Inertial manifolds for reaction diffusion equations in higher space dimensions

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.