Eventual monotonicity and convergence to travelling fronts for the solutions of parabolic equations in cylinders
Annales de l'I.H.P. Analyse non linéaire, Tome 14 (1997) no. 4, pp. 499-552.
@article{AIHPC_1997__14_4_499_0,
author = {Roquejoffre, Jean-Michel},
title = {Eventual monotonicity and convergence to travelling fronts for the solutions of parabolic equations in cylinders},
journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
pages = {499--552},
publisher = {Gauthier-Villars},
volume = {14},
number = {4},
year = {1997},
zbl = {0884.35013},
mrnumber = {1464532},
language = {en},
url = {www.numdam.org/item/AIHPC_1997__14_4_499_0/}
}
Roquejoffre, Jean-Michel. Eventual monotonicity and convergence to travelling fronts for the solutions of parabolic equations in cylinders. Annales de l'I.H.P. Analyse non linéaire, Tome 14 (1997) no. 4, pp. 499-552. http://www.numdam.org/item/AIHPC_1997__14_4_499_0/

[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 155203 | Zbl 0117.10001

[2] D.G. Aronson and H.F. Weinberger, Nonlinear diffusion in population genetics, combustion and nerve propagation, Partial differential equations and related topics, Lect. Notes in Math., Vol. 446, Springer Verlag, New-York, 1975, pp. 5-49. | MR 427837 | Zbl 0325.35050

[3] D.G. Aronson and H.F. Weinberger, Multidimensional diffusion arising in population genetics, Adv. Math., Vol. 30, 1978, pp. 33-58. | MR 511740 | Zbl 0407.92014

[4] H. Berestycki, L.A. Caffarelli and L. Nirenberg, Uniform estimates for regularisations of free boundary problems, Analysis and partial differential equations, C. Sadosky & M. Decker eds., 1990, pp. 567-617. | Zbl 0702.35252

[5] H. Berestycki and B. Larrouturou, Planar travelling front solutions of reaction-diffusion problems, to appear.

[6] H. Berestycki, B. Larrouturou and P.L. Lions, Multidimensional travelling wave solutions of a flame propagation model, Arch. Rat. Mech. Anal., Vol. 111, 1990, pp. 33-49. | MR 1051478 | Zbl 0711.35066

[7] H. Berestycki, B. Larrouturou and J.M. Roquejoffre, Stability of travelling fronts in a model for flame propagation, Part I: Linear analysis, Arch. Rat. Mech. Anal., Vol. 117, 1992, pp. 97-117. | MR 1145107 | Zbl 0763.76033

[8] H. Berestycki and L. Nirenberg, Some qualitative properties of solutions of semilinear equations in cylindrical domains, Analysis et Cetera (dedicated to J. Moser), P. H. Rabinowitz & E. Zehnder eds., Academic Press, New-York, 1990, pp. 115-164. | Zbl 0705.35004

[9] H. Berestycki and L. Nirenberg, Travelling fronts in cylinders, Ann. IHP, Analyse non linéaire, Vol. 9, 1992, pp. 497-573. | Numdam | MR 1191008 | Zbl 0799.35073

[10] H. Berestycki and L. Nirenberg, On the method of moving planes and the sliding method, Bol. da So. Brasileira de Matematica, Vol. 22, 1991, pp. 1-37 | MR 1159383 | Zbl 0784.35025

[11] P.C. Fife and J.B. Mcleod, The approach of solutions of nonlinear diffusion equations by travelling front solutions, Arch. Rat. Mech. Anal., Vol. 65, 1977, pp. 335-361. | MR 442480 | Zbl 0361.35035

[12] A. Friedman, Partial differential equations of parabolic type, Prentice Hall, 1964. | MR 181836 | Zbl 0144.34903

[13] B. Gidas, W.M. Ni And L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys., Vol. 68, 1979, pp. 209-243. | MR 544879 | Zbl 0425.35020

[14] L. Glangetas and J.M. Roquejoffre, Bifurcations of travelling waves in the thermo-diffusive model for flame propagation, Arch. Rat. Mech. Anal., Vol. 134, 1996, pp. 341-402. | MR 1414292 | Zbl 0920.76091

