Longtime behavior for a generalized Cahn-Hilliard system with fractional operators
Pierluigi Colli, Gianni Gilardi, J\"urgen Sprekels
TL;DR
This paper analyzes the long-term behavior of solutions to a fractional Cahn-Hilliard system, characterizing the omega-limit set based on the eigenvalues of involved operators, with implications for stationary solutions and chemical potential behavior.
Contribution
It provides a complete characterization of the omega-limit set for the fractional Cahn-Hilliard system, depending on the first eigenvalue of the operator, extending previous well-posedness results.
Findings
Omega-limit set characterized by eigenvalues
Chemical potential vanishes when eigenvalue is positive
Conditions for uniqueness and constancy of related functions
Abstract
In this contribution, we deal with the longtime behavior of the solutions to the fractional variant of the Cahn-Hilliard system, with possibly singular potentials, that we have recently investigated in the paper `Well-posedness and regularity for a generalized fractional Cahn-Hilliard system' (see arXiv:1804.11290). More precisely, we study the omega-limit of the phase parameter and characterize it completely. Our characterization depends on the first eigenvalue of one of the operators involved: if it is positive, then the chemical potential vanishes at infinity and every element of the omega-limit is a stationary solution to the phase equation; if, instead, the first eigenvalue is 0, then every element of the omega-limit satisfies a problem containing a real function related to the chemical potential. Such a function is nonunique and time dependent, in general, as we show by an…
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document
Longtime behavior
for a generalized Cahn–Hilliard system
with fractional operators
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centerPierluigi Colli*(1)*
e-mail: [email protected]
Gianni Gilardi*(1)*
e-mail: [email protected]
Jürgen Sprekels*(2)*
e-mail: [email protected]
(1) Dipartimento di Matematica “F. Casorati”, Università di Pavia
and Research Associate at the IMATI – C.N.R. Pavia
via Ferrata 5, 27100 Pavia, Italy
(2) Department of Mathematics
Humboldt-Universität zu Berlin
Unter den Linden 6, 10099 Berlin, Germany
and
Weierstrass Institute for Applied Analysis and Stochastics
Mohrenstrasse 39, 10117 Berlin, Germany