# Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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## Inertial manifolds for reaction diffusion equations in higher space dimensionsHTML articles powered by AMS MathViewer

by John Mallet-Paret and George R. Sell
J. Amer. Math. Soc. 1 (1988), 805-866 Request permission

## Abstract:

In this paper we show that the scalar reaction diffusion equation ${u_t} = \nu \Delta u + f(x,u),\qquad u \in R$ with $x \in {\Omega _n} \subset {R^n}\quad (n = 2,3)$ and with Dirichlet, Neumann, or periodic boundary conditions, has an inertial manifold when (1) the equation is dissipative, and (2) $f$ is of class ${C^3}$ and for ${\Omega _3} = {(0,2\pi )^3}$ or ${\Omega _2} = (0,2\pi /{a_1}) \times (0,2\pi /{a_2})$, where ${a_1}$ and ${a_2}$ are positive. The proof is based on an (abstract) Invariant Manifold Theorem for flows on a Hilbert space. It is significant that on ${\Omega _3}$ the spectrum of the Laplacian $\Delta$ does not have arbitrary large gaps, as required in other theories of inertial manifolds. Our proof is based on a crucial property of the Schroedinger operator $\Delta + \upsilon (x)$, which is valid only in space dimension $n \leq 3$. This property says that $\Delta + \upsilon (x)$ can be well approximated by the constant coefficient problem $\Delta + \bar \upsilon$ over large segments of the Hilbert space ${L^2}(\Omega )$, where $\bar \upsilon = {({\text {vol}}\Omega )^{ - 1}}\int _\Omega {\upsilon \;dx}$ is the average value of $\upsilon$. We call this property the Principle of Spatial Averaging. The proof that the Schroedinger operator satisfies the Principle of Spatial Averaging on the regions ${\Omega _2}$ and ${\Omega _3}$ described above follows from a gap theorem for finite families of quadratic forms, which we present in an Appendix to this paper.
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