On the dynamics of a class of nonclassical parabolic equations

We consider the first initial and boundary value problem of nonclassical parabolic equations ut − μΔut − Δu + g(u) = f (x) on a bounded domain Ω, where μ ∈ [0, 1]. First, we establish some uniform decay estimates for the solutions of the problem which are independent of the parameter μ. Then we prove the continuity of solutions as μ→0. Finally we show that the problem has a unique global attractor Aμ in V2 = H2(Ω) ∩ H1 0 (Ω) in the topology of H2(Ω); moreover, Aμ →A0 in the sense of Hausdorff semidistance in H1 0 (Ω) as μ goes to 0. © 2005 Elsevier Inc. All rights reserved.