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May  2017, 37(5): 2681-2704. doi: 10.3934/dcds.2017115

Multidimensional stability of time-periodic planar traveling fronts in bistable reaction-diffusion equations

1. 

Department of Mathematics, Harbin Institute of Technology, Harbin, Heilongjiang 150001, China

2. 

School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, China

Corresponding author

Received  April 2015 Revised  December 2016 Published  February 2017

This paper is concerned with the stability of time periodic planar traveling fronts of bistable reaction-diffusion equations in multidimensional space. We first show that time periodic planar traveling fronts are asymptotically stable under spatially decaying initial perturbations. In particular, we show that such fronts are algebraically stable when the initial perturbations belong to $L^1$ in a certain sense. Then we further prove that there exists a solution that oscillates permanently between two time periodic planar traveling fronts, which reveals that time periodic planar traveling fronts are not always asymptotically stable under general bounded perturbations. Finally, we address the asymptotic stability of time periodic planar traveling fronts under almost periodic initial perturbations.

Citation: Wei-Jie Sheng, Wan-Tong Li. Multidimensional stability of time-periodic planar traveling fronts in bistable reaction-diffusion equations. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2681-2704. doi: 10.3934/dcds.2017115
References:
[1]

N. AlikakosP. W. Bates and X. Chen, Periodic traveling waves and locating oscillating patterns in multidimensional domains, Trans. Amer. Math. Soc., 351 (1999), 2777-2805.  doi: 10.1090/S0002-9947-99-02134-0.  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. Chen and R. Yuan, Multidimensional stability of disturbed pyramidal traveling fronts in the Allen-Cahn equation, Discrete Contin. Dyn. Syst. B, 20 (2015), 1015-1029.  doi: 10.3934/dcdsb.2015.20.1015.  Google Scholar

[4]

R. A. Fisher, The advance of advantageous genes, Ann. Eugenics, 7 (1937), 355-369.  doi: 10.1111/j.1469-1809.1937.tb02153.x.  Google Scholar

[5]

A. Friedman, Partial Differential Equations of Parabolic Type, Englewood Cliffs, NJ: Prentice-Hall, 1964.  Google Scholar

[6]

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

[7]

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

[8]

F. HamelR. Monneau and J.-M. Roquejoffre, Stability of travelling waves in a model for conical flames in two space dimensions, Ann. Sci. Ec. Norm. Sup., 37 (2004), 469-506.  doi: 10.1016/j.ansens.2004.03.001.  Google Scholar

[9]

T. Kapitula, Multidimensional stability of planar traveling waves, Trans. Amer. Math. Soc., 349 (1997), 257-269.  doi: 10.1090/S0002-9947-97-01668-1.  Google Scholar

[10]

A. N. KolmogorovI. G. Petrovsky and N. S. Piskunov, A study of the equation of diffusion with increase in the quantity of matter, and its application to a biological problem, Bjul. Moskovskogo Gos. Univ., 1 (1937), 1-26.   Google Scholar

[11]

C. D. Levermore and J. Xin, Multidimensional stability of traveling waves in a bistable reaction-diffusion equation Ⅱ, Comm. Partial Differential Equations, 17 (1992), 1901-1924.  doi: 10.1080/03605309208820908.  Google Scholar

[12]

G. Lv and M. Wang, Stability of planar waves in monostable reaction-diffusion equations, Proc. Amer. Math. Soc., 139 (2011), 3611-3621.  doi: 10.1090/S0002-9939-2011-10767-6.  Google Scholar

[13]

G. Lv and M. Wang, Stability of planar waves in reaction-diffusion system, Sci China Math, 54 (2011), 1403-1419.  doi: 10.1007/s11425-011-4210-0.  Google Scholar

[14]

H. Matano and M. Nara, Large time behavior of disturbed planar fronts in the Allen-Cahn equation, J. Differential Equations, 251 (2011), 3522-3557.  doi: 10.1016/j.jde.2011.08.029.  Google Scholar

[15]

H. MatanoM. Nara and M. Taniguchi, Stability of planar waves in the Allen-Cahn equation, Comm. Partial Differential Equations, 34 (2009), 976-1002.  doi: 10.1080/03605300902963500.  Google Scholar

[16]

M. Nara and M. Taniguchi, Stability of a traveling wave in curvature flows for spatially non-decaying perturbations, Discrete Contin. Dyn. Syst., 14 (2006), 203-220.  doi: 10.3934/dcds.2006.14.203.  Google Scholar

[17]

M. Nara and M. Taniguchi, Convergence to V-shaped fronts for spatially non-decaying inital perturbations, Discrete Contin. Dyn. Syst., 16 (2006), 137-156.  doi: 10.3934/dcds.2006.16.137.  Google Scholar

