Complicated dynamics in scalar semilinear parabolic equations in higher space dimension

We study the dynamics of the boundary value problem u_t - Lu = g(x, u, ▽u), x ε{lunate} Ω, (1) u |_∂Ω = 0, (2) where L is a second order uniformly elliptic operator and Ω ⊂R^N is diffeomorphic to the ball in R^N, N≥2. The main result asserts that given any C^k-vector field V on R^N+1 with V(0) = 0 one can adjust coefficients of L and the function g such that the corresponding problem (1), (2) has an N+ 1-dimensional invariant manifold through the equilibrium u ≡ 0 and the Taylor expansion at u ≡ 0 of the vector field representing the flow on this manifold coincides (in appropriate coordinates) with the Taylor expansion of V, up to k-th order terms. This result implies that a hyperbolic invariant N-torus can be found in (1), (2) (if L and g are appropriately chosen). This result also indicates that "chaotic dynamics" is likely to occur for some choices of L and g.