Smooth extensions for inertial manifolds of semilinear parabolic equations

The paper is devoted to a comprehensive study of smoothness of inertial manifolds (IMs) for abstract semilinear parabolic problems. It is well known that in general we cannot expect more than $C^{1,𝜀}$-regularity for such manifolds (for some positive, but small $𝜀$). Nevertheless, as shown in the paper, under 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 $C^{1,𝜀}$-smooth IM of a minimal dimension). The proof is strongly based on the Whitney extension theorem.