[18]

J. M. Roquejoffre and V. Roussier-Michon, Nontrivial large-time behaviour in bistable reaction-diffusion equations, Ann. Mat. Pura Appl., 188 (2009), 207-233.  doi: 10.1007/s10231-008-0072-7.  Google Scholar

[19]

W. Shen, Traveling waves in time almost periodic structures governed by bistable nonlinearities, Ⅰ. Stability and uniqueness, J. Differential Equations, 159 (1999), 1-54.  doi: 10.1006/jdeq.1999.3651.  Google Scholar

[20]

W. Shen, Traveling waves in time almost periodic structures governed by bistable nonlinearities, Ⅱ. Existence, J. Differential Equations, 159 (1999), 55-101.  doi: 10.1006/jdeq.1999.3652.  Google Scholar

[21]

W. Shen, Dynamical systems and traveling waves in almost periodic structures, J. Differential Equations, 169 (2001), 493-548.  doi: 10.1006/jdeq.2000.3906.  Google Scholar

[22]

W. Shen, Traveling waves in time dependent bistable media, Differential Integral Equations, 19 (2006), 241-278.   Google Scholar

[23]

W. Shen, Variational principle for spatial spreading speeds and generalized propagating speeds in time almost periodic and space periodic KPP models, Trans. Amer. Math. Soc., 362 (2010), 5125-5168.  doi: 10.1090/S0002-9947-10-04950-0.  Google Scholar

[24]

W. J. Sheng, Time periodic traveling curved fronts of bistable reaction-diffusion equations in $\mathbb{R}^3$, Ann. Mat. Pura Appl., (2016).  doi: 10.1007/s10231-016-0589-0.  Google Scholar

[25]

W. J. ShengW. 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

[26]

W. J. ShengW. 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

[27]

A. I. Volpert, V. A. Volpert and V. A. Volpert, Travelling Wave Solutions of Parabolic Systems, Translations of Mathematical Monographs, vol. 140. American Mathematical Society, Providence, RI, 1994.  Google Scholar

[28]

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

[29]

J. Xin, Multidimensional stability of traveling waves in a bistable reaction-diffusion equation Ⅰ, Comm. Partial Differential Equations, 17 (1992), 1889-1899.  doi: 10.1080/03605309208820907.  Google Scholar

[30]

G. Zhao, Multidimensional periodic traveling waves in infinite cylinders, Discrete Contnu. Dyn. Syst., 24 (2009), 1025-1045.  doi: 10.3934/dcds.2009.24.1025.  Google Scholar

[31]

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

[32]

G. Zhao and S. Ruan, Time periodic traveling wave solutions for periodic advection-reaction-diffusion systems, J. Differential Equations, 257 (2014), 1078-1147.  doi: 10.1016/j.jde.2014.05.001.  Google Scholar

show all references

References:
[1]

N. AlikakosP. W. Bates and X. Chen, Periodic traveling waves and locating oscillating patterns in multidimensional domains, Trans. Amer. Math. Soc., 351 (1999), 2777-2805.  doi: 10.1090/S0002-9947-99-02134-0.  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. Chen and R. Yuan, Multidimensional stability of disturbed pyramidal traveling fronts in the Allen-Cahn equation, Discrete Contin. Dyn. Syst. B, 20 (2015), 1015-1029.  doi: 10.3934/dcdsb.2015.20.1015.  Google Scholar

[4]

R. A. Fisher, The advance of advantageous genes, Ann. Eugenics, 7 (1937), 355-369.  doi: 10.1111/j.1469-1809.1937.tb02153.x.  Google Scholar

[5]

A. Friedman, Partial Differential Equations of Parabolic Type, Englewood Cliffs, NJ: Prentice-Hall, 1964.  Google Scholar

[6]

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

[7]

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

[8]

F. HamelR. Monneau and J.-M. Roquejoffre, Stability of travelling waves in a model for conical flames in two space dimensions, Ann. Sci. Ec. Norm. Sup., 37 (2004), 469-506.  doi: 10.1016/j.ansens.2004.03.001.  Google Scholar

[9]

T. Kapitula, Multidimensional stability of planar traveling waves, Trans. Amer. Math. Soc., 349 (1997), 257-269.  doi: 10.1090/S0002-9947-97-01668-1.  Google Scholar

[10]

A. N. KolmogorovI. G. Petrovsky and N. S. Piskunov, A study of the equation of diffusion with increase in the quantity of matter, and its application to a biological problem, Bjul. Moskovskogo Gos. Univ., 1 (1937), 1-26.   Google Scholar

