where , , , is a positive constant and the functions , , have values in , is considered. The function is assumed to be nonlinear and to satisfy natural smoothness and growth conditions and to vanish at zero. The function belongs to weighted Sobolev space , , with the norm defined by
The weighted Sobolev spaces , , are defined with the norms .
Babin and Vishik showed that there exists a unique solution to (1) with initial conditions , which belongs to , . The semigroup has an absorbing invariant set which is bounded in ; is continuous on in the topology of .
The authors consider the restriction of this semigroup to the invariant set , 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.