ASYMPTOTICS OF THE ELEMENTS OF ATTRACTORS CORRESPONDING TO SINGULARLYPERTURBED PARABOLIC EQUATIONS

In a domain OMEGA(n) sub-set-of R(n) we consider the first boundary va lue problem for a quasilinear parabolic fourth-order equation with a s mall parameter epsilon in the highest derivatives, which degenerates f or epsilon = 0 into a second order equation. It is well known that the semigroup corresponding to this problem has an attractor, that is, an invariant attracting set in the phase space. In this paper we investi gate the structure of this attractor by means of an asymptotic expansi on in epsilon. The dominant term of the asymptotics is the solution of a second-order equation. The asymptotic expansion also contains bound ary layer functions, which are responsible for the deterioration of th e differential properties of the elements of the attractor near the bo undary. The asymptotics constructed in this way (with an estimate of t he remainder) enable us to study the differential properties of attrac tors and their behavior as epsilon --> 0 in any interior subdomain OME GA', OMEGA'BAR subset-of OMEGA. For simplicity, the investigation is c arried out in the case when OMEGA is a bounded cylindrical domain. The generalization to OMEGA sub-set-of R(n) does not present any difficul ties.