Realization of Arbitrary Vector Fields on Center Manifolds of Parabolic Dirichlet BVPs

Consider the following parabolic equation: ut − Lu = σ(x, u, u), t ≥ 0, x Ωu(x, t) = 0, t ≥ 0, x ∂ΩHere Ω RN is a smooth bounded domain, L is a second-order self-adjoint uniformly elliptic differential operator and σ: × R × RN→ R is some nonlinearity which explicitly depends on the gradient of the solution. Using the Nash-Moser implicit function theorem we prove in this paper that if L and Ω satisfy the so-called Polá ik condition and X1 denotes the kernel of L with Dirichlet boundary conditions (in this case the dimension of X1 is necessarily N or N + 1) then for every sufficiently smooth and "small" vector field υ defined in a neighborhood of zero in X1 there exists a nonlinearity σ and a local center manifold Mσ of (Pσ) such that υ is exactly the reduced vector field of (Pσ) on Mσ. This result implies, in particular, that arbitrary chaotic behavior of solutions of ODEs is also observable in suitable scalar parabolic equations with Dirichlet boundary conditions and gradient dependent nonlinearities.