This paper is mainly concerned with the dynamical behavior of the following nonclassical parabolic equation:
where is an open bounded set of with sufficiently regular boundary , Nonclassical parabolic equations arise as models to describe physical phenomena such as non-Newtonian flow, soil mechanics and heat conduction, etc.
The main aim of this paper is as follows. First, some uniform decay estimates for (1)–(3) which are independent of are established. These estimates are particularly useful in understanding the effects of the term to the dynamics of the equation as . Secondly, the continuous dependence of solutions of (1)–(3) on as is considered. Let . Then it is shown that for some constant , for any with where is the solution semigroup of (1)–(3). Finally, the existence of the global attractor for the system is established and the upper semicontinuity of at is proved.