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

Abstract Consider the following parabolic equation: u t − Lu = σ(x, u, ∇u), t ≥ 0, x ∈ Ω u(x, t) = 0, t ≥ 0, x ∈ ∂Ω Here Ω ⊂ R N is a smooth bounded domain, L is a second-order self-adjoint uniformly elliptic differential operator and σ: Ω × R × R N → 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 Polacik condition and X 1 denotes the kernel of L with Dirichlet boundary conditions (in this case the dimension of X 1 is necessarily N or N + 1) then for every sufficiently smooth and "small" vector field υ defined in a neighborhood of zero in X 1 there exists a nonlinearity σ and a local center manifold M σ of ( P σ ) such that υ is exactly the reduced vector field of ( P σ) on M σ . 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.