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

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

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 $N\geq4$.
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:
[1]

D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics,, \emph{Adv. in Math.}, 30 (1978), 33. doi: 10.1016/0001-8708(78)90130-5.

[2]

X.-X. Bao and Z.-C. Wang, Existence and stability of time periodic traveling waves for a periodic bistable Lotka-Volterra competition system,, \emph{J. Differential Equations, 255 (2013), 2402. doi: 10.1016/j.jde.2013.06.024.

[3]

H. Berestycki and F. Hamel, Front propagation in periodic excitable media,, \emph{Comm. Pure Appl. Math., 55 (2002), 949. doi: 10.1002/cpa.3022.

[4]

A. Bonnet and F. Hamel, Existence of nonplanar solutions of a simple model of premixed Bunsen flames,, \emph{SIAM J. Math. Anal.}, 31 (1999), 80. doi: 10.1137/S0036141097316391.

[5]

M. El Smaily, F. Hamel and R. Huang, Two-dimensional curved fronts in a periodic shear flow,, \emph{Nonlinear Analysis TMA}, 74 (2011), 6469. doi: 10.1016/j.na.2011.06.030.

[6]

P. C. Fife, Mathematical Aspects of Reacting and Diffusing Systems,, Lecture Notes in Biomathematics, (1979).

[7]

A. Friedman, Partial Differential Equations of Parabolic Type,, Printic-Hall, (1964).

[8]

J. Földes and P. Poláčik, On cooperative parabolic systems: Harnack inequalities and asymptotic symmmetry,, \emph{Discrete Contin. Dyn. Syst., 25 (2009), 133. doi: 10.3934/dcds.2009.25.133.

[9]

F. Hamel and R. Monneau, Solutions of semilinear elliptic equations in $R^N$ with conical-shaped level sets,, \emph{Comm. Partial Differential Equations}, 25 (2000), 769. doi: 10.1080/03605300008821532.

[10]

F. Hamel and N. Nadirashvili, Travelling fronts and entire solutions of the Fisher-KPP equation in $R^N$,, \emph{Arch. Rational Mech. Anal.}, 157 (2001), 91. doi: 10.1007/PL00004238.

[11]

F. Hamel, R. Monneau and J.-M. Roquejoffre, Stability of travelling waves in a model for conical flames in two space dimensions,, \emph{Ann. Sci. Ecole Norm. Sup.}, 37 (2004), 469. doi: 10.1016/j.ansens.2004.03.001.

[12]

F. Hamel, R. Monneau and J.-M. Roquejoffre, Existence and qualitative properties of multidimensional conical bistable fronts,, \emph{Discrete Contin. Dynam. Syst.}, 13 (2005), 1069. doi: 10.3934/dcds.2005.13.1069.

[13]

F. Hamel, R. Monneau and J.-M. Roquejoffre, Asymptotic properties and classification of bistable fronts with Lipschitz level sets,, \emph{Discrete Contin. Dynam. Syst.}, 14 (2006), 75.

[14]

F. Hamel, Qualitative properties of monostable pulsating fronts: exponential decay and monotonicity,, \emph{J. Math. Pures Appl., 89 (2008), 355. doi: 10.1016/j.matpur.2007.12.005.

[15]

F. Hamel and L. Roques, Uniqueness and stability properties of monostable pulsating fronts,, \emph{J. Eur. Math. Soc., 13 (2011), 345. doi: 10.4171/JEMS/256.

[16]

R. Huang, Stability of travelling fronts of the Fisher-KPP equation in $R^N$,, \emph{NoDEA Nonlinear Differential Equations Appl.}, 15 (2008), 599. doi: 10.1007/s00030-008-7041-0.

[17]

N. V. Krylov and M. V. Safonov, An estimate for the probability of a diffusion process hitting a set of positive measure,, \emph{Dokl. Akad. Nauk SSSR, 245 (1979), 18.

[18]

Y. Kurokawa and M. Taniguchi, Multi-dimensional pyramidal travelling fronts in the Allen-Cahn equations,, \emph{Proc. Roy. Soc. Edinburgh Sect. A}, 141 (2011), 1031. doi: 10.1017/S0308210510001253.

