Approximate inertial manifolds for reaction-diffusion equations in high space dimension. (English) Zbl 0702.35127

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


35K57 Reaction-diffusion equations
35B40 Asymptotic behavior of solutions to PDEs
35P10 Completeness of eigenfunctions and eigenfunction expansions in context of PDEs


Zbl 0624.76072
Full Text: DOI


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