Reaction-diffusion systems with prescribed large time behaviour
Vakulenko, S. A.
Annales de l'I.H.P. Physique théorique, Tome 67 (1997), p. 373-410 / Harvested from Numdam
Publié le : 1997-01-01
@article{AIHPA_1997__66_4_373_0,
     author = {Vakulenko, S. A.},
     title = {Reaction-diffusion systems with prescribed large time behaviour},
     journal = {Annales de l'I.H.P. Physique th\'eorique},
     volume = {67},
     year = {1997},
     pages = {373-410},
     mrnumber = {1459513},
     zbl = {0894.35048},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIHPA_1997__66_4_373_0}
}
Vakulenko, S. A. Reaction-diffusion systems with prescribed large time behaviour. Annales de l'I.H.P. Physique théorique, Tome 67 (1997) pp. 373-410. http://gdmltest.u-ga.fr/item/AIHPA_1997__66_4_373_0/

[1] I. Prigogine and G. Nicolis, Self-Organization in Nonequilibrium Systems (Wiley, New York, 1977). | MR 522141 | Zbl 0363.93005

[2] H. Haken, Synergetics. An Introduction (Springer, Heidelberg, New York, 1977). | MR 471840 | Zbl 0396.93001

[3] G. Foias and G. Prodi, Sur le comportement global des solutions nonstationaire des équations de Navier-Stokes en dimensional 2, Rend. Semin. Math. Univ. di Padova, Vol. 39, 1967, pp. 1-34. | Numdam | MR 223716 | Zbl 0176.54103

[4] A.B. Babin, M.I. Vishik, Regular attractors of semigroups and evolution equations, J. Math. Pures Appl., Vol. 62, 1983, pp. 441-491. | MR 735932 | Zbl 0565.47045

[5] O.A. Ladygenskaya, Finding minimal global attractors for Navier-Stokes equations and other partial differential equations, Uspechi Mat. Nauk, Vol. 42, 1987, pp. 25-60. | MR 933994 | Zbl 0687.35072

[6] Yu.S. Il'Ashenko, Weakly contracting systems and attractors of Galerkin approximation for Navier-Stokes equation on two-dimensional torus, Uspechi Mechanics, Vol. 1, 1982, pp. 31-63. | Zbl 0789.35132

[7] J.K. Hale, 1988, Asymptotic behavior of dissipative systems (Providence: American Mathematical Society). | MR 941371 | Zbl 0642.58013

[8] I.D. Chueshov, Global attractors in nonlinear problems, Uspechi Mat. Nauk, Vol. 48, 1993, pp. 135-162. | MR 1243614 | Zbl 0805.58042

[9] P. Constantin, C. Foias, B. Nicolaenko and R. Temam, Integrable manifolds and inertial manifolds for dissipative differential equations, 1989. | Zbl 0683.58002

[10] R. Mane, Reduction of semilinear parabolic equations to finite dimensional C1-flow, Geometry and Topology, Lecture Notes in Mathematics No. 597, Springer-Verlag, New York, 1977, pp. 361-378. | MR 451309 | Zbl 0412.35049

[11] J. Mallet-Paret and G.R. Sell, Inertial manifolds for reaction-diffusion equations in higher space dimensions, J. Amer. Math. Soc., Vol. 1, 1988, pp. 805-866. | MR 943276 | Zbl 0674.35049

[12] R. Temam, Infinite dimensional dynamical systems in mechanics and physics, New York etc., (Springer, 1988). | MR 953967 | Zbl 0662.35001

[13] M. W. Hirsch, Stability and convergence in strongly monotone dynamical systems, J. Reine Angew Math., Vol. 383, 1988, pp. 1-58. | MR 921986 | Zbl 0624.58017

[14] H.L. Smith and H.R. Thieme, Convergence for strongly order preserving semiflows SIAM, J. Math. Anal.., Vol. 22, 1991, pp. 1081-1101. | MR 1112067 | Zbl 0739.34040

[15] P. Polacik and I. Terescak, Convergence to cycles as a typical asyptotic behavior in smooth discrete-time strongly monotone dynamical systems, Arch. Rat. Mech. Anal., Vol. 116, 1991, pp. 339-360. | MR 1132766 | Zbl 0755.58039

[16] P. Polacik and I. Terescak, Exponential separation and invariant bundles for maps in ordered Banach spaces with applications to parabolic equations, J. Dynamics Diff. Equations, Vol. 5, 1993, pp. 279-303. | MR 1223450 | Zbl 0786.58002

