\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

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

  • Corresponding author

    Corresponding author 
Abstract Full Text(HTML) Related Papers Cited by
  • 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.

    Mathematics Subject Classification: Primary: 35B35, 35K57; Secondary: 35C07, 35B10.

    Citation:

    \begin{equation} \\ \end{equation}
  • 加载中
  • [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.
    [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. 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.
    [4] R. A. Fisher, The advance of advantageous genes, Ann. Eugenics, 7 (1937), 355-369.  doi: 10.1111/j.1469-1809.1937.tb02153.x.
    [5] A. Friedman, Partial Differential Equations of Parabolic Type, Englewood Cliffs, NJ: Prentice-Hall, 1964.
    [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.
    [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.
    [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.
    [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.
    [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. 
    [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.
    [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.
    [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.
    [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.
    [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.
    [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.
    [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.
    [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.
    [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.
    [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.
    [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.
    [22] W. Shen, Traveling waves in time dependent bistable media, Differential Integral Equations, 19 (2006), 241-278. 
    [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.
    [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.
    [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.
    [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.
    [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.
    [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.
    [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.
    [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.
    [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.
    [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.
  • 加载中
SHARE

Article Metrics

HTML views(1863) PDF downloads(481) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return