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A generalization of the Allen–Cahn equation
Miranville, Alain; Quintanilla de Latorre, Ramón
Universitat Politècnica de Catalunya. Departament de Matemàtica Aplicada II; Universitat Politècnica de Catalunya. GRAA - Grup de Recerca en Anàlisi Aplicada
This is a pre-copyedited, author-produced PDF of an article accepted for publication in IMA Journal of Applied Mathematics following peer review. The version of record: Miranville, A.; Quintanilla, R. A generalization of the Allen–Cahn equation. "IMA Journal of Applied Mathematics", 01 Abril 2015, vol. 80, núm. 2, p. 410-430 is available online at:http://imamat.oxfordjournals.org/content/80/2/410.
Our aim in this paper is to study generalizations of the Allen–Cahn equation based on a modification of the Ginzburg–Landau free energy proposed in S. Torabi et al. (2009, A new phase-field model for strongly anisotropic systems. Proc. R. Soc. A, 465, 1337–1359). In particular, the free energy contains an additional term called Willmore regularization. We prove the existence, uniqueness and regularity of solutions, as well as the existence of the global attractor. Furthermore, we study the convergence to the Allen–Cahn equation, when the Willmore regularization goes to zero. We finally study the spatial behaviour of solutions in a semi-infinite cylinder, assuming that such solutions exist.
Peer Reviewed
Àrees temàtiques de la UPC::Matemàtiques i estadística::Equacions diferencials i integrals::Equacions en derivades parcials
Àrees temàtiques de la UPC::Matemàtiques i estadística::Equacions diferencials i integrals
Differential equations, Parabolic
Differential equations, Partial
Allen–Cahn equation
Willmore regularization
well-posedness
dissipativity
global attractor
spatial behaviour
Equacions diferencials parabòliques
Equacions en derivades parcials
Classificació AMS::35 Partial differential equations::35B Qualitative properties of solutions
Classificació AMS::35 Partial differential equations::35K Parabolic equations and systems
info:eu-repo/semantics/submittedVersion
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