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Exponential attractors of reaction-diffusion systems in an unbounded domain. (English) Zbl 0846.35061

The system

${\partial }_{t}u-{\Delta }u+f\left(u\right)+{\lambda }_{0}u-g=0,\phantom{\rule{2.em}{0ex}}\left(1\right)$

where $u=u\left(t,x\right)$, $t>0$, $x\in {ℝ}^{n}$, ${\lambda }_{0}$ is a positive constant and the functions $u$, $f$, $g$ have values in ${ℝ}^{n}$, 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

${|u|}_{0,\gamma }^{2}={\int }_{{ℝ}^{n}}{\left(1+|x|}^{2}{\right)}^{\gamma }{|u\left(x\right)|}^{2}dx·$

The weighted Sobolev spaces ${H}_{l,\gamma }$, $l=1,2$, are defined with the norms ${|u|}_{l,\gamma }^{2}={\sum }_{|\alpha |\le l}{|{\partial }^{\alpha }u|}_{0,\gamma }^{2}$.

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 ${L}_{2}\left(\left[0,T\right]$, ${H}_{2,\gamma }\right)\cap {L}_{\infty }\left(\left[0,T\right],{H}_{1,\gamma }\right)$. The semigroup ${S}_{t}:{H}_{0,\gamma }\to {H}_{0,\gamma }$ $\left({S}_{t}{u}_{0}=u\left(t\right)\right)$ 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 ${H}_{0,\gamma }$.

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.

MSC:
 35K57 Reaction-diffusion equations 35B40 Asymptotic behavior of solutions of PDE 35K15 Second order parabolic equations, initial value problems 47H20 Semigroups of nonlinear operators
References:
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