[19]

O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type,, Amer. Math. Soc., (1968).

[20]

J. D. Murray, Mathematical Biology,, Springer-Verlag, (1989). doi: 10.1007/978-3-662-08539-4.

[21]

G. Nadin, Traveling fronts in space-time periodic media,, \emph{J. Math. Pures Appl., 92 (2009), 232. doi: 10.1016/j.matpur.2009.04.002.

[22]

G. Nadin, The principal eigenvalue of a space-time periodic parabolic operator,, \emph{Ann. Mat. Pura Appl., 188 (2009), 269. doi: 10.1007/s10231-008-0075-4.

[23]

G. Nadin, Existence and uniqueness of the solution of a space-time periodic reaction-diffusion equation,, \emph{J. Differential Equations, 249 (2010), 1288. doi: 10.1016/j.jde.2010.05.007.

[24]

G. Nadin, Some depence results between the spreading speed and the cofficients of the space-time periodic Fisher-KPP equation,, \emph{European J. Appl. Math., 22 (2011), 169. doi: 10.1017/S0956792511000027.

[25]

W.-M. Ni and M. Taniguchi, Traveling fronts of pyramidal shapes in competition-diffusion systems,, \emph{Netw. Heterog. Media}, 8 (2013), 379. doi: 10.3934/nhm.2013.8.379.

[26]

H. Ninomiya and M. Taniguchi, Global stability of traveling curved fronts in the Allen-Cahn equations,, \emph{Discrete Contin. Dynam. Syst.}, 15 (2006), 819. doi: 10.3934/dcds.2006.15.819.

[27]

H. Ninomiya and M. Taniguchi, Existence and global stability of traveling curved fronts in the Allen-Cahn equations,, \emph{J. Differential Equations}, 213 (2005), 204. doi: 10.1016/j.jde.2004.06.011.

[28]

J. Nolen, M. Rudd and J. Xin, Existence of KPP type fronts in spatially-temporally periodic advection and variational principle for propagation speeds,, \emph{Dyn. Partial Differ. Equ., 2 (2005), 1. doi: 10.4310/DPDE.2005.v2.n1.a1.

[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,, \emph{Discrete Contin. Dyn. Syst., 13 (2005), 1217. doi: 10.3934/dcds.2005.13.1217.

[30]

W.-J. Sheng, W.-T. Li and Z.-C. Wang, Periodic pyramidal traveling fronts of bistable reaction-diffusion equations with time-periodic nonlinearity,, \emph{J. Differential Equations}, 252 (2012), 2388. doi: 10.1016/j.jde.2011.09.016.

[31]

W.-J. Sheng, W.-T. Li and Z.-C. Wang, Multidimensional stability of V-shaped traveling fronts in the Allen-Cahn equation,, \emph{Sci. China Math., 56 (2013), 1969. doi: 10.1007/s11425-013-4699-5.

[32]

M. Taniguchi, Traveling fronts of pyramidal shapes in the Allen-Cahn equations,, \emph{SIAM J. Math. Anal.}, 39 (2007), 319. doi: 10.1137/060661788.

[33]

M. Taniguchi, The uniqueness and asymptotic stability of pyramidal traveling fronts in the Allen-Cahn equations,, \emph{J. Differential Euqations}, 246 (2009), 2103. doi: 10.1016/j.jde.2008.06.037.

[34]

Z.-C. Wang and J. Wu, Periodic traveling curved fronts in reaction-diffusion equation with bistable time-periodic nonlinearity,, \emph{J. Differential Equations, 250 (2011), 3196. doi: 10.1016/j.jde.2011.01.017.

[35]

Z.-C. Wang, Traveling curved fronts in monotone bistable systems,, \emph{Discrete Contin. Dynam. Syst.}, 32 (2012), 2339. doi: 10.3934/dcds.2012.32.2339.

[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.

