J.evol.equ. 5 (2005) 465 – 483 1424–3199/05/040465 – 19 DOI 10.1007/s00028-005-0199-6 © Birkh¨ auser Verlag, Basel, 2005 Robust exponential attractors for a phase-field system with memory Maurizio Grasselli, Vittorino Pata Abstract. H.G. Rotstein et al. proposed a nonconserved phase-field system characterized by the presence of memory terms both in the heat conduction and in the order parameter dynamics. These hereditary effects are represented by time convolution integrals whose relaxation kernels k and h are nonnegative, smooth and decreasing. Rescaling k and h properly, we obtain a system of coupled partial integrodifferential equations depending on two relaxation times ε and σ . When ε and σ tend to 0, the formal limiting system is the well-known nonconserved phase-field model proposed by G. Caginalp. Assuming the exponential decay of the relaxation kernels, the rescaled system, endowed with homogeneous Neumann boundary conditions, generates a dissipative strongly continuous semigroup S ε,σ (t) on a suitable phase space, which accounts for the past histories of the temperature as well as of the order parameter. Our main result consists in proving the existence of a family of exponential attractors E ε,σ for S ε,σ (t), with ε, σ ∈ [0, 1], whose symmetric Hausdorff distance from E 0,0 tends to 0 in an explicitly controlled way. 1. Introduction H.G. Rotstein et al. [19] have recently proposed a phenomenological model which describes first order phase transitions, occurring when configurational modes, resolving themselves on a slow time scale, are present (see also [8, 17, 18]). More precisely, letting  ⊂ R 3 be a bounded domain with smooth boundary ∂, and indicating by ϑ = ϑ(t) the (relative) temperature and by χ = χ(t) the order parameter (or phase-field), the model is represented by the system of integro-partial differential equations in  × R + , with R + = (0, ∞),  (ϑ + χ) t -  ∞ 0 k(s)ϑ(t - s)ds = 0, χ t +  ∞ 0 h(s)[-χ(t - s) + φ(χ(t - s)) - ϑ(t - s)]ds = 0, (1.1) This work was partially supported by the Italian MIUR PRIN Research Projects Modellizzazione Matematica ed Analisi dei Problemi a Frontiera Libera and Aspetti Teorici e Applicativi di Equazioni a Derivate Parziali, and by the Italian MIUR FIRB Research Project Analisi di Equazioni a Derivate Parziali, Lineari e Non Lineari: Aspetti Metodologici, Modellistica, Applicazioni. Mathematics Subject Classification (2000): 35B41, 37L25, 37L30, 45K05, 80A22. Key words: Phase-field models, memory kernels, strongly continuous semigroups, absorbing sets, robust exponential attractors, global attractors.