Existence and Stability of Travelling Front Solutions for General Auto-catalytic Chemical Reaction Systems

Y. Li¹ ; Y. Wu²

¹ Department of Mathematics, Xi’an Jiaotong University, Xi’an 710049, P.R. China and Department of Mathematics and Statistics, Wright State University, Dayton, OH45435, USA
² College of Mathematical Sciences, Capital Normal University, Beijing, 100048, P.R. China

Mathematical modelling of natural phenomena, Tome 8 (2013) no. 3, pp. 104-132

Résumé

This paper is concerned with the existence and stability of travelling front solutions for more general autocatalytic chemical reaction systems ut = duxx − uf(v), vt = vxx + uf(v) with d > 0 and d ≠ 1, where f(v) has super-linear or linear degeneracy at v = 0. By applying Lyapunov-Schmidt decomposition method in some appropriate exponentially weighted spaces, we obtain the existence and continuous dependence of wave fronts with some critical speeds and with exponential spatial decay for d near 1. By applying special phase plane analysis and approximate center manifold theorem, the existence of traveling waves with algebraic spatial decay or with some lower exponential decay is also obtained for d > 0. Further, by spectral estimates and Evans function method, the wave fronts with exponential spatial decay are proved to be spectrally or linearly stable in some suitable exponentially weighted spaces. Finally, by adopting the main idea of proof in [12] and some similar arguments as in [21], the waves with critical speeds or with non-critical speeds are proved to be locally exponentially stable in some exponentially weighted spaces and Lyapunov stable in Cunif(ℝ) space, if the initial perturbation of the waves is small in both the weighted and unweighted norms; the perturbation of the waves also stays small in L1(ℝ) norm and decays algebraically in Cunif(ℝ) norm, if the initial perturbation is in addition small in L1 norm.

Détail

DOI : 10.1051/mmnp/20138308

Bibliographie

[1] S. Ai, W. Huang Proc. Royal Soc. Edin. A 2007 671 700

[2] J. Alexander, R. Gardner, C. Jones J. Reine Angew. Math. 1990 167 212

[3] N.J. Balmforth, R.V. Crastev, J.A. Malham Proc. R. Soc. Lond. A 1999 1401 1433

[4] J.W. Bebernes, C.-M. Li, Y. Li Rocky Mountain J. Math. 1997 123 150

[5] H. Berestycki, L. Nirenberg Ann. Institute. Henri Poincare, Analysis non lineaire 1992 497 572

[6] J. Billingham, D.J. Needham Dyn. Stab. Syst. 1991 33 49

[7] M. Bramson Mem. Amer. Math. Soc. 1983

[8] J. Carr. Application of centre manifold theory. Applied Mathematical Sciences, Vol.35, Springer-Verlag, New York, 1981.

[9] X.F. Chen, Y.W. Qi SIAM J.Math. Anal. 2007 437 448

[10] X.F. Chen, Y.W. Qi J. Differential Equations 2009 3038 3057

[11] S. Focant, Th. Galley Physica D 1998 346 368

[12] A. Ghazaryan, Y. Latushkin, S. Schecter Stability of traveling waves for a class of reaction-diffusion systems that arise in chemical reaction models SIAM J. Math. Anal. 2010 2434 2472

[13] F. Hamel, L. Roques J. Differential Equations 2010 1726 1745

[14] D. Henry. Geometric theory of semilinear parabolic equations. Lecture Notes in Mathematics 840, Springer-Verlag, New York, 1981.

[15] Y. Hosono Japan J. Indust. Appl. Math. 2007 79 104

[16] X. Hou, Y. Li Discrete Contin. Dyn. Syst. 2006 681 701

[17] T. Kato. Peturbation theory for linear operators. corrected printing of the second edition, Springer-Verlag, Berlin, 1980.

[18] K. Kirchgässner, G. Raugel J. Differential Equations 1998 399 456

[19] M. Marion Nonl. Anal. TMA 1998 1269 1292

[20] M.J. Metcalf, J.H. Merkin, S.K. Scott Proc. R. Soc. Lond. B 1994 155 174

[21] Y. Li, Y. Wu SIAM J. Math. Anal. 2012 1474 1521

[22] A. De Pablo, J. L. Vazquez Euro. J. Applied Mathematics 1998 285 304

[23] R.L. Pego, M.I. Weinstein Philos. Trans. Roy. Soc. London Ser. A 1992 47 94

[24] F. Rothe Proc. Roy. Soc. Edinburgh, Sect A 1978 213 234

[25] D.H. Sattinger Advances in Math. 1976 312 255

[26] J.A. Sherratt, B.P. Marchant IMA J. Appl. Math. 1996 289 302

[27] H. Takase, B.D. Sleeman Proc. R. Soc. Lond. A 1999 1561 1598

[28] K. Uchiyama J. Math. Kyoto. Univ. 1978 453 508

[29] Y. Wu, Y.X. Wu. Asymptotic behavior of solution to degeneate Fisher equations with algebraic decaying initial values. peprint.

[30] Y. Wu, X. Xing Discrete Contin. Dyn. Syst. A 2008 1123 1139

[31] Y. Wu, X. Xing J. Math. Anal. Appl. 2005 698 711

[32] Y. Wu, X. Xing, Q. Ye Discrete Contin. Dyn. Syst. A 2006 47 66