[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,, \emph{J. Math. Pures Appl.}, 95 (2011), 627. doi: 10.1016/j.matpur.2010.11.005.

show all references

References:
[1]

D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics,, \emph{Adv. in Math.}, 30 (1978), 33. doi: 10.1016/0001-8708(78)90130-5.

[2]

X.-X. Bao and Z.-C. Wang, Existence and stability of time periodic traveling waves for a periodic bistable Lotka-Volterra competition system,, \emph{J. Differential Equations, 255 (2013), 2402. doi: 10.1016/j.jde.2013.06.024.

[3]

H. Berestycki and F. Hamel, Front propagation in periodic excitable media,, \emph{Comm. Pure Appl. Math., 55 (2002), 949. doi: 10.1002/cpa.3022.

[4]

A. Bonnet and F. Hamel, Existence of nonplanar solutions of a simple model of premixed Bunsen flames,, \emph{SIAM J. Math. Anal.}, 31 (1999), 80. doi: 10.1137/S0036141097316391.

[5]

M. El Smaily, F. Hamel and R. Huang, Two-dimensional curved fronts in a periodic shear flow,, \emph{Nonlinear Analysis TMA}, 74 (2011), 6469. doi: 10.1016/j.na.2011.06.030.

[6]

P. C. Fife, Mathematical Aspects of Reacting and Diffusing Systems,, Lecture Notes in Biomathematics, (1979).

[7]

A. Friedman, Partial Differential Equations of Parabolic Type,, Printic-Hall, (1964).

[8]

J. Földes and P. Poláčik, On cooperative parabolic systems: Harnack inequalities and asymptotic symmmetry,, \emph{Discrete Contin. Dyn. Syst., 25 (2009), 133. doi: 10.3934/dcds.2009.25.133.

[9]

F. Hamel and R. Monneau, Solutions of semilinear elliptic equations in $R^N$ with conical-shaped level sets,, \emph{Comm. Partial Differential Equations}, 25 (2000), 769. doi: 10.1080/03605300008821532.

[10]

F. Hamel and N. Nadirashvili, Travelling fronts and entire solutions of the Fisher-KPP equation in $R^N$,, \emph{Arch. Rational Mech. Anal.}, 157 (2001), 91. doi: 10.1007/PL00004238.

[11]

F. Hamel, R. Monneau and J.-M. Roquejoffre, Stability of travelling waves in a model for conical flames in two space dimensions,, \emph{Ann. Sci. Ecole Norm. Sup.}, 37 (2004), 469. doi: 10.1016/j.ansens.2004.03.001.

[12]

F. Hamel, R. Monneau and J.-M. Roquejoffre, Existence and qualitative properties of multidimensional conical bistable fronts,, \emph{Discrete Contin. Dynam. Syst.}, 13 (2005), 1069. doi: 10.3934/dcds.2005.13.1069.

[13]

F. Hamel, R. Monneau and J.-M. Roquejoffre, Asymptotic properties and classification of bistable fronts with Lipschitz level sets,, \emph{Discrete Contin. Dynam. Syst.}, 14 (2006), 75.

[14]

F. Hamel, Qualitative properties of monostable pulsating fronts: exponential decay and monotonicity,, \emph{J. Math. Pures Appl., 89 (2008), 355. doi: 10.1016/j.matpur.2007.12.005.

[15]

F. Hamel and L. Roques, Uniqueness and stability properties of monostable pulsating fronts,, \emph{J. Eur. Math. Soc., 13 (2011), 345. doi: 10.4171/JEMS/256.

[16]

R. Huang, Stability of travelling fronts of the Fisher-KPP equation in $R^N$,, \emph{NoDEA Nonlinear Differential Equations Appl.}, 15 (2008), 599. doi: 10.1007/s00030-008-7041-0.

[17]

N. V. Krylov and M. V. Safonov, An estimate for the probability of a diffusion process hitting a set of positive measure,, \emph{Dokl. Akad. Nauk SSSR, 245 (1979), 18.

[18]

Y. Kurokawa and M. Taniguchi, Multi-dimensional pyramidal travelling fronts in the Allen-Cahn equations,, \emph{Proc. Roy. Soc. Edinburgh Sect. A}, 141 (2011), 1031. doi: 10.1017/S0308210510001253.

