Weak solutions of the Shigesaka-Kawasaki-Teramoto equations and their   attractors
  We derive the global existence of weak solutions of the Shigesada-Kawasaki-Teramoto systems in space dimension less or equal to 4 with a rather general condition on the coefficients. The existence is established using finite differences in time with truncations and an argument of Stampachia's maximum principle to show the positivity of the solutions. We derive also the existence of a weak global attractor. 
Du Pham and Roger Temam
math.AP