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

The system

t u-Δu+f(u)+λ 0 u-g=0,(1)

where u=u(t,x), t>0, x n , λ 0 is a positive constant and the functions u, f, g have values in 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,γ , γ>0, with the norm defined by

|u| 0,γ 2 = n (1+|x| 2 ) γ |u(x)| 2 dx·

The weighted Sobolev spaces H l,γ , l=1,2, are defined with the norms |u| l,γ 2 = |α|l | α u| 0,γ 2 .

Babin and Vishik showed that there exists a unique solution to (1) with initial conditions u| t=0 =u 0 H 1,γ , which belongs to L 2 ([0,T], H 2,γ )L ([0,T],H 1,γ ). The semigroup S t :H 0,γ H 0,γ (S t u 0 =u(t)) has an absorbing invariant set B 4 which is bounded in H 2,γ ; S t is continuous on B 4 in the topology of H 0,γ .

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:
35K57Reaction-diffusion equations
35B40Asymptotic behavior of solutions of PDE
35K15Second order parabolic equations, initial value problems
47H20Semigroups of nonlinear operators
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