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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 |
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. |
[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. |
[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. |
[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. |
[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. |
[6] |
P. C. Fife, Mathematical Aspects of Reacting and Diffusing Systems, Lecture Notes in Biomathematics, 28. Springer-Verlag, Berlin-New York, 1979. |
[7] |
A. Friedman, Partial Differential Equations of Parabolic Type, Printic-Hall, Inc., Englewood Cliffs, N.J., 1964. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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, J. Math. Pures Appl., 92 (2009), 232-262.
doi: 10.1016/j.matpur.2009.04.002. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[36] |
Z.-C. Wang, Cylindrically symmetric traveling fronts in periodic reaction-diffusion equation with bistable nonlinearity, Proc. Royal Soc. Edinburgh, Sect. A Math.,
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, J. Math. Pures Appl., 95 (2011), 627-671.
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, Adv. in Math., 30 (1978), 33-76.
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, J. Differential Equations, 255 (2013), 2402-2435.
doi: 10.1016/j.jde.2013.06.024. |
[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. |
[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. |
[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. |
[6] |
P. C. Fife, Mathematical Aspects of Reacting and Diffusing Systems, Lecture Notes in Biomathematics, 28. Springer-Verlag, Berlin-New York, 1979. |
[7] |
A. Friedman, Partial Differential Equations of Parabolic Type, Printic-Hall, Inc., Englewood Cliffs, N.J., 1964. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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, J. Math. Pures Appl., 92 (2009), 232-262.
doi: 10.1016/j.matpur.2009.04.002. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[36] |
Z.-C. Wang, Cylindrically symmetric traveling fronts in periodic reaction-diffusion equation with bistable nonlinearity, Proc. Royal Soc. Edinburgh, Sect. A Math.,
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, J. Math. Pures Appl., 95 (2011), 627-671.
doi: 10.1016/j.matpur.2010.11.005. |
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