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Explicit construction of an inertial manifold for a reaction diffusion equation. (English) Zbl 0691.35049

An inertial manifold is constructed for the scalar reaction-diffusion equation \(u_ t=vu_{xx}+f(u)\) with a cubic nonlinearity. Uniform bounds are obtained for the number of zeros along solutions to the variational equations satisfied by the difference of two elements on the unstable manifolds of equilibria. This uniformity leads to the global parametrization of the attractor as a function defined in the linear unstable manifold of the least stable equilibrium. By the introduction of local techniques near each equilibrium, we succeed in constructing an inertial manifold of lowest possible dimension.
Reviewer: M.Krüger

MSC:

35K57 Reaction-diffusion equations
35B99 Qualitative properties of solutions to partial differential equations
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[1] Angement, S. B., The Morse-Smale property for a similinear parabolic equation, J. Differential Equations, 62, 427-442 (1986) · Zbl 0581.58026
[2] Brunovsky, P.; Fiedler, B., Zero numbers on invariant manifolds in scalar reaction diffusion equations, Nonlinear Anal. TMA, 10, 178-193 (1986) · Zbl 0594.35056
[3] Chafee, N.; Infante, E., A bifurcation problem for a nonlinear parabolic equation, J. Appl. Anal., 4, 17-37 (1974) · Zbl 0296.35046
[4] Coddinoton, E. A.; Levinson, N., Ordinary Differential Equations (1955), McGraw-Hill: McGraw-Hill New York · Zbl 0064.33002
[5] P. Constantin, “Proceedings of the AMS/SIAM Conference on the Connection between Finite and Infinite Dimensional Systems” (C. Foias, B. Nicolaenko and R. Temam, Eds.), Contemporary Mathematics, in press.; P. Constantin, “Proceedings of the AMS/SIAM Conference on the Connection between Finite and Infinite Dimensional Systems” (C. Foias, B. Nicolaenko and R. Temam, Eds.), Contemporary Mathematics, in press.
[6] P. Constantin, C. Foias, B. Nicolaenko, and R. Temam, Integral manifolds and inertial manifolds for dissipative partial differential equations, Inst. Appl. Math. Sci. Comp. preprint series No. 2, Indiana University at Bloomington.; P. Constantin, C. Foias, B. Nicolaenko, and R. Temam, Integral manifolds and inertial manifolds for dissipative partial differential equations, Inst. Appl. Math. Sci. Comp. preprint series No. 2, Indiana University at Bloomington. · Zbl 0683.58002
[7] Doering, C. R.; Gibbon, J. D.; Holm, D. D.; Nicolaenko, B., Low dimensional behavior in the complex Ginzburg-Landau equation, Nonlinearity, 1, 279-309 (1988) · Zbl 0655.58021
[8] Foias, C.; Sell, G. R.; Temam, R., Inertial manifolds for nonlinear evolutionary equations, J. Differential Equations, 73, 309-353 (1988) · Zbl 0643.58004
[9] Foias, C.; Nicolaenko, B.; Sell, G. R.; Temam, R., Inertial manifolds for the Kuramoto-Sivashinsky equation and an estimate of their lowest dimensions, J. Math. Pures Appl., 67, 197-226 (1988) · Zbl 0694.35028
[10] Hale, J. K., Ordinary Differential Equations (1980), Krieger: Krieger Malabar, Florida · Zbl 0186.40901
[11] Henry, D., Geometric Theory of Semilinear Parabolic Equations, (Lecture Notes in Mathematics, No. 840 (1981), Springer-Verlag: Springer-Verlag New York/Berlin) · Zbl 0456.35001
[12] Henry, D., Some infinite-dimensional Morse-Smale systems difined by parabolic partial diffeential equations, J. Differential Equations, 59, 165-205 (1985) · Zbl 0572.58012
[13] Istrǎţescu, V. I., Fixed Point Theory (1981), D. Reidel: D. Reidel Dordrecht, Holland · Zbl 0465.47035
[14] Lukacs, E., Applications of Faà di Bruno’s formula in mathematical statistics, Amer. Math. Monthly, 62, 340-348 (1955) · Zbl 0064.38402
[15] Mallet-Paret, J.; Sell, G. R., Inertial Manifolds for reaction diffusion equations in higher space dimensions, J. Amer. Math. Soc., 1, 805-866 (1988) · Zbl 0674.35049
[16] Matano, H., Nonincrease of the lap number of a solution for a one dimensional semi-linear parabolic equation, Pub. Fac. Sci. Univ. Tokyo Sec. IA, 29, 401-441 (1982) · Zbl 0496.35011
[17] Matjowsky, B. J., A simple nonlinear dynamical stability problem, Bull. Amer. Math. Soc. (N.S.), 76, 620-625 (1970) · Zbl 0195.11102
[18] Nickel, K., Gestaltaussagen über lösungen parabolischer differentialgleichungen, J. Reine Angew. Math., 211, 78-94 (1962) · Zbl 0127.31801
[19] B. Nicolaenko, B. Scheurer, and R. Temam, Some global dynamical properties of a class of pattern formation equations, Inst. Math. Appl. preprint series No. 381, University of Minnesota.; B. Nicolaenko, B. Scheurer, and R. Temam, Some global dynamical properties of a class of pattern formation equations, Inst. Math. Appl. preprint series No. 381, University of Minnesota. · Zbl 0691.35019
[20] Redheffer, R. M.; Walter, W., The total variation of solutions of parabolic differential equations and a maximum principle in unbounded domains, Math. Ann., 209, 57-67 (1974) · Zbl 0267.35053
[21] Sell, G. R., Nonautonomous differential equations and topological dynamics. II. Limiting equations, Trans. Amer. Math. Soc., 127, 263-283 (1967) · Zbl 0189.39602
[22] Sell, G. R., The structure of a flow in the vicinity of an almost periodic motion, J. Differential Equations, 27, 359-393 (1978) · Zbl 0382.34017
[23] Smoller, J., Shock Waves and Reaction Diffusion Equations (1983), Springer-Verlag: Springer-Verlag New York · Zbl 0508.35002
[24] Ghidaglia, J. M.; Héron, B., Dimension of the attractors associated to the Ginzburg-Landau partial differential equation, Physica, 28D, 282-304 (1987) · Zbl 0623.58049
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