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 ∈ ‫ގ‬ 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.1. Introduction 499 2. Preliminaries, I: Taylor expansions and the Whitney extension theorem 504 3. Preliminaries, II: Spectral gaps and the construction of an inertial manifold 508 4. Main result 515 5. Examples and concluding remarks 522 Appendix: Verifying the compatibility conditions 527