  The paper is devoted to a comprehensive study of smoothness of inertial
manifolds for abstract semilinear parabolic problems. It is well known that in
general we cannot expect more than $C^{1,\varepsilon}$-regularity for such
manifolds (for some positive, but small $\varepsilon$). Nevertheless, as shown
in the paper, under the natural assumptions, the obstacles to the existence of
a $C^n$-smooth inertial manifold (where $n\in\mathbb N$ is any given number)
can be removed by increasing the dimension and by modifying properly the
nonlinearity outside of the global attractor (or even outside the
$C^{1,\varepsilon}$-smooth IM of a minimal dimension). The proof is strongly
based on the Whitney extension theorem.
