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On the dynamics of a class of nonclassical parabolic equations. (English) Zbl 1092.35016

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

$\begin{array}{cc}\hfill {u}_{t}-\mu {\Delta }{u}_{t}-{\Delta }u+g\left(u\right)=f\left(x\right),& \phantom{\rule{1.em}{0ex}}\text{in}\phantom{\rule{4pt}{0ex}}{\Omega }×{ℝ}_{+},\hfill \\ \hfill u\left(t,x\right)=0,& \phantom{\rule{1.em}{0ex}}\text{for}\phantom{\rule{4pt}{0ex}}x\in \partial {\Omega },\hfill \\ \hfill u\left(0,x\right)={u}_{0},& \phantom{\rule{1.em}{0ex}}\text{for}\phantom{\rule{4pt}{0ex}}x\in {\Omega },\hfill \end{array}$

where ${\Omega }$ is an open bounded set of ${ℝ}^{n}$ with sufficiently regular boundary $\partial {\Omega }$, $\mu \in \left[0,1\right]·$ 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