Three counterexamples in the theory of inertial manifolds

An example of a dissipative semilinear parabolic equation in a Hilbert space without smooth inertial manifolds is constructed. Moreover, the attractor of this equation can be embedded in no finite-dimensional C-1 invariant submanifold of the phase space. The class of scalar reaction-diffusion equations in bounded domains Omega subset of R-m without inertial manifolds M subset of L-2(Omega) with the property of absolute normal hyperbolicity on the set E of stationary points of the phase semiflow is described. Such equations may have inertial manifolds with the weaker property of normal hyperbolicity on E. Three-dimensional reaction-diffusion systems without inertial manifolds normally hyperbolic at stationary points are found.