Existência de atrator para um sistema de equações de evolução

Abstract

In this work, we study the behavior at infinity of solutions of (1.1) − (1.3). We recall that these equations do not have smoothing property and the usual dip between Sobolev spaces loses compactness due to lack of regularity in the domain. To overcome these difficulties, we use some Strichartz-type inequalities, the decomposition technique and also an idea of ​​concentrated compactness. We use Strichartz-type inequalities and an energy conservation method to show the existence and continuity of semigroups generated by these equations. We obtain the asymptotic smoothing effect by decomposing the semigroup into a decaying part and a more regular part. Taking the limits when t goes to infinity, we have to overcome the incompactness of Sobolev's dips. We borrow the idea of ​​the principle of concentrated compactness and use the “nullification” and the “dichotomy” via a cut function.