Pullback attractors for a class of non-autonomous nonclassical diffusion equations

We consider the first initial boundary value problem for the non-autonomous nonclassical diffusion equation ut−εΔut−Δu+f(u)=g(t)ut−εΔut−Δu+f(u)=g(t), ε∈[0,1]ε∈[0,1], in a bounded domain in RNRN. Under a Sobolev growth rate of the nonlinearity ff and a suitable exponential growth of the external force gg, using the asymptotic a priori   estimate method, we prove the existence of pullback DD-attractors Aεˆ in the space H01(Ω) and the upper semicontinuity of Aεˆ at ε=0ε=0.