[11]

C. D. Levermore and J. Xin, Multidimensional stability of traveling waves in a bistable reaction-diffusion equation Ⅱ, Comm. Partial Differential Equations, 17 (1992), 1901-1924.  doi: 10.1080/03605309208820908.  Google Scholar

[12]

G. Lv and M. Wang, Stability of planar waves in monostable reaction-diffusion equations, Proc. Amer. Math. Soc., 139 (2011), 3611-3621.  doi: 10.1090/S0002-9939-2011-10767-6.  Google Scholar

[13]

G. Lv and M. Wang, Stability of planar waves in reaction-diffusion system, Sci China Math, 54 (2011), 1403-1419.  doi: 10.1007/s11425-011-4210-0.  Google Scholar

[14]

H. Matano and M. Nara, Large time behavior of disturbed planar fronts in the Allen-Cahn equation, J. Differential Equations, 251 (2011), 3522-3557.  doi: 10.1016/j.jde.2011.08.029.  Google Scholar

[15]

H. MatanoM. Nara and M. Taniguchi, Stability of planar waves in the Allen-Cahn equation, Comm. Partial Differential Equations, 34 (2009), 976-1002.  doi: 10.1080/03605300902963500.  Google Scholar

[16]

M. Nara and M. Taniguchi, Stability of a traveling wave in curvature flows for spatially non-decaying perturbations, Discrete Contin. Dyn. Syst., 14 (2006), 203-220.  doi: 10.3934/dcds.2006.14.203.  Google Scholar

[17]

M. Nara and M. Taniguchi, Convergence to V-shaped fronts for spatially non-decaying inital perturbations, Discrete Contin. Dyn. Syst., 16 (2006), 137-156.  doi: 10.3934/dcds.2006.16.137.  Google Scholar

[18]

J. M. Roquejoffre and V. Roussier-Michon, Nontrivial large-time behaviour in bistable reaction-diffusion equations, Ann. Mat. Pura Appl., 188 (2009), 207-233.  doi: 10.1007/s10231-008-0072-7.  Google Scholar

[19]

W. Shen, Traveling waves in time almost periodic structures governed by bistable nonlinearities, Ⅰ. Stability and uniqueness, J. Differential Equations, 159 (1999), 1-54.  doi: 10.1006/jdeq.1999.3651.  Google Scholar

[20]

W. Shen, Traveling waves in time almost periodic structures governed by bistable nonlinearities, Ⅱ. Existence, J. Differential Equations, 159 (1999), 55-101.  doi: 10.1006/jdeq.1999.3652.  Google Scholar

[21]

W. Shen, Dynamical systems and traveling waves in almost periodic structures, J. Differential Equations, 169 (2001), 493-548.  doi: 10.1006/jdeq.2000.3906.  Google Scholar

[22]

W. Shen, Traveling waves in time dependent bistable media, Differential Integral Equations, 19 (2006), 241-278.   Google Scholar

[23]

W. Shen, Variational principle for spatial spreading speeds and generalized propagating speeds in time almost periodic and space periodic KPP models, Trans. Amer. Math. Soc., 362 (2010), 5125-5168.  doi: 10.1090/S0002-9947-10-04950-0.  Google Scholar

[24]

W. J. Sheng, Time periodic traveling curved fronts of bistable reaction-diffusion equations in $\mathbb{R}^3$, Ann. Mat. Pura Appl., (2016).  doi: 10.1007/s10231-016-0589-0.  Google Scholar

[25]

W. J. ShengW. 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

[26]

W. J. ShengW. 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

[27]

A. I. Volpert, V. A. Volpert and V. A. Volpert, Travelling Wave Solutions of Parabolic Systems, Translations of Mathematical Monographs, vol. 140. American Mathematical Society, Providence, RI, 1994.  Google Scholar

[28]

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

[29]

J. Xin, Multidimensional stability of traveling waves in a bistable reaction-diffusion equation Ⅰ, Comm. Partial Differential Equations, 17 (1992), 1889-1899.  doi: 10.1080/03605309208820907.  Google Scholar

[30]

G. Zhao, Multidimensional periodic traveling waves in infinite cylinders, Discrete Contnu. Dyn. Syst., 24 (2009), 1025-1045.  doi: 10.3934/dcds.2009.24.1025.  Google Scholar

[31]

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

[32]

G. Zhao and S. Ruan, Time periodic traveling wave solutions for periodic advection-reaction-diffusion systems, J. Differential Equations, 257 (2014), 1078-1147.  doi: 10.1016/j.jde.2014.05.001.  Google Scholar

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