Convergence of solutions for the fractional Cahn–Hilliard system

This paper deals with the Cauchy–Dirichlet problem for the fractional Cahn–Hilliard equation. The main results consist of global (in time) existence of weak solutions, characterization of parabolic smoothing effects (implying under proper condition eventual boundedness of trajectories), and converge...

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Bibliographic Details
Published in:Journal of functional analysis 2019-05, Vol.276 (9), p.2663-2715
Main Authors: Akagi, Goro, Schimperna, Giulio, Segatti, Antonio
Format: Article
Language:English
Subjects:
Cahn–Hilliard equation
Fractional (Dirichlet) Laplacian
Long-time behavior of solutions
Łojasiewicz–Simon's inequality
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Summary:This paper deals with the Cauchy–Dirichlet problem for the fractional Cahn–Hilliard equation. The main results consist of global (in time) existence of weak solutions, characterization of parabolic smoothing effects (implying under proper condition eventual boundedness of trajectories), and convergence of each solution to a (single) equilibrium. In particular, to prove the convergence result, a variant of the so-called Łojasiewicz–Simon inequality is provided for the fractional Dirichlet Laplacian and (possibly) non-analytic (but C1) nonlinearities.
ISSN:0022-1236
1096-0783
DOI:10.1016/j.jfa.2019.01.006