi: 10.1017/S0308210510001253. Google Scholar
[19]	O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, Amer. Math. Soc., Providence, R.I., 1968. Google Scholar
[20]	J. D. Murray, Mathematical Biology, Springer-Verlag, 1989. doi: 10.1007/978-3-662-08539-4. Google Scholar
[21]	G. Nadin, Traveling fronts in space-time periodic media, J. Math. Pures Appl., 92 (2009), 232-262. doi: 10.1016/j.matpur.2009.04.002. Google Scholar
[22]	G. Nadin, The principal eigenvalue of a space-time periodic parabolic operator, Ann. Mat. Pura Appl., 188 (2009), 269-295. doi: 10.1007/s10231-008-0075-4. Google Scholar
[23]	G. Nadin, Existence and uniqueness of the solution of a space-time periodic reaction-diffusion equation, J. Differential Equations, 249 (2010), 1288-1304. doi: 10.1016/j.jde.2010.05.007. Google Scholar
[24]	G. Nadin, Some depence results between the spreading speed and the cofficients of the space-time periodic Fisher-KPP equation, European J. Appl. Math., 22 (2011), 169-185. doi: 10.1017/S0956792511000027. Google Scholar
[25]	W.-M. Ni and M. Taniguchi, Traveling fronts of pyramidal shapes in competition-diffusion systems, Netw. Heterog. Media, 8 (2013), 379-395. doi: 10.3934/nhm.2013.8.379. Google Scholar
[26]	H. Ninomiya and M. Taniguchi, Global stability of traveling curved fronts in the Allen-Cahn equations, Discrete Contin. Dynam. Syst., 15 (2006), 819-832. doi: 10.3934/dcds.2006.15.819. Google Scholar
[27]	H. Ninomiya and M. Taniguchi, Existence and global stability of traveling curved fronts in the Allen-Cahn equations, J. Differential Equations, 213 (2005), 204-233. doi: 10.1016/j.jde.2004.06.011. Google Scholar
[28]	J. Nolen, M. Rudd and J. Xin, Existence of KPP type fronts in spatially-temporally periodic advection and variational principle for propagation speeds, Dyn. Partial Differ. Equ., 2 (2005), 1-24. doi: 10.4310/DPDE.2005.v2.n1.a1. Google Scholar
[29]	J. Nolen and J. Xin, Existence of KPP type fronts in space-time periodic shear flows and a study of minimal speeds based on variational principle, Discrete Contin. Dyn. Syst., 13 (2005), 1217-1234. doi: 10.3934/dcds.2005.13.1217. Google Scholar
[30]	W.-J. Sheng, W.-T. Li and Z.-C. Wang, Periodic pyramidal traveling fronts of bistable reaction-diffusion equations with time-periodic nonlinearity, J. Differential Equations, 252 (2012), 2388-2424. doi: 10.1016/j.jde.2011.09.016. Google Scholar
[31]	W.-J. Sheng, W.-T. Li and Z.-C. Wang, Multidimensional stability of V-shaped traveling fronts in the Allen-Cahn equation, Sci. China Math., 56 (2013), 1969-1982. doi: 10.1007/s11425-013-4699-5. Google Scholar
[32]	M. Taniguchi, Traveling fronts of pyramidal shapes in the Allen-Cahn equations, SIAM J. Math. Anal., 39 (2007), 319-344. doi: 10.1137/060661788. Google Scholar
[33]	M. Taniguchi, The uniqueness and asymptotic stability of pyramidal traveling fronts in the Allen-Cahn equations, J. Differential Euqations, 246 (2009), 2103-2130. doi: 10.1016/j.jde.2008.06.037. Google Scholar
[34]	Z.-C. Wang and J. Wu, Periodic traveling curved fronts in reaction-diffusion equation with bistable time-periodic nonlinearity, J. Differential Equations, 250 (2011), 3196-3229. doi: 10.1016/j.jde.2011.01.017. Google Scholar
[35]	Z.-C. Wang, Traveling curved fronts in monotone bistable systems, Discrete Contin. Dynam. Syst., 32 (2012), 2339-2374. doi: 10.3934/dcds.2012.32.2339. Google Scholar
[36]	Z.-C. Wang, Cylindrically symmetric traveling fronts in periodic reaction-diffusion equation with bistable nonlinearity,, \emph{Proc. Royal Soc. Edinburgh, (). doi: 10.1017/S0308210515000268. Google Scholar
[37]	G. Zhao and S. Ruan, Existence, uniqueness and asymptotic stability of time periodic traveling waves for a periodic Lotka-Volterra competition system with diffusion, J. Math. Pures Appl., 95 (2011), 627-671. doi: 10.1016/j.matpur.2010.11.005. Google Scholar

show all references

References:

