Analysis and Applications, Vol. 11, No. 6 (2013) 1350024 (31 pages) c  World Scientific Publishing Company DOI: 10.1142/S0219530513500243 ATTRACTORS FOR A CAGINALP MODEL WITH A LOGARITHMIC POTENTIAL AND COUPLED DYNAMIC BOUNDARY CONDITIONS MONICA CONTI ∗ , STEFANIA GATTI † and ALAIN MIRANVILLE ‡ ∗ Dipartimento di Matematica “F. Brioschi”, Politecnico di Milano Via Bonardi 9, I-20133 Milano, Italy monica.conti@polimi.it † Dipartimento di Matematica, Universit` a di Modena e Reggio Emilia Via Campi 213/B, I-41125 Modena, Italy stefania.gatti@unimore.it ‡ Laboratoire de Math´ ematiques et Applications Universit´ e de Poitiers UMR CNRS 7348 — SP2MI Boulevard Marie et Pierre Curie — T´ el´ eport 2 F-86962 Chasseneuil Futuroscope Cedex, France Alain.Miranville@math.univ-poitiers.fr Received 16 May 2012 Accepted 13 November 2012 Published 21 October 2013 We study the longtime behavior of the Caginalp phase-field model with a logarithmic potential and dynamic boundary conditions for both the order parameter and the tem- perature. Due to the possible lack of distributional solutions, we deal with a suitable definition of solutions based on variational inequalities, for which we prove well-posedness and the existence of global and exponential attractors with finite fractal dimension. Keywords : Caginalp equations; dynamic boundary conditions; logarithmic potential; global attractor; exponential attractor. Mathematics Subject Classification 2010: 35K55, 35J60, 80A22 1. Introduction In this paper, we consider a Caginalp phase-field system for the kinetics of phase transitions, introduced in [1] (see also [5, 22]) to model melting-solidification phe- nomena in certain materials: more precisely, denoting by ϑ and u the relative tem- perature and the order parameter, respectively, we have a system of two parabolic equations,  ∂ t u - ∆u + f (u) - λu = ϑ, in Ω, t> 0, ∂ t ϑ - ∆ϑ = -∂ t u, in Ω, t> 0. (1.1) 1350024-1 Anal. Appl. 2013.11. Downloaded from www.worldscientific.com by 222.165.197.234 on 06/22/14. For personal use only.