Reaction-diffusion problems in cylinders with no invariance by translation. Part II: Monotone perturbations

Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire

Volumn 14, Issue 5, 1997, Pages 555-596

  Hamel, François ^a  

^a SORBONNE UNIVERSITÉ   (France)

Author keywords

Asymptotic behaviours; Monotonicity properties; Nonlinear PDE's; Sliding method; Sub and Supersolutions

Indexed keywords

EID: 0039835893     PISSN: 02941449     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0294-1449(97)80126-6     Document Type: Article

References (26)

1. S. AGMON and L. NIRENBERG, Properties of solutions of ordinary differential equations in Banach space, Comm. Pure Appl. Math., Vol. 16, 1963, pp. 121-239.
2. D. G. ARONSON and H. F. WEINBERGER, Multidimensional nonlinear diffusions arising in population genetics, Adv. in Math., Vol. 30, 1978, pp. 33-58.
3. H. BERESTYCKI and B. LARROUTUROU, Quelques aspects mathématiques de la propagation des flammes prémélangées, Nonlinear p.d.e. and their applications, Collège de France seminar, Vol. 10, Brézis and Lions eds, Pitman Longman, Harbow, UK, 1990.
4. H. BERESTYCKI and B. LARROUTUROU, A semilinear elliptic equation in a strip arising in a two-dimensional flame propagation model, J. Reine Angew. Math., Vol. 396, 1989, pp. 14-40.
5. H. BERESTYCKI, B. LARROUTUROU and P. L. LIONS, Multidimensional traveling-wave solutions of a flame propagation model, Arch. Rat. Mech. Anal., Vol. 111, 1990, pp. 33-49.
6. H. BERESTYCKI, B. LARROUTUROU and J. M. ROQUEJOFFRE, Stability of traveling fronts in a curved flame model, Part I, Linear Analysis, Arch. Rat. Mech. Anal., Vol. 117, 1992, pp. 97-117.
7. H. BERESTYCKI and L. NIRENBERG, Travelling fronts in cylinders, Ann. Inst. H. Poincaré, Analyse Non Linéaire, Vol. 9, 5, 1992, pp. 497-572.
8. H. BERESTYCKI and L. NIRENBERG, On the method of moving planes and the sliding method, Bol. da Soc. Braseleira de Matematica, Vol. 22, 1991, pp. 1-37.
9. P. C. FIFE and J. B. MCLEOD, The approach of solutions of non-linear diffusion equations to traveling front solutions, Arch. Rat. Mech. Anal., Vol. 65, 1977, pp. 335-361.
10. F. HAMEL, Reaction-diffusion problems in cylinders with no invariance by translation, part I: Small perturbations, Ann. Inst. H. Poincaré, Analyse Non Linéaire, Vol. 14, 1997, pp. 457-498.
11. F. HAMEL, Formules min-max pour les vitesses d'ondes progressives multidimensionnelles, preprint Labo. Ana. Num., no 96032, Univ. Paris VI, France, 1996.
12. Ja. I. KANEL'. Stabilization of solution of the Cauchy problem for equations encountred in combustion theory, Mat. Sbornik., Vol. 59, 1962, pp. 245-288.
13. A. N. KOLMOGOROV, I. G. PETROVSKY and N. S. PISKUNOV, Etude de l'équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique, Bulletin Université d'Etat à Moscow (Bjul. Moskowskogo Gos. Univ.), Série internationale, section A.1, 1937, pp. 1-26.
14. A. LIÑAN, Fluid Dynamic Aspects of Combustion Theory, course of IAC. Mauro Picone, Rome, 1989.
15. A. PAZY, Asymptotic expansions of solutions of ordinary differential equations in Hubert space, Arch. Rat. Mech. Anal., Vol. 24, 1967, pp. 193-218.
16. G. I. SIVASHINSKY, Instabilities, pattern formation and turbulence in flames, Ann. Rev. Fluid Mech., Vol. 15, 1983, pp. 179-199.
17. J. M. VEGA, On the uniqueness of multidimensional travelling fronts of some semilinear equations, J. Math. Anal. Appl., Vol. 177, 1993, pp. 481-490.
18. J. M. VEGA, The asymptotic behavior of the solutions of some semilinear elliptic equations in cylindrical domains, J. Diff. Eq., Vol. 102, 1993, pp. 119-152.
19. V. A. VOLPERT and A. I. VOLPERT, Existence and Stability of Stationary Solutions for a Class of Semilinear Parabolic Systems, Comm. in Part. Diff. Eq., Vol. 18, 1993, pp. 2051-2069.
20. F. WILLIAMS, Combustion Theory, Addison-Wesley, reading MA, 1983.
21. X. XIN, Existence and Uniqueness of Travelling Waves in a Reaction-Diffusion Equation with Combustion Nonlinearity, Idiana Univ. Math. J., Vol. 40, No 3, 1991.
22. X. XIN, Existence of planar flame fronts in convective-diffusive periodic media, Arch. Rat. Mech. Analysis, Vol. 121, 1992, pp. 205-233.
23. X. XIN, Existence and stability of travelling waves in periodic media governed by a bistable nonlinearity, J. Dyn. Diff. Eq., Vol. 3, 1991, pp. 541-573.
24. X. XIN, Existence and non existence of travelling waves and reaction-diffusion front propagation in periodic media, J. Statist. Phys., Vol. 73, 1993, pp. 893-926.
25. J. B. ZELDOVICH and D. A. FRANK-KAMENETSKII, A theory of thermal propagation of flame, Acta physiochimica URSS, Vol. 9, 1938. English translation in Dynamics of curved fronts, R. Pelcé ed., Perspectives in Physics Series, Academic Press, New York, 1988, pp. 131-140.
26. J. B. ZELDOVICH and D. A. FRANK-KAMENETSKII, A theory of thermal propagation of flame, Acta physiochimica URSS, Vol. 9, 1938. English translation in Dynamics of curved fronts, R. Pelcé ed., Perspectives in Physics Series, Academic Press, New York, 1988, pp. 131-140.