## 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
Full Text: