Asymptotic behaviour of nonlocal reaction–diffusion equations

The existence of a global attractor in L2(Ω)L2(Ω) is established for a reaction–diffusion equation on a bounded domain ΩΩ in RdRd with Dirichlet boundary conditions, where the reaction term contains an operator F:L2(Ω)→L2(Ω)F:L2(Ω)→L2(Ω) which is nonlocal and possibly nonlinear. Existence of weak solutions is established, but uniqueness is not required. Compactness of the multivalued flow is obtained via estimates obtained from limits of Galerkin approximations. In contrast with the usual situation, these limits apply for all and not just for almost all time instants.