• Previous Article
    Periodic solutions for a prescribed-energy problem of singular Hamiltonian systems
  • DCDS Home
  • This Issue
  • Next Article
    Equivalent formulations for steady periodic water waves of fixed mean-depth with discontinuous vorticity
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

[1]

Masaharu Taniguchi. Axisymmetric traveling fronts in balanced bistable reaction-diffusion equations. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3981-3995. doi: 10.3934/dcds.2020126

[2]

Mohammad Ghani, Jingyu Li, Kaijun Zhang. Asymptotic stability of traveling fronts to a chemotaxis model with nonlinear diffusion. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021017

[3]

Maho Endo, Yuki Kaneko, Yoshio Yamada. Free boundary problem for a reaction-diffusion equation with positive bistable nonlinearity. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3375-3394. doi: 10.3934/dcds.2020033

[4]

Chungang Shi, Wei Wang, Dafeng Chen. Weak time discretization for slow-fast stochastic reaction-diffusion equations. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021019

[5]

Ting Liu, Guo-Bao Zhang. Global stability of traveling waves for a spatially discrete diffusion system with time delay. Electronic Research Archive, , () : -. doi: 10.3934/era.2021003

[6]

Raffaele Folino, Ramón G. Plaza, Marta Strani. Long time dynamics of solutions to $ p $-Laplacian diffusion problems with bistable reaction terms. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020403

[7]

Hideki Murakawa. Fast reaction limit of reaction-diffusion systems. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1047-1062. doi: 10.3934/dcdss.2020405

[8]

Klemens Fellner, Jeff Morgan, Bao Quoc Tang. Uniform-in-time bounds for quadratic reaction-diffusion systems with mass dissipation in higher dimensions. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 635-651. doi: 10.3934/dcdss.2020334

[9]

Jonathan J. Wylie, Robert M. Miura, Huaxiong Huang. Systems of coupled diffusion equations with degenerate nonlinear source terms: Linear stability and traveling waves. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 561-569. doi: 10.3934/dcds.2009.23.561

[10]

Lin Shi, Xuemin Wang, Dingshi Li. Limiting behavior of non-autonomous stochastic reaction-diffusion equations with colored noise on unbounded thin domains. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5367-5386. doi: 10.3934/cpaa.2020242

[11]

Nabahats Dib-Baghdadli, Rabah Labbas, Tewfik Mahdjoub, Ahmed Medeghri. On some reaction-diffusion equations generated by non-domiciliated triatominae, vectors of Chagas disease. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2021004

[12]

Kaixuan Zhu, Ji Li, Yongqin Xie, Mingji Zhang. Dynamics of non-autonomous fractional reaction-diffusion equations on $ \mathbb{R}^{N} $ driven by multiplicative noise. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020376

[13]

Fang-Di Dong, Wan-Tong Li, Shi-Liang Wu, Li Zhang. Entire solutions originating from monotone fronts for nonlocal dispersal equations with bistable nonlinearity. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 1031-1060. doi: 10.3934/dcdsb.2020152

[14]

Weiwei Liu, Jinliang Wang, Yuming Chen. Threshold dynamics of a delayed nonlocal reaction-diffusion cholera model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020316

[15]

Abdelghafour Atlas, Mostafa Bendahmane, Fahd Karami, Driss Meskine, Omar Oubbih. A nonlinear fractional reaction-diffusion system applied to image denoising and decomposition. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020321

[16]

Shin-Ichiro Ei, Shyuh-Yaur Tzeng. Spike solutions for a mass conservation reaction-diffusion system. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3357-3374. doi: 10.3934/dcds.2020049

[17]

Chihiro Aida, Chao-Nien Chen, Kousuke Kuto, Hirokazu Ninomiya. Bifurcation from infinity with applications to reaction-diffusion systems. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3031-3055. doi: 10.3934/dcds.2020053

[18]

Leilei Wei, Yinnian He. A fully discrete local discontinuous Galerkin method with the generalized numerical flux to solve the tempered fractional reaction-diffusion equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020319

[19]

H. M. Srivastava, H. I. Abdel-Gawad, Khaled Mohammed Saad. Oscillatory states and patterns formation in a two-cell cubic autocatalytic reaction-diffusion model subjected to the Dirichlet conditions. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020433

[20]

Guillaume Cantin, M. A. Aziz-Alaoui. Dimension estimate of attractors for complex networks of reaction-diffusion systems applied to an ecological model. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020283

2019 Impact Factor: 1.338

Metrics

  • PDF downloads (423)
  • HTML views (52)
  • Cited by (1)

Other articles
by authors

[Back to Top]