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 in the topology of H2(Ω); moreover, Aμ→A0 in the sense of Hausdorff semidistance in as μ goes to 0.