Set-Valued Analysis 8: 375–403, 2000. © 2000 Kluwer Academic Publishers. Printed in the Netherlands. 375 On Global Attractors of Multivalued Semiprocesses and Nonautonomous Evolution Inclusions ⋆ VALERY S. MELNIK 1 and JOSÉ VALERO 2 1 Institute of System Applied Analysis, Pr. Pobedy 37, 252056 Kiev, Ukraine 2 CEU San Pablo-Elche, Comisario 3, 03203-Elche, Alicante, Spain (Received: 1 February 2000) Abstract. In this paper we define multivalued semiprocesses and give theorems providing the existence of global attractors for such systems. This theory generalizes the construction of nonau- tonomous dynamical systems given by V. V. Chepyzhov and M. I. Vishik to the case where the system is not supposed to have a unique solution for each initial state. Further, we apply these theorems to nonautonomous differential inclusions of reaction–diffusion type. Mathematics Subject Classifications (2000): 35B40, 35K55, 35K90. Key words: global attractor, nonautonomous multivalued dynamical systems, differential inclusions. 1. Introduction In recent years, several works have been published concerning the existence of global attractors for autonomous multivalued dynamical systems [3, 4, 16–18, 21, 22]. In [21, 22, 25] the existence and properties of global attractors were studied for autonomous evolution inclusions. In [26], the results of [22] were extended to a class of differential inclusions generated by a difference of subdifferential maps. In all these works, the equations or inclusions generating the dynamical system are not supposed to have a unique global solution for each initial state. Some works were also devoted to the approximations of multivalued semiflows and their attractors [11–13] and in [14] the dependence on a parameter of global attractors of autonomous evolution inclusions was studied. Concerning nonautonomous dynamical systems a nice theory was constructed in [7] for abstract processes and semiprocesses with applications to nonautonomous reaction–diffusion equations, Navier–Stokes equations and hyperbolic equations. In [19, 20] this theory was generalized for multivalued semiprocesses and applied to reaction–diffusion equations in the case where the convergence to the attractor is considered in a weak sense. In [6] was studied a random multivalued system generated by a differential inclusion with random perturbations. ⋆ This work has been supported by PB-2-FS-97 grant (Fundaci´ on S´ eneca (Comunidad Aut´ onoma de Murcia)).