Attractors for non-autonomous parabolic problems with singular initial data

Abstract

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 W 1 , r ( Ω ) ( 1 < r < N ), respectively. When the initial data belongs to L r ( Ω ) , we establish the existence of uniform attractors in L r ( Ω ) for the family of processes with external forces being translation bounded but not translation compact in L loc p ( R ; L r ( Ω ) ) . When we consider the existence of uniform attractors in H 0 1 ( 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.