[1]	D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics, Adv. in Math., 30 (1978), 33-76. doi: 10.1016/0001-8708(78)90130-5. Google Scholar
[2]	X.-X. Bao and Z.-C. Wang, Existence and stability of time periodic traveling waves for a periodic bistable Lotka-Volterra competition system, J. Differential Equations, 255 (2013), 2402-2435. doi: 10.1016/j.jde.2013.06.024. Google Scholar
[3]	H. Berestycki and F. Hamel, Front propagation in periodic excitable media, Comm. Pure Appl. Math., 55 (2002), 949-1032. doi: 10.1002/cpa.3022. Google Scholar
[4]	A. Bonnet and F. Hamel, Existence of nonplanar solutions of a simple model of premixed Bunsen flames, SIAM J. Math. Anal., 31 (1999), 80-118. doi: 10.1137/S0036141097316391. Google Scholar
[5]	M. El Smaily, F. Hamel and R. Huang, Two-dimensional curved fronts in a periodic shear flow, Nonlinear Analysis TMA, 74 (2011), 6469-6486. doi: 10.1016/j.na.2011.06.030. Google Scholar
[6]	P. C. Fife, Mathematical Aspects of Reacting and Diffusing Systems, Lecture Notes in Biomathematics, 28. Springer-Verlag, Berlin-New York, 1979. Google Scholar
[7]	A. Friedman, Partial Differential Equations of Parabolic Type, Printic-Hall, Inc., Englewood Cliffs, N.J., 1964. Google Scholar
[8]	J. Földes and P. Poláčik, On cooperative parabolic systems: Harnack inequalities and asymptotic symmmetry, Discrete Contin. Dyn. Syst., 25 (2009), 133-157. doi: 10.3934/dcds.2009.25.133. Google Scholar
[9]	F. Hamel and R. Monneau, Solutions of semilinear elliptic equations in $R^N$ with conical-shaped level sets, Comm. Partial Differential Equations, 25 (2000), 769-819. doi: 10.1080/03605300008821532. Google Scholar
[10]	F. Hamel and N. Nadirashvili, Travelling fronts and entire solutions of the Fisher-KPP equation in $R^N$, Arch. Rational Mech. Anal., 157 (2001), 91-163. doi: 10.1007/PL00004238. Google Scholar
[11]	F. Hamel, R. Monneau and J.-M. Roquejoffre, Stability of travelling waves in a model for conical flames in two space dimensions, Ann. Sci. Ecole Norm. Sup., 37 (2004), 469-506. doi: 10.1016/j.ansens.2004.03.001. Google Scholar
[12]	F. Hamel, R. Monneau and J.-M. Roquejoffre, Existence and qualitative properties of multidimensional conical bistable fronts, Discrete Contin. Dynam. Syst., 13 (2005), 1069-1096. doi: 10.3934/dcds.2005.13.1069. Google Scholar
[13]	F. Hamel, R. Monneau and J.-M. Roquejoffre, Asymptotic properties and classification of bistable fronts with Lipschitz level sets, Discrete Contin. Dynam. Syst., 14 (2006), 75-92. Google Scholar
[14]	F. Hamel, Qualitative properties of monostable pulsating fronts: exponential decay and monotonicity, J. Math. Pures Appl., 89 (2008), 355-399. doi: 10.1016/j.matpur.2007.12.005. Google Scholar
[15]	F. Hamel and L. Roques, Uniqueness and stability properties of monostable pulsating fronts, J. Eur. Math. Soc., 13 (2011), 345-390. doi: 10.4171/JEMS/256. Google Scholar
[16]	R. Huang, Stability of travelling fronts of the Fisher-KPP equation in $R^N$, NoDEA Nonlinear Differential Equations Appl., 15 (2008), 599-622. doi: 10.1007/s00030-008-7041-0. Google Scholar
[17]	N. V. Krylov and M. V. Safonov, An estimate for the probability of a diffusion process hitting a set of positive measure, Dokl. Akad. Nauk SSSR, 245 (1979), 18-20. Google Scholar
[18]	Y. Kurokawa and M. Taniguchi, Multi-dimensional pyramidal travelling fronts in the Allen-Cahn equations, Proc. Roy. Soc. Edinburgh Sect. A, 141 (2011), 1031-1054. doi: 10.1017/S0308210510001253. Google Scholar
[19]	O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, Amer. Math. Soc., Providence, R.I., 1968. Google Scholar
[20]	J. D. Murray, Mathematical Biology, Springer-Verlag, 1989. doi: 10.1007/978-3-662-08539-4. Google Scholar
