Fractional Cahn–Hilliard, Allen–Cahn and porous medium equations

Goro Akagi, Giulio Schimperna, Antonio Segatti

Research output: Contribution to journalArticle

36 Citations (Scopus)

Abstract

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∖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.

Original languageEnglish
Pages (from-to)2935-2985
Number of pages51
JournalJournal of Differential Equations
Volume261
Issue number6
DOIs
Publication statusPublished - 2016 Sep 15

Keywords

  • Cahn–Hilliard equation
  • Fractional Laplacian
  • Fractional porous medium equation
  • Singular limit
  • Stationary solution

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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