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doi: 10.3934/era.2021058
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Multidimensional stability of pyramidal traveling fronts in degenerate Fisher-KPP monostable and combustion equations

College of Science, Northwest A&F University, Yangling, Shaanxi 712100, China

* Corresponding author: Zhen-Hui Bu

Received  April 2021 Revised  July 2021 Early access August 2021

Fund Project: This paper was supported by Natural Science Basic Research Program in Shaanxi Province of China (2020JQ-236, 2019JQ-246)

In this paper, multidimensional stability of pyramidal traveling fronts are studied to the reaction-diffusion equations with degenerate Fisher-KPP monostable and combustion nonlinearities. By constructing supersolutions and subsolutions coupled with the comparison principle, we firstly prove that under any initial perturbation (possibly large) decaying at space infinity, the three-dimensional pyramidal traveling fronts are asymptotically stable in weighted $ L^{\infty} $ spaces on $ \mathbb{R}^{n}\; (n\geq4) $. Secondly, we show that under general bounded perturbations (even very small), the pyramidal traveling fronts are not asymptotically stable by constructing a solution which oscillates permanently between two three-dimensional pyramidal traveling fronts on $ \mathbb{R}^{4} $.

Citation: Denghui Wu, Zhen-Hui Bu. Multidimensional stability of pyramidal traveling fronts in degenerate Fisher-KPP monostable and combustion equations. Electronic Research Archive, doi: 10.3934/era.2021058
References:
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Z.-H. Bu and Z.-C. Wang, Stability of pyramidal traveling fronts in the degenerate monostable and combustion equations I, Discrete Contin. Dyn. Syst., 37 (2017), 2395-2430.  doi: 10.3934/dcds.2017104.  Google Scholar

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Z.-H. Bu and Z.-C. Wang, Global stability of V-shaped traveling fronts in combustion and degenerate monostable equations, Discrete Contin. Dyn. Syst., 38 (2018), 2251-2286.  doi: 10.3934/dcds.2018093.  Google Scholar

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Z.-H. Bu and Z.-C. Wang, Multidimensional stability of traveling fronts in combustion and non-KPP monostable equations, Z. Angew. Math. Phys., 69 (2018), Paper No. 12, 27 pp. doi: 10.1007/s00033-017-0906-5.  Google Scholar

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H. Cheng and R. Yuan, Multidimensional stability of disturbed pyramidal traveling fronts in the Allen-Cahn equation, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 1015-1029.  doi: 10.3934/dcdsb.2015.20.1015.  Google Scholar

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F. HamelR. Monneau and J.-M. Roquejoffre, Existence and qualitative properties of multidimensional conical bistable fronts, Discrete Contin. Dyn. Syst., 13 (2005), 1069-1096.  doi: 10.3934/dcds.2005.13.1069.  Google Scholar

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J. A. LeachD. J. Needham and A. L. Kay, The evolution of reaction-diffusion waves in a class of scalar reaction-diffusion equations: algebraic decay rates, Phys. D, 167 (2002), 153-182.  doi: 10.1016/S0167-2789(02)00428-1.  Google Scholar

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C. D. Levermore and J. X. 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

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

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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

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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

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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.  Google Scholar

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W. ShengW. Li and Z. 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

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M. Taniguchi, Traveling fronts of pyramidal shapes in the Allen-Cahn equations, SIAM J. Math. Anal., 39 (2007), 319-344.  doi: 10.1137/060661788.  Google Scholar

[18]

M. Taniguchi, The uniqueness and asymptotic stability of pyramidal traveling fronts in the Allen-Cahn equations, J. Differential Equations, 246 (2009), 2103-2130.  doi: 10.1016/j.jde.2008.06.037.  Google Scholar

[19]

Z.-C. Wang and Z.-H. Bu, Nonplanar traveling fronts in reaction-diffusion equations with combustion and degenerate Fisher-KPP nonlinearities, J. Differential Equations, 260 (2016), 6405-6450.  doi: 10.1016/j.jde.2015.12.045.  Google Scholar

[20]

J. X. 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

[21]

H. Zeng, Multidimensional stability of traveling fronts in monostable reaction-diffusion equations with complex perturbations, Sci. China Math., 57 (2014), 353-366.  doi: 10.1007/s11425-013-4617-x.  Google Scholar