[21]	G. Nadin, Traveling fronts in space-time periodic media, J. Math. Pures Appl., 92 (2009), 232-262. doi: 10.1016/j.matpur.2009.04.002. Google Scholar
[22]	G. Nadin, The principal eigenvalue of a space-time periodic parabolic operator, Ann. Mat. Pura Appl., 188 (2009), 269-295. doi: 10.1007/s10231-008-0075-4. Google Scholar
[23]	G. Nadin, Existence and uniqueness of the solution of a space-time periodic reaction-diffusion equation, J. Differential Equations, 249 (2010), 1288-1304. doi: 10.1016/j.jde.2010.05.007. Google Scholar
[24]	G. Nadin, Some depence results between the spreading speed and the cofficients of the space-time periodic Fisher-KPP equation, European J. Appl. Math., 22 (2011), 169-185. doi: 10.1017/S0956792511000027. Google Scholar
[25]	W.-M. Ni and M. Taniguchi, Traveling fronts of pyramidal shapes in competition-diffusion systems, Netw. Heterog. Media, 8 (2013), 379-395. doi: 10.3934/nhm.2013.8.379. Google Scholar
[26]	H. Ninomiya and M. Taniguchi, Global stability of traveling curved fronts in the Allen-Cahn equations, Discrete Contin. Dynam. Syst., 15 (2006), 819-832. doi: 10.3934/dcds.2006.15.819. Google Scholar
[27]	H. Ninomiya and M. Taniguchi, Existence and global stability of traveling curved fronts in the Allen-Cahn equations, J. Differential Equations, 213 (2005), 204-233. doi: 10.1016/j.jde.2004.06.011. Google Scholar
[28]	J. Nolen, M. Rudd and J. Xin, Existence of KPP type fronts in spatially-temporally periodic advection and variational principle for propagation speeds, Dyn. Partial Differ. Equ., 2 (2005), 1-24. doi: 10.4310/DPDE.2005.v2.n1.a1. Google Scholar
[29]	J. Nolen and J. Xin, Existence of KPP type fronts in space-time periodic shear flows and a study of minimal speeds based on variational principle, Discrete Contin. Dyn. Syst., 13 (2005), 1217-1234. doi: 10.3934/dcds.2005.13.1217. Google Scholar
[30]	W.-J. Sheng, W.-T. Li and Z.-C. Wang, Periodic pyramidal traveling fronts of bistable reaction-diffusion equations with time-periodic nonlinearity, J. Differential Equations, 252 (2012), 2388-2424. doi: 10.1016/j.jde.2011.09.016. Google Scholar
[31]	W.-J. Sheng, W.-T. Li and Z.-C. Wang, Multidimensional stability of V-shaped traveling fronts in the Allen-Cahn equation, Sci. China Math., 56 (2013), 1969-1982. doi: 10.1007/s11425-013-4699-5. Google Scholar
[32]	M. Taniguchi, Traveling fronts of pyramidal shapes in the Allen-Cahn equations, SIAM J. Math. Anal., 39 (2007), 319-344. doi: 10.1137/060661788. Google Scholar
[33]	M. Taniguchi, The uniqueness and asymptotic stability of pyramidal traveling fronts in the Allen-Cahn equations, J. Differential Euqations, 246 (2009), 2103-2130. doi: 10.1016/j.jde.2008.06.037. Google Scholar
[34]	Z.-C. Wang and J. Wu, Periodic traveling curved fronts in reaction-diffusion equation with bistable time-periodic nonlinearity, J. Differential Equations, 250 (2011), 3196-3229. doi: 10.1016/j.jde.2011.01.017. Google Scholar
[35]	Z.-C. Wang, Traveling curved fronts in monotone bistable systems, Discrete Contin. Dynam. Syst., 32 (2012), 2339-2374. doi: 10.3934/dcds.2012.32.2339. Google Scholar
[36]	Z.-C. Wang, Cylindrically symmetric traveling fronts in periodic reaction-diffusion equation with bistable nonlinearity,, \emph{Proc. Royal Soc. Edinburgh, (). doi: 10.1017/S0308210515000268. Google Scholar
[37]	G. Zhao and S. Ruan, Existence, uniqueness and asymptotic stability of time periodic traveling waves for a periodic Lotka-Volterra competition system with diffusion, J. Math. Pures Appl., 95 (2011), 627-671. doi: 10.1016/j.matpur.2010.11.005. Google Scholar

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Curved fronts of monostable reaction-advection-diffusion equations in space-time periodic media

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