*(English)*Zbl 1092.35016

This paper is mainly concerned with the dynamical behavior of the following nonclassical parabolic equation:

where ${\Omega}$ is an open bounded set of ${\mathbb{R}}^{n}$ with sufficiently regular boundary $\partial {\Omega}$, $\mu \in [0,1]\xb7$ 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 $\mu \in \left[0,1\right]$ are established. These estimates are particularly useful in understanding the effects of the term $\mu {\Delta}{u}_{t}$ to the dynamics of the equation as $\mu \to 0$. Secondly, the continuous dependence of solutions of (1)–(3) on $\mu $ as $\mu \to 0$ is considered. Let $R,T>0$. Then it is shown that for some constant ${C}_{T}\left(R\right)>0$, $\parallel {S}_{\mu}\left(t\right){u}_{0}-{S}_{0}\left(t\right){u}_{0}{\parallel}_{1}\le {C}_{T}\left(R\right)\sqrt{\mu},\forall t\in \left[0,T\right],$ for any ${u}_{0}\in {H}^{2}\left({\Omega}\right)\cap {H}_{0}^{1}\left({\Omega}\right)$ with $\parallel {u}_{0}{\parallel}_{2}\le R,$ where ${S}_{\mu}\left(t\right)$ is the solution semigroup of (1)–(3). Finally, the existence of the global attractor ${A}_{\mu}$ for the system is established and the upper semicontinuity of ${A}_{\mu}$ at $\mu =0$ is proved.

##### MSC:

35B41 | Attractors (PDE) |

35Q35 | PDEs in connection with fluid mechanics |

35B40 | Asymptotic behavior of solutions of PDE |

35K60 | Nonlinear initial value problems for linear parabolic equations |