[19]

O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type,, Amer. Math. Soc., (1968).

[20]

J. D. Murray, Mathematical Biology,, Springer-Verlag, (1989). doi: 10.1007/978-3-662-08539-4.

[21]

G. Nadin, Traveling fronts in space-time periodic media,, \emph{J. Math. Pures Appl., 92 (2009), 232. doi: 10.1016/j.matpur.2009.04.002.

[22]

G. Nadin, The principal eigenvalue of a space-time periodic parabolic operator,, \emph{Ann. Mat. Pura Appl., 188 (2009), 269. doi: 10.1007/s10231-008-0075-4.

[23]

G. Nadin, Existence and uniqueness of the solution of a space-time periodic reaction-diffusion equation,, \emph{J. Differential Equations, 249 (2010), 1288. doi: 10.1016/j.jde.2010.05.007.

[24]

G. Nadin, Some depence results between the spreading speed and the cofficients of the space-time periodic Fisher-KPP equation,, \emph{European J. Appl. Math., 22 (2011), 169. doi: 10.1017/S0956792511000027.

[25]

W.-M. Ni and M. Taniguchi, Traveling fronts of pyramidal shapes in competition-diffusion systems,, \emph{Netw. Heterog. Media}, 8 (2013), 379. doi: 10.3934/nhm.2013.8.379.

[26]

H. Ninomiya and M. Taniguchi, Global stability of traveling curved fronts in the Allen-Cahn equations,, \emph{Discrete Contin. Dynam. Syst.}, 15 (2006), 819. doi: 10.3934/dcds.2006.15.819.

[27]

H. Ninomiya and M. Taniguchi, Existence and global stability of traveling curved fronts in the Allen-Cahn equations,, \emph{J. Differential Equations}, 213 (2005), 204. doi: 10.1016/j.jde.2004.06.011.

[28]

J. Nolen, M. Rudd and J. Xin, Existence of KPP type fronts in spatially-temporally periodic advection and variational principle for propagation speeds,, \emph{Dyn. Partial Differ. Equ., 2 (2005), 1. doi: 10.4310/DPDE.2005.v2.n1.a1.

[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,, \emph{Discrete Contin. Dyn. Syst., 13 (2005), 1217. doi: 10.3934/dcds.2005.13.1217.

[30]

W.-J. Sheng, W.-T. Li and Z.-C. Wang, Periodic pyramidal traveling fronts of bistable reaction-diffusion equations with time-periodic nonlinearity,, \emph{J. Differential Equations}, 252 (2012), 2388. doi: 10.1016/j.jde.2011.09.016.

[31]

W.-J. Sheng, W.-T. Li and Z.-C. Wang, Multidimensional stability of V-shaped traveling fronts in the Allen-Cahn equation,, \emph{Sci. China Math., 56 (2013), 1969. doi: 10.1007/s11425-013-4699-5.

[32]

M. Taniguchi, Traveling fronts of pyramidal shapes in the Allen-Cahn equations,, \emph{SIAM J. Math. Anal.}, 39 (2007), 319. doi: 10.1137/060661788.

[33]

M. Taniguchi, The uniqueness and asymptotic stability of pyramidal traveling fronts in the Allen-Cahn equations,, \emph{J. Differential Euqations}, 246 (2009), 2103. doi: 10.1016/j.jde.2008.06.037.

[34]

Z.-C. Wang and J. Wu, Periodic traveling curved fronts in reaction-diffusion equation with bistable time-periodic nonlinearity,, \emph{J. Differential Equations, 250 (2011), 3196. doi: 10.1016/j.jde.2011.01.017.

[35]

Z.-C. Wang, Traveling curved fronts in monotone bistable systems,, \emph{Discrete Contin. Dynam. Syst.}, 32 (2012), 2339. doi: 10.3934/dcds.2012.32.2339.

[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.

[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,, \emph{J. Math. Pures Appl.}, 95 (2011), 627. doi: 10.1016/j.matpur.2010.11.005.

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