Sharp estimates of the dimension of inertial manifolds for nonlinear parabolic equations

Sufficient conditions are obtained for the existence of a $k$-dimensional invariant manifold that attracts as $t\to\infty$ all solutions $u(t)$ of the evolution equation $\dot u=-Au+F(u)$ in a Hilbert space, where $A$ is a linear selfadjoint operator, semibounded from below, with compact resolvent, and $F$ is a uniformly Lipschitz (in suitable norms) nonlinearity; these conditions sharpen previously known conditions and cannot be improved.