[22]

H. Zeng, Stability of planar travelling waves for bistable reaction-diffusion equations in multiple dimensions, Appl. Anal., 93 (2014), 653-664.  doi: 10.1080/00036811.2013.797075.  Google Scholar

show all references

References:
[1]

D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusions arising in population genetics, Adv. Math., 30 (1978), 33-76.  doi: 10.1016/0001-8708(78)90130-5.  Google Scholar

[2]

Z.-H. Bu and Z.-C. Wang, Stability of pyramidal traveling fronts in the degenerate monostable and combustion equations I, Discrete Contin. Dyn. Syst., 37 (2017), 2395-2430.  doi: 10.3934/dcds.2017104.  Google Scholar

[3]

Z.-H. Bu and Z.-C. Wang, Global stability of V-shaped traveling fronts in combustion and degenerate monostable equations, Discrete Contin. Dyn. Syst., 38 (2018), 2251-2286.  doi: 10.3934/dcds.2018093.  Google Scholar

[4]

Z.-H. Bu and Z.-C. Wang, Multidimensional stability of traveling fronts in combustion and non-KPP monostable equations, Z. Angew. Math. Phys., 69 (2018), Paper No. 12, 27 pp. doi: 10.1007/s00033-017-0906-5.  Google Scholar

[5]

H. Cheng and R. Yuan, Multidimensional stability of disturbed pyramidal traveling fronts in the Allen-Cahn equation, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 1015-1029.  doi: 10.3934/dcdsb.2015.20.1015.  Google Scholar

[6]

F. Hamel and R. Monneau, Solutions of semilinear elliptic equations in RN with conical-shaped level sets, Comm. Partial Differential Equations, 25 (2000), 769-819.  doi: 10.1080/03605300008821532.  Google Scholar

[7]

F. HamelR. Monneau and J.-M. Roquejoffre, Existence and qualitative properties of multidimensional conical bistable fronts, Discrete Contin. Dyn. Syst., 13 (2005), 1069-1096.  doi: 10.3934/dcds.2005.13.1069.  Google Scholar

[8]

J. He and Y. Wu, Spatial decay and stability of traveling fronts for degenerate Fisher type equations in cylinder, J. Differential Equations, 265 (2018), 5066-5114.  doi: 10.1016/j.jde.2018.06.031.  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]

J. A. LeachD. J. Needham and A. L. Kay, The evolution of reaction-diffusion waves in a class of scalar reaction-diffusion equations: algebraic decay rates, Phys. D, 167 (2002), 153-182.  doi: 10.1016/S0167-2789(02)00428-1.  Google Scholar

[11]

C. D. Levermore and J. X. 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 mono-stable reaction-diffusion equations, Proc. Amer. Math. Soc., 139 (2011), 3611-3621.  doi: 10.1090/S0002-9939-2011-10767-6.  Google Scholar

[13]

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

[14]

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

[15]

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.  Google Scholar

[16]

W. ShengW. Li and Z. 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

[17]

M. Taniguchi, Traveling fronts of pyramidal shapes in the Allen-Cahn equations, SIAM J. Math. Anal., 39 (2007), 319-344.  doi: 10.1137/060661788.  Google Scholar

[18]

M. Taniguchi, The uniqueness and asymptotic stability of pyramidal traveling fronts in the Allen-Cahn equations, J. Differential Equations, 246 (2009), 2103-2130.  doi: 10.1016/j.jde.2008.06.037.  Google Scholar

[19]

Z.-C. Wang and Z.-H. Bu, Nonplanar traveling fronts in reaction-diffusion equations with combustion and degenerate Fisher-KPP nonlinearities, J. Differential Equations, 260 (2016), 6405-6450.  doi: 10.1016/j.jde.2015.12.045.  Google Scholar

[20]

J. X. 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

[21]

H. Zeng, Multidimensional stability of traveling fronts in monostable reaction-diffusion equations with complex perturbations, Sci. China Math., 57 (2014), 353-366.  doi: 10.1007/s11425-013-4617-x.  Google Scholar

[22]

H. Zeng, Stability of planar travelling waves for bistable reaction-diffusion equations in multiple dimensions, Appl. Anal., 93 (2014), 653-664.  doi: 10.1080/00036811.2013.797075.  Google Scholar

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