Dynamics of the nonclassical diffusion equations

We consider the dynamical behavior of the nonclassical diffusion equation with critical nonlinearity for both autonomous and nonautonomous cases. For the autonomous case, we obtain the existence of a global attractor when the forcing term only belongs to H-1, this result simultaneously resolves a problem in Acta Mathematicae Applicatae Sinica 18 ( 2002), 273-276 related to the critical exponent. For the nonautonomous case, assumed that the time-dependent forcing term is translation bounded instead of translation compact, we first prove the asymptotic regularity of solutions, then the existence of a compact uniform attractor together with its structure and regularity has been obtained; finally, we show the existence of ( nonautonomous) exponential attractors.