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Inertial manifolds for reaction diffusion equations in higher space dimensions

Authors: John Mallet-Paret and George R. Sell
Journal: J. Amer. Math. Soc. 1 (1988), 805-866
MSC: Primary 58F12; Secondary 35K57, 47H20, 58D25
MathSciNet review: 943276
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Abstract: In this paper we show that the scalar reaction diffusion equation

$\displaystyle {u_t} = \nu \Delta u + f(x,u),\qquad u \in R$

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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Keywords: Averaging, gaps, inertial manifolds, invariant manifolds, quadratic forms, reaction diffusion equations, spatial averaging
Article copyright: © Copyright 1988 American Mathematical Society

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