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Akagi, G.* ; Schimperna, G.* ; Segatti, A.*

Fractional Cahn-Hilliard, Allen-Cahn and porous medium equations.

J. Differ. Equations 261, 2935-2985 (2015)
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We introduce a fractional variant of the Cahn-Hilliard equation settled in a bounded domain Ω⊂RN and complemented with homogeneous Dirichlet boundary conditions of solid type (i.e., imposed in the whole of RN(set minus)Ω). After setting a proper functional framework, we prove existence and uniqueness of weak solutions to the related initial-boundary value problem. Then, we investigate some significant singular limits obtained as the order of either of the fractional Laplacians appearing in the equation is let tend to 0. In particular, we can rigorously prove that the fractional Allen-Cahn, fractional porous medium, and fractional fast-diffusion equations can be obtained in the limit. Finally, in the last part of the paper, we discuss existence and qualitative properties of stationary solutions of our problem and of its singular limits.
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Publikationstyp Artikel: Journalartikel
Dokumenttyp Wissenschaftlicher Artikel
Schlagwörter Cahn-hilliard Equation ; Fractional Laplacian ; Fractional Porous Medium Equation ; Singular Limit ; Stationary Solution; Degenerate Diffusion-equations; Nonlinear Diffusion; Bounded Domains; Sobolev Spaces; Laplacian; Operators; Uniqueness; Behavior; Systems; Brezis
ISSN (print) / ISBN 0022-0396
e-ISSN 1090-2732
Quellenangaben Band: 261, Heft: 6, Seiten: 2935-2985 Artikelnummer: , Supplement: ,
Verlag Elsevier
Verlagsort San Diego