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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: $$\align u_t - \mu \Delta u_t - \Delta u + g(u) = f(x), &\quad\text{in}\ \Omega \times \Bbb R_ + ,\tag 1\\ u({t,x}) = 0, &\quad\text{for}\ x \in \partial \Omega ,\tag2\\ u({0,x}) = u_0 , &\quad\text{for}\ x \in \Omega,\tag 3\endalign$$ where $\Omega $ is an open bounded set of $\Bbb R^n $ with sufficiently regular boundary $\partial \Omega$, $\mu \in [0,1].$ 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 $\mu \in [{0,1}]$ 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 $C_T (R) > 0$, $\Vert {S_\mu (t)u_0 - S_0 (t)u_0} \Vert _1 \leq C_T (R)\sqrt \mu, \forall t \in [{0,T}], $ for any $u_0 \in H^2 (\Omega) \cap H_0^1 (\Omega)$ with $\Vert {u_0} \Vert _2 \leq R,$ where $S_\mu (t)$ 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:
35B41Attractors (PDE)
35Q35PDEs in connection with fluid mechanics
35B40Asymptotic behavior of solutions of PDE
35K60Nonlinear initial value problems for linear parabolic equations
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References:
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