[17] B. Fiedler and J. Mallet-Paret, A Poincare-Bendixson theorem for scalar reaction-diffusion equations, Arch. Rat. Mech. and Anal., Vol. 107, 1989, pp. 325-345. | MR 1004714 | Zbl 0704.35070

[18] T.I. Zelenyak, Stabilization of solution of boundary nonlinear problems for a second order parabolic equations with one space variable. Diff. Eq., Vol. 4, 1968, pp. 17-22. | Zbl 0232.35053

[19] P. Polacik, Realization of any finite jet in a scalar semilinear parabolic equation on the ball in R2, Annali Scuola Norm Pisa, Vol. 17, 1991, pp. 83-102. | Numdam | MR 1118222 | Zbl 0774.35041

[20] P. Polacik, Complicated dynamics in Scalar Semilinear Parabolic Equations, In Higher Space Dimensions, J. of Diff. Eq., Vol. 89, 1991, pp. 244-271. | MR 1091478 | Zbl 0738.35027

[21] N.V. Nikolenko, Invariant asymptotically stable tori for perturbed KdV, Uspechi Mat. Nauk, Vol. 35, 1980, pp. 121-181. | MR 595143 | Zbl 0451.35074

[22] B. Fiedler and P. Polacik, Complicated dynamics of scalar reaction-diffusion equations with a nonlocal term, Proc. Roy. Soc., Edinburgh, Vol. 434A, 1990, pp. 167-192. | MR 1059652 | Zbl 0726.35060

[23] S.A. Vakulenko, The oscillating wave fronts, Nonlinear Analysis TMA, Vol. 19, 1992, pp. 1033-1046. | MR 1194143 | Zbl 0801.35042

[24] S.A. Vakulenko, Existence of Ruelle-Takens transition to for some evolution equations, C.R.A.S. Paris, Vol. 316, serie I, 1993, pp. 1015-1018. | MR 1222964 | Zbl 0799.35102

[25] S.A. Vakulenko, The existence of chemical waves with complex front movement, Zh. Vychisl. Mov. i Mat. Fiz., Vol. 31, 1991, pp. 735-744. | MR 1120014

[26] V.I. Arnol'D, Geometric methods in Theory of Ordinary Differential Equations, 2nd ed. (New York: Springer 1988). | MR 947141

[27] S. Smale, Dynamics retrospective: great problems, attempts that failed, Physica D, Vol. 51, 1991, pp. 267-273. | MR 1128817 | Zbl 0745.58018

[28] S. Smale, Mathematics of Time (Springer, N. Y. 1980). | MR 607330

[29] D.V. Anosov, S.X. Aranson et al., Dynamical systems with hyperbolic behaviour Itogi Nauki i techniki Sovr. Prob. Mat. VINITI, Vol. 66, 1991. | MR 1136551

[30] D. Ruelle and F. Takens, On the nature of turbulence, Comm. Math. Phys., Vol. 20, 1971, pp.167-192. | MR 284067 | Zbl 0223.76041

[31] R. Newhouse, D. Ruelle and F. Takens, Occurence of strange axiom A attractors from quasi periodic flows, Comm. Math. Phys., Vol. 64, 1978, pp. 35-40. | MR 516994 | Zbl 0396.58029

[32] D. Ruelle, Elements of differentiable dynamics and bifurcation theory (Acad. Press, Boston etc., 1989). | MR 982930 | Zbl 0684.58001

[33] Z. Nitecki, Differentiable Dynamics (M.I.T. Press, Cambridge, etc., 1971). | MR 649788

[34] D. Henry, Geometric theory of semilinear parabolic equations, Lecture Notes in Mathematics 840 (Berlin: Springer 1981). | MR 610244 | Zbl 0456.35001

[35] V.I. Arnol'D, Kolmogorov's hydrodynamic attractors, Proc. Roy. Soc. London, Ser. A, Vol. 434, 1991, pp. 19-22. | MR 1124924 | Zbl 0726.76045

[36] L.D. Meshalkin and Yu.G. Sinai, The study of the stability of a stationary solution of the systems of equations of the plane motion of the imcompressible vicsous fluid., Appl. Math. Mech., Vol. 6, 1961, pp. 1140-1143. | Zbl 0108.39501

[37] V.I. Arnol'D, Mathematical methods in Classical Mechanics (Moscow 1974). | Zbl 0647.70001

[38] Y. Kuramoto, Chemical oscillations, waves and turbulence (Springer, Berlin, etc., 1984). | MR 762432 | Zbl 0558.76051