Inverse Problems and Chaotic Dynamics of Parabolic Equations on Arbitrary Spatial Domains

LetΩ Nbe an arbitrary smooth bounded domain. We prove the existence of a polynomial functiona(x) on Nsuch that an arbitrary (and sufficiently small) vector field on N+1can be realized on the center manifold of the semilinear parabolic equation[formula]with an appropriate nonlinearityg: (x, s, w) Ω× × N g(x, s, w) . This extends earlier results of Polá ik and Rybakowski and shows that arbitrary chaotic behavior is possible for semilinear scalar parabolic BVPs on arbitrary bounded domainsΩ. To prove this we establish some spectral perturbation and convergence results and solve an abstract inverse eigenvalue problem.