Journal of Dynamics and Differential Equations, Vol. 13, No. 4, October 2001 ( 2001) Attractors of Parabolic Equations Without Uniqueness Jose Valero 1 Received April 1, 1999; revised October 30, 2000 In this paper we study the existence of global compact attractors for nonlinear parabolic equations of the reaction-diffusion type and variational inequalities. The studied equations are generated by a difference of subdifferential maps and are not assumed to have a unique solution for each initial state. Applications are given to inclusions modeling combustion in porous media and processes of transmission of electrical impulses in nerve axons. KEY WORDS: Global attractors; differential inclusions; multivalued dynami- cal systems; reaction-diffusion equations. AMS Subject Classifications (2000): 35B40, 35B41, 35K55, 35K57. 1. INTRODUCTION The theory of global attractors for semigroups generated by nonlinear PDE is well known (see [4, 17, 19, 31]) and has been applied to a great number of equations. The main characteristic of these systems is the unique (and global in time) solvability of the equations for each initial state of the system. However, there are systems for which more than one solution can exist for each initial state. Hence, they are multivalued. In the last years several papers have been written concerning such systems (see [1, 7, 12, 20, 2326, 28, 30). The main goal of this paper is to study the existence of global compact attractors for parabolic equations without uniqueness of solutions instead of known results in which the unique solvability is guaranteed (see [26, 13, 14, 18, 21, 22, 31, 34, 35]). 711 1040-7294011000-071119.500  2001 Plenum Publishing Corporation 1 Universidad Cardenal Herrera, Comisario 3, 03203-Elche (Alicante), Spain.