Inertial Manifolds for Reaction Diffusion Equations in Higher Space Dimensions

In this paper we show that the scalar reaction diffusion equation \[ {u_t} = ν Δ u + f(x,u), u ∈ R\] with x ∈ Ω ₙ ⊂ Rⁿ (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 C³ and for Ω ₃ = (0,2π )³ or Ω ₂ = (0,2π /a₁) × (0,2π /a₂), where a₁ and a₂ are positive. The proof is based on an (abstract) Invariant Manifold Theorem for flows on a Hilbert space. It is significant that on Ω ₃ the spectrum of the Laplacian Δ 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 Δ + υ (x), which is valid only in space dimension n ≤ 3. This property says that Δ + υ (x) can be well approximated by the constant coefficient problem Δ + υ over large segments of the Hilbert space L²(Ω ), where υ = (volΩ )- 1∫ _Ω υ \;dx is the average value of υ. We call this property the Principle of Spatial Averaging. The proof that the Schroedinger operator satisfies the Principle of Spatial Averaging on the regions Ω ₂ and Ω ₃ described above follows from a gap theorem for finite families of quadratic forms, which we present in an Appendix to this paper.