[15] 1. C. Gohberg and M.G. Krein, Introduction to the theory of linear nonselfadjoint operators, Transl. Math. Monog. Vol. 18, Am. Math. Soc., Providence, R.I., 1969. | MR 246142 | Zbl 0181.13504

[16] D. Henry, Geometric theory of semilinear parabolic equations, Lecture Notes in Mathematics, Springer Verlag, New-York, 1981. | MR 610244 | Zbl 0456.35001

[17] M.W. Hirsch, Stability and convergence in strongly monotone dynamical systems, J. Reine Angew. Math., Vol. 383, 1988, pp. 1-53. | MR 921986 | Zbl 0624.58017

[18] A.V. Ivanov, The Harnack inequality for weak solutions of quasilinear parabolic equations of second order, Soviet Math. Dokl., Vol. 8, 1967, pp. 463-466. | Zbl 0163.33902

[19] C.K.R.T. Jones, Asymptotic behaviour of a reaction-diffusion equation in higher dimensions, Rocky Mountain J. Math., Vol. 13, 1983, pp. 355-364. | MR 702830 | Zbl 0528.35054

[20] O.A. Ladyzhenskaya, V.A. Solonnikov and N.N. Uralceva, Linear and quasilinear equations of parabolic type, Transl. Math. Monog., Vol. 23, Am. Math. Soc., Providence, R.I, 1968. | Zbl 0174.15403

[21] P.L. Lions, Structure of the set of steady-state solutions and asymptotic behaviour of semilinear heat equations, J. Diff. Eq., Vol. 53, 1984, pp. 362-386. | MR 752205 | Zbl 0491.35057

[22] J.F. Mallordy and J.M. Roquejoffre, A parabolic equation of KPP type, SIAM J. Math. Anal., Vol. 26, 1995, pp. 1-20. | MR 1311879 | Zbl 0813.35041

[23] H. Matano, Strong comparison principles in nonlinear parabolic equations, Nonlinear parabolic equations: Qualitative properties of solutions, L. Boccardo & A. Tesei eds., Pitman Longman 1987. | Zbl 0664.35048

[24] H. Matano, Existence of nontrivial unstable sets for equilibriums in strongly order-preserving dynamical systems, J. Fac. Sci. Univ. Tokyo, Vol. 30, 1983, pp. 645-673. | MR 731522 | Zbl 0545.35042

[25] A. Pazy, Asymptotic expansions of solutions of ordinary differential equations in Hilbert space, Arch. Rat. Mech. Anal., Vol. 24 1967, pp. 193-218. | MR 209618 | Zbl 0147.12303

[26] J.M. Roquejoffre, Stability of travelling fronts in a model for flame propagation, Part II: Nonlinear stability, Arch. Rat. Mech. Anal., Vol. 117, 1992, pp. 119-153. | MR 1145108 | Zbl 0763.76034

[27] J.M. Roquejoffre, Convergence to travelling waves for solutions of a class of semilinear parabolic equations, J. Diff. Eq., Vol. 108, 1994, pp. 262-295. | MR 1270581 | Zbl 0806.35093

[28] J.M. Roquejoffre and D. Terman, On the stability of steady planar premixed flames, Nonlinear analysis, TMA, Vol. 22, 1994, pp. 137-154. | MR 1258953 | Zbl 0802.35071

[29] D.H. Sattinger, Stability of waves of nonlinear parabolic systems, Adv. Math., Vol. 22, 1976, pp. 312-355. | MR 435602 | Zbl 0344.35051

[30] J. Smoller, Shock waves and reaction diffusion equations, Grund. math. Wiss., Vol. 258, SPRINGER Verlag. | MR 688146 | Zbl 0508.35002

[31] H.B. Stewart, Generation of analytic semigroups by strongly elliptic operators under general boundary conditions, Trans. Am. Math. Soc., Vol. 259, 1980, pp. 299-310. | MR 561838 | Zbl 0451.35033

[32] J.M. Vega, Multidimensional travelling wave fronts in a model from combustion theory and in related problems, Differential and Integral equations, Vol. 6 1993, pp. 131-155. | Zbl 0786.35080

[33] J.B. Zeldovich and D.A. Frank-Kamenetskii, A theory of thermal propagation of flame, Acta Physiochimica URSS, Vol. 9, 1938.