Reaction-diffusion systems with prescribed large time behaviour

Vakulenko, S. A.

Annales de l'I.H.P. Physique théorique, Tome 66 (1997) no. 4, pp. 373-410.

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

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

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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

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

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

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

[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 | Zbl

[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 | Zbl

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

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

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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

[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 | Zbl

[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 | Zbl

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

[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 | Zbl

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

[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 | Zbl

[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

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

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

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

[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

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

[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 | Zbl

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

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

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

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

[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

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

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