Reaction-diffusion problems in cylinders with no invariance by translation. Part I : small perturbations

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1. [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. Zbl0117.10001MR155203
2. [2] D.G. Aronson and H.F. Weinberger, Multidimensional nonlinear diffusions arising in population genetics, Adv. in Math., Vol. 30, 1978, pp. 33-76. Zbl0407.92014MR511740
3. [3] D.L. Barrow and P.W. Bates, Bifurcation and stability of periodic travelling wave for a reaction-diffusion system, J. Diff. Eqns., Vol. 50, 1983 , pp. 218-233. Zbl0494.35056MR719447
4. [4] 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, PitmanLongman, Harbow, UK, 1990. Zbl0755.35090MR1131819
5. [5] 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. Zbl0658.35036MR988546
6. [6] 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. Zbl0711.35066MR1051478
7. [7] H. Berestycki, B. Larrouturou, P.L. Lions and J.-M. Roquejoffre, An elliptic system modelling the propagation of a multidimensional flame, preprint. 
8. [8] 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. Zbl0763.76033MR1145107
9. [9] H. Berestycki and L. Nirenberg, Travelling fronts in cylinders, Ann. Inst. H.Poincaré, Analyse Non Linéaire, Vol. 9, 5, 1992, pp. 497-572. Zbl0799.35073MR1191008
10. [10] H. Berestycki and G.I. Sivashinsky, Flame extinction by a periodic flow field, SIAM, J. Appl. Math., Vol. 51, 1991, pp. 344-350. Zbl0717.76108MR1095023
11. [11] J.D. Buckmaster and G.S.S. Ludford, Lectures on Mathematical Combustion, CBMS-NSF Conf. Series in Applied Math., Vol. 43, SIAM, 1983. Zbl0574.76001MR765073
12. [12] 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. Zbl0361.35035MR442480
13. [13] P.S. Hagan, Travelling waves and multiple travelling waves solutions of parabolic equations, SIAM J. Math. Anal., Vol. 13, 1982, pp. 717-738. Zbl0504.35050
14. [14] F. Hamel, Reaction-diffusion problems in cylinders with no invariance by translation. Part II: Monotone perturbations, Ann. Inst. H. Poincaré, Vol. 14, 1997. Zbl0902.35036MR1470782
15. [15] F. Hamel, Problèmes de réaction-diffusion sans invariance par translation dans des cylindres infinis, C. R. Acad. Sci. Paris, Vol. 321, I, 1995, pp. 855-860. Zbl0839.35045MR1355841
16. [16] F. Hamel, Quelques problèmes d'ondes progressives dans les équations aux dérivées partielles et applications à la théorie de la combustion, Ph. D. Thesis, Univ. Paris VI, France, 1996. 
17. [17] D. Henry, Geometric theory of semilinear parabolic equations, Lectures Notes in Math., Springer Verlag, New York, 1981. Zbl0456.35001MR610244
18. [18] Ja.I. Kanel, On the stability of solutions of the equations of combustion theory for finite initial functions, Mat Sbornik, Vol. 65, (107), 1964, pp. 398-413. Zbl0168.36301MR177209
19. [19] 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 Moscou (Bjul. Moskowskogo Gos. Univ.), Série internationale, section A.l, 1937, pp. 1-26. Zbl0018.32106
20. [20] M.G. Krein and M.A. Rutman, Linear operators leaving invariant a cone in a Banach space, AMS Translation, Vol. 26, 1950. Zbl0030.12902MR38008
21. [21] A. Liñan, Fluid Dynamic Aspects of Combustion Theory, course of IAC. Mauro Picone, Rome, 1989. 
22. [22] G. Papanicolaou and X. Xin, Reaction-diffusion Fronts in Periodically Layered media, J. Statist. Phys., Vol. 63, 1991, pp. 915-931. MR1116041
23. [23] A. Pazy, Asymptotic expansions of solutions of ordinary differential equations in Hilbert space, Arch. Rat. Mech. Anal., Vol. 24, 1967, pp. 193-218. Zbl0147.12303MR209618
24. [24] J.M. Roquejoffre, Stability of traveling fronts in a curved flame model, Part II: Non-linear Orbital Stability, Arch. Rat. Mech. Anal., Vol. 117, 1992, pp. 119-153. Zbl0763.76034MR1145108
25. [25] D.H. Sattinger, Stability of waves of nonlinear parabolic systems, Adv. Math., Vol. 22, 1976, pp. 312-355. Zbl0344.35051MR435602
26. [26] G.I. Sivashinsky, Instabilities, pattern formation and turbulence in flames, Ann. Rev. Fluid Mech., Vol. 15, 1983, pp. 179-199. Zbl0538.76053
27. [27] J.M. Vega, Multidimensional travelling fronts in a model from combustion theory and related problems, Diff. and Int. Eq., Vol. 6, 1993, pp. 131-155. Zbl0786.35080MR1190170
28. [28] F. Williams, Combustion Theory, Addison-Wesley, Reading MA, 1983. 
29. [29] 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. Zbl0727.35070MR1129338
30. [30] X. Xin, Existence of planar flame fronts in convective-diffusive periodic media, Arch. Rat. Mech. Analysis, Vol. 121, 1992, pp. 205-233. Zbl0764.76074MR1188981
31. [3 1 ] 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. Zbl0769.35033MR1129560
32. [32] 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. Zbl1102.35340MR1251222
33. [33] J.B. Zeldovic and D.A. Frank-Kamenetskii, A theory of thermal propagation of flame, Acta physiochimica URSS, Vol. 9, 1938, pp. 341-350, English translation in Dynamics of curved fronts, R. Pelcé ed., Perspectives in Physics Series, Academic Press, New York1988, pp. 131-140. 

1. François Hamel, Reaction-diffusion problems in cylinders with no invariance by translation. Part II : monotone perturbations