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Smooth extensions for inertial manifolds of semilinear parabolic equations

Anna Kostianko and Sergey Zelik

Vol. 17 (2024), No. 2, 499–533
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 C1,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 Cn-smooth inertial manifold (where 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 C1,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
Authors
Anna Kostianko
Department of Mathematics
Zhejiang Normal University
Zhejiang
China
Sergey Zelik
Department of Mathematics
Zhejiang Normal University
Zhejiang
China
Department of Mathematics
University of Surrey
Guildford
United Kingdom
Keldysh Institute of Applied Mathematics
Moscow
Russia

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