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

Hamel, François

Annales de l'I.H.P. Analyse non linéaire, Tome 14 (1997) no. 4, pp. 457-498.

[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. | MR | Zbl

[2] D.G. Aronson and H.F. Weinberger, Multidimensional nonlinear diffusions arising in population genetics, Adv. in Math., Vol. 30, 1978, pp. 33-76. | MR | Zbl

[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. | MR | Zbl

[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, Pitman Longman, Harbow, UK, 1990. | MR | Zbl

[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. | EuDML | MR | Zbl

[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. | MR | Zbl

[7] H. Berestycki, B. Larrouturou, P.L. Lions and J.-M. Roquejoffre, An elliptic system modelling the propagation of a multidimensional flame, preprint.

[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. | MR | Zbl

[9] H. Berestycki and L. Nirenberg, Travelling fronts in cylinders, Ann. Inst. H.Poincaré, Analyse Non Linéaire, Vol. 9, 5, 1992, pp. 497-572. | EuDML | Numdam | MR | Zbl

[10] H. Berestycki and G.I. Sivashinsky, Flame extinction by a periodic flow field, SIAM, J. Appl. Math., Vol. 51, 1991, pp. 344-350. | MR | Zbl

[11] J.D. Buckmaster and G.S.S. Ludford, Lectures on Mathematical Combustion, CBMS-NSF Conf. Series in Applied Math., Vol. 43, SIAM, 1983. | MR | Zbl

[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. | MR | Zbl

[13] P.S. Hagan, Travelling waves and multiple travelling waves solutions of parabolic equations, SIAM J. Math. Anal., Vol. 13, 1982, pp. 717-738. | Zbl

[14] F. Hamel, Reaction-diffusion problems in cylinders with no invariance by translation. Part II: Monotone perturbations, Ann. Inst. H. Poincaré, Vol. 14, 1997. | Numdam | MR | Zbl

[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. | MR | Zbl

[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] D. Henry, Geometric theory of semilinear parabolic equations, Lectures Notes in Math., Springer Verlag, New York, 1981. | MR | Zbl

[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. | MR | Zbl

[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. | Zbl

[20] M.G. Krein and M.A. Rutman, Linear operators leaving invariant a cone in a Banach space, AMS Translation, Vol. 26, 1950. | MR

[21] A. Liñan, Fluid Dynamic Aspects of Combustion Theory, course of IAC. Mauro Picone, Rome, 1989.

[22] G. Papanicolaou and X. Xin, Reaction-diffusion Fronts in Periodically Layered media, J. Statist. Phys., Vol. 63, 1991, pp. 915-931. | MR

[23] A. Pazy, Asymptotic expansions of solutions of ordinary differential equations in Hilbert space, Arch. Rat. Mech. Anal., Vol. 24, 1967, pp. 193-218. | MR | Zbl

[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. | MR | Zbl

[25] D.H. Sattinger, Stability of waves of nonlinear parabolic systems, Adv. Math., Vol. 22, 1976, pp. 312-355. | MR | Zbl

[26] G.I. Sivashinsky, Instabilities, pattern formation and turbulence in flames, Ann. Rev. Fluid Mech., Vol. 15, 1983, pp. 179-199. | Zbl

[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. | MR | Zbl

[28] F. Williams, Combustion Theory, Addison-Wesley, Reading MA, 1983.

[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. | MR | Zbl

[30] X. Xin, Existence of planar flame fronts in convective-diffusive periodic media, Arch. Rat. Mech. Analysis, Vol. 121, 1992, pp. 205-233. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[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 York 1988, pp. 131-140.