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Abstract
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
∈
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.
Keywords
inertial manifolds, finite-dimensional reduction,
smoothness, Whitney extension theorem
Mathematical Subject Classification
Primary: 35B40, 35B42, 37D10, 37L25
Milestones
Received: 20 June 2021
Accepted: 26 July 2022
Published: 6 March 2024
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