Global attractor of the multivalued semigroup associated with a phase-field model of grain boundary motion with constraint

Pages: 824 - 833, Issue Special, September 2011

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Nobuyuki Kenmochi - Department of Education, School of Education, Bukkyo University, 96 Kitahananobo-cho, Murasakino, Kita-ku, Kyoto, 603-8301, Japan (email)
Noriaki Yamazaki - Department of Mathematics, Faculty of Engineering, Kanagawa University, 3-27-1 Rokkakubashi, Kanagawa-ku, 221-8686, Japan (email)

Abstract: We consider a phase-fi eld model of grain boundary motion with constraint, which is a nonlinear system of Kobayashi-Warren-Carter type: a nonlinear parabolic partial diff erential equation and a nonlinear parabolic variational inequality. Recently the existence of solutions to our system was shown in the N-dimensional case. Also the uniqueness was proved in the case when the space dimensional is one and initial data are good. In this paper we study the asymptotic stability of our model without uniqueness. In fact we shall construct global attractors for multivalued semigroups (multivalued semiflows) associated with our system in the N-dimensional case.

Keywords:  Global attractor, multivalued semigroups, grain boundary motion, singular di usion equation
Mathematics Subject Classification:  Primary: 35B40, 35R35; Secondary: 35K55

Received: July 2010;      Revised: August 2010;      Published: October 2011.