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Inertial manifolds for reaction diffusion equations in higher space dimensions. (English) Zbl 0674.35049

Scalar reaction-diffusion equations of the type \(u_ t=\nu \Delta u+f(x,u)\) with Dirichlet, Neumann or periodic boundary conditions on \(\Omega_ 3=(0,2\pi)^ 3\) or \(\Omega_ 2=(0,2\pi /a_ 1)\times (0,2\pi /a_ 2)\) are considered. The existence of an inertial manifold is proved under the assumption that the problem is dissipative and f is of the class \(C^ 3\). Former methods based on the spectral gap condition cannot be used because the Laplacian does not satisfy this condition on \(\Omega_ 3\). An abstract invariant manifold theorem for flows on a Hilbert space is given and it is used for the proof of the result mentioned above. The method cannot be extended to a higher space dimension because a certain basic property of the Schrödinger operator which is valid only for \(n\leq 3\) is essentially used.
Reviewer: M.Kučera

MSC:

35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35K57 Reaction-diffusion equations
34C30 Manifolds of solutions of ODE (MSC2000)
35P20 Asymptotic distributions of eigenvalues in context of PDEs
11B05 Density, gaps, topology
11E99 Forms and linear algebraic groups
47H10 Fixed-point theorems
11N05 Distribution of primes
35B40 Asymptotic behavior of solutions to PDEs
34C29 Averaging method for ordinary differential equations
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