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Stability of planar waves in mono-stable reaction-diffusion equations
Authors:
Guangying Lv and Mingxin Wang
Journal:
Proc. Amer. Math. Soc. 139 (2011), 3611-3621
MSC (2010):
Primary 35B35; Secondary 35K57
Posted:
February 18, 2011
MathSciNet review:
2813391
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Additional Information
Abstract: This paper is concerned with the asymptotic stability of planar waves in mono-stable reaction-diffusion equations in  , where  . Under initial perturbation that decays at space infinity, the perturbed solution converges to planar waves as  . The convergence is uniform in  .
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- X. F. Chen and J.-S. Guo, Existence and asymptotic stability of traveling waves of discrete quasilinear monostable equations, J. Differential Equations 184 (2002), 549-569. MR 1929888 (2003h:35107)
- 2.
- R. A. Fisher, The advance of advantageous genes, Ann. Eugenics 7 (1937), 355-369.
- 3.
- T. Kapitula, Multidimensional stability of planar traveling waves, Trans. Amer. Math. Soc. 349 (1997), 257-269. MR 1360225 (97d:35104)
- 4.
- A. N. Kolmogorov, I. G. Petrovsky and N. S. Piskunov, Etude de l'equation de la diffusion avec croissance de la quantite de matiere et son application a un probleme biologique, Bull. Univ. Moskov. Ser. Internat. Sect. 1 (1937), 1-25.
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- G. Y. Lv and M. X. Wang, Nonlinear stability of travelling wave fronts for delayed reaction diffusion equations, Nonlinearity 23 (2010), 845-873. MR 2602017
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- H. Matano, M. Nara and M. Taniguchi, Stability of planar waves in the Allen-Cahn equation, Comm. Partial Differential Equations 34 (2009), 976-1002. MR 2560308 (2010j:35289)
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- G. Raugel and K. Kirchgässner, Stability of fronts for a KPP-system, II: The critical case, J. Differential Equations 146 (1998), 399-456. MR 1631295 (2000j:35133)
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- D. H. Sattinger, On the stability of waves of nonlinear parabolic systems, Adv. Math 22 (1976), 312-355. MR 0435602 (55:8561)
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- A. I. Volpert, V. A. Volpert and V. A. Volpert, Traveling wave solutions of parabolic systems, Translation of Mathematicial Monographs, Vol. 140, Amer. Math. Soc., Providence, RI, 1994. MR 1297766 (96c:35092)
- 11.
- J. X. Xin, Multidimensional stability of traveling waves in a bistable reaction-diffusion equation I, Comm. Partial Differential Equations 17 (1992) 1889-1899. MR 1194743 (94c:35104)
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Additional Information
Guangying Lv
Affiliation:
Department of Mathematics, Southeast University, Nanjing 210018, People’s Republic of China
Email:
gyLvmaths@126.com
Mingxin Wang
Affiliation:
Science Research Center, Harbin Institute of Technology, Harbin 150080, People’s Republic of China
Email:
mxwang@hit.edu.cn
DOI:
http://dx.doi.org/10.1090/S0002-9939-2011-10767-6
PII:
S 0002-9939(2011)10767-6
Keywords:
Traveling wave fronts,
stability,
upper and lower solutions,
reaction-diffusion equation.
Received by editor(s):
March 7, 2010
Received by editor(s) in revised form:
March 20, 2010, August 16, 2010, and August 29, 2010
Posted:
February 18, 2011
Additional Notes:
The first author is supported by the JSPS Innovation Program CX09$B_{-}$044Z and the Scientific Research Foundation of the Graduate School of Southeast University (YBJJ1009).
The second author is supported by PRC Grants NSFC 10771032 and 11071049.
Communicated by:
Yingfei Yi
Article copyright:
© Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
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