Attractors for non-autonomous parabolic problems with singular initial data

In this paper, we study the asymptotic behavior of solutions of non-autonomous parabolic problems with singular initial data. We first establish the well-posedness of the equation when the initial data belongs to L^r(Ω) (1<r<∞) and W1,r(Ω) (1<r<N), respectively. When the initial data belongs to Lr(Ω), we establish the existence of uniform attractors in Lr(Ω) for the family of processes with external forces being translation bounded but not translation compact in L_locp(R{double-struck};L^r(Ω)). When we consider the existence of uniform attractors in H01(Ω), the solution of equation lacks the higher regularity, so we introduce a new type of solution and prove the existence result. For the long time behavior of solutions of the equation in W^1,r(Ω), we only obtain the uniform attracting property in the weak topology.