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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:

u t -μΔu t -Δu+g(u)=f(x),inΩ× + ,u(t,x)=0,forxΩ,u(0,x)=u 0 ,forxΩ,

where Ω is an open bounded set of n with sufficiently regular boundary Ω, μ[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 μ[0,1] are established. These estimates are particularly useful in understanding the effects of the term μΔu t to the dynamics of the equation as μ0. Secondly, the continuous dependence of solutions of (1)–(3) on μ as μ0 is considered. Let R,T>0. Then it is shown that for some constant C T (R)>0, S μ (t)u 0 -S 0 (t)u 0 1 C T (R)μ,t[0,T], for any u 0 H 2 (Ω)H 0 1 (Ω) with u 0 2 R, where S μ (t) is the solution semigroup of (1)–(3). Finally, the existence of the global attractor A μ for the system is established and the upper semicontinuity of A μ at μ=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