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Zbl 0846.35061
Babin, Anatoli; Nicolaenko, Basil
Exponential attractors of reaction-diffusion systems in an unbounded domain.
(English)
[J] J. Dyn. Differ. Equations 7, No.4, 567-590 (1995). ISSN 1040-7294; ISSN 1572-9222/e

The system $$\partial_t u- \Delta u+ f(u)+ \lambda_0 u- g= 0,\tag1$$ where $u= u(t, x)$, $t> 0$, $x\in \bbfR^n$, $\lambda_0$ is a positive constant and the functions $u$, $f$, $g$ have values in $\bbfR^n$, is considered. The function $f$ is assumed to be nonlinear and to satisfy natural smoothness and growth conditions and to vanish at zero. The function $g$ belongs to weighted Sobolev space $H_{0, \gamma}$, $\gamma> 0$, with the norm defined by $$|u|^2_{0, \gamma}= \int_{\bbfR^n} (1+ |x|^2)^\gamma |u(x)|^2 dx.$$ The weighted Sobolev spaces $H_{l, \gamma}$, $l= 1, 2$, are defined with the norms $|u|^2_{l, \gamma}= \sum_{|\alpha|\le l} |\partial^\alpha u|^2_{0, \gamma}$.\par Babin and Vishik showed that there exists a unique solution to (1) with initial conditions $u|_{t= 0}= u_0\in H_{1, \gamma}$, which belongs to $L_2([0, T]$, $H_{2, \gamma})\cap L_\infty([0, T], H_{1, \gamma})$. The semigroup $S_t: H_{0, \gamma}\to H_{0, \gamma}$ $(S_t u_0= u(t))$ has an absorbing invariant set $B_4$ which is bounded in $H_{2,\gamma}$; $S_t$ is continuous on $B_4$ in the topology of $H_{0, \gamma}$.\par The authors consider the restriction of this semigroup to the invariant set $B_4$, and prove that it possesses an exponential attractor. Exponential attractors (also called inertial sets) have finite fractal dimension and contain global attractors. A very important property of exponential attractors is their lower and upper semicontinuity with respect to Galerkin approximations.
[A.Cichocka (Katowice)]
MSC 2000:
*35K57 Reaction-diffusion equations
35B40 Asymptotic behavior of solutions of PDE
35K15 Second order parabolic equations, initial value problems
47H20 Semigroups of nonlinear operators

Keywords: weighted Sobolev space; exponential attractor; inertial sets; fractal dimension; Galerkin approximations

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