## 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

### MSC:

 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
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### References:

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