DISCRETE AND CONTINUOUS doi:10.3934/dcdsb.2009.11.205 DYNAMICAL SYSTEMS SERIES B Volume 11, Number 2, March 2009 pp. 205–231 ATTRACTORS FOR A NON-LINEAR PARABOLIC EQUATION MODELLING SUSPENSION FLOWS Jos´ e M. Amig´ o Universidad Miguel Hern´ andez, Centro de Investigaci´ on Operativa Avda. Universidad s/n, Elche (Alicante), 03202, Spain Isabelle Catto CEREMADE, UMR CNRS 7534, Universit´ e Paris IX-Dauphine ´ Angel Gim´ enez and Jos´ e Valero Universidad Miguel Hern´ andez, Centro de Investigaci´ on Operativa Avda. Universidad s/n, Elche (Alicante), 03202, Spain (Communicated by Miguel Sanjuan) Abstract. In this paper we prove the existence of a global attractor with respect to the weak topology of a suitable Banach space for a parabolic scalar differential equation describing a non-Newtonian flow. More precisely, we study a model proposed by H´ ebraud and Lequeux for concentrated suspensions. 1. Introduction. Non-Newtonian (or complex) fluids are ubiquitous in nature and industry, appearing for instance in foods, biofluids, personal care products, phar- macology and bioengineering, electronics and optical materials, energy and plastic production, etc. In fact, one could say that Newtonian (or simple) fluids, i.e., those fluids whose stress-tensor is given by the Navier-Stokes ansatz, are rather an excep- tion (if not an idealization), even though they include such a prominent member as water. Attending to their rheologic properties, complex fluids are classified in different categories, including suspensions, colloids, melt polymers, liquid crystals, gels and foams, among others. Needless to say, coping with such a broad diversity of fluids requires physical insight, mathematical sophistication, and a lot of ingenuity. Non-Newtonian fluids are notoriously difficult to model and to analyze. To begin with, these fluids display very nonlinear flow properties (such as memory effects and discontinuities) that are far from being understood from first principles. As a result one has to resort in general to phenomenological (or macroscopic) descriptions or, in some cases, to mesoscopic models describing the interaction of different types of microstructures (hard or soft spheres, rods, dumb bells, etc.) much larger than the atomic scale. Elaborated mesoscopic models are being successfully used in polymers, liquid crystals and suspensions. 2000 Mathematics Subject Classification. Primary: 35B40, 35B41, 35K55; Secondary: 37B25, 58C06. Key words and phrases. Non-Newtonian fluids, set-valued dynamical system, global attractor. This work has been mainly supported by the Generalitat Valenciana, grant GV05/064. Also, it was partially supported by the MCYT (Ministerio de Educaci´ on y Ciencia, Spain) and FEDER (Fondo Europeo de Desarrollo Regional), grants MTM2005-01412, MTM2005-03868. 205