Marion, Martine Approximate inertial manifolds for reaction-diffusion equations in high space dimension. (English) Zbl 0702.35127 J. Dyn. Differ. Equations 1, No. 3, 245-267 (1989). This paper deals with a reaction-diffusion equation \[ (*)\quad u_ t- d\Delta u+g(u)=0\text{ on } \Omega \times {\mathbb{R}}_+ \] (d\(>0\), \(g\in C^ 1({\mathbb{R}})\) satisfying a polynomial growth condition, \(\Omega \subset {\mathbb{R}}^ n\) a bounded domain) subject to either a homogeneous Dirichlet or Neumann boundary condition or a periodic one in case that \(\Omega =(0,L)^ n\), \(L>0\). The author establishes the existence of an approximate inertial manifold in the sense of C. Foias, O. Manley, and R. Temam [C. R. Acad. Sci., Paris, Ser. I 305, 497-500 (1987; Zbl 0624.76072)]. Very roughly, this is a smooth, finite- dimensional manifold, which forms the “zero-section” of a “tubular neighborhood” that absorbs each orbit of (*). The “thickness” of the neighborhood depends on the dimension m of the manifold, and it is shown for the general construction that this quantity converges to 0 at an order \((\mu_ 1/\mu_ m)^ 2\) as \(m\to \infty\), whereas for \(n\leq 12\) a manifold can be found with a convergence order \((\mu_ 1/\mu_ m)^ 3\). \(\mu_ j\) denotes the j-th eigenvalue of \(- d\Delta w+w=\mu w\) subject to the same boundary condition as given for (*). Reviewer: G.Hetzer Cited in 42 Documents MSC: 35K57 Reaction-diffusion equations 35B40 Asymptotic behavior of solutions to PDEs 35P10 Completeness of eigenfunctions and eigenfunction expansions in context of PDEs Keywords:Dirichlet; Neumann boundary condition; approximate inertial manifold; thickness; eigenvalue Citations:Zbl 0624.76072 PDFBibTeX XMLCite \textit{M. Marion}, J. Dyn. Differ. Equations 1, No. 3, 245--267 (1989; Zbl 0702.35127) Full Text: DOI References: [1] Babin, A. V., and Vishik, M. I. (1983). Attractors of partial differential equations and estimates of their dimension.Russ. Math. Surv. 38:4, 151-213. · Zbl 0541.35038 [2] Foias, C., Sell, G. R., and Temam, R. (1985). Variétés inertielles des équations différentielles dissipatives.C. R. Acad. Sci. Ser. I 301, 139-141; and Inertial manifolds for nonlinear evolutionary equations.J. Diff. Eg. 73, 309-353 (1988). [3] Foias, C., Manley, O., and Temam, R. (1987). Sur l’interaction des petits et grands tourbillons dans des écoulements turbulents.C. R. Acad. Sci. Ser. I 305, 497-500. · Zbl 0624.76072 [4] Mallet-Paret, J., and Sell, G. R. (1989). Inertial manifolds for reaction-diffusion equations in higher space dimension.J. Amer. Math. Soc. 1, 805-866. · Zbl 0674.35049 [5] Mallet-Paret, J., and Sell, G. R. (1989). (In preparation.) [6] Marion, M. (1987). Attractors for reaction-diffusion equations: existence and estimate of their dimension.Appl. Anal. 25, 101-147. · Zbl 0609.35009 [7] Marion, M. (1989). Inertial manifolds associated to partly dissipative reaction-diffusion systems.J. Math. Anal. Appl. (In press). · Zbl 0689.58039 [8] Mora, X. (1983). Finite dimensional attracting manifolds in reaction-diffusion equations.Contemp. Math. 17, 353-360. · Zbl 0525.35046 [9] Rothe, F. (1984).Global Solutions of Reaction-Diffusion Systems, Springer-Verlag, Berlin. · Zbl 0546.35003 [10] Temam, R. (1988).Infinite Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, Berlin. · Zbl 0662.35001 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.