The critical forms of the attractors for semigroups and the existence of critical attractors

被引:14
作者
De, L [1 ]
机构
[1] Inner Mongolia Univ, Dept Math, Huhehot, Peoples R China
来源
PROCEEDINGS OF THE ROYAL SOCIETY A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES | 1998年 / 454卷 / 1976期
关键词
critical form of attractor; semigroup with B-ACP; B-bounded semigroup; pi-limit set; global attractor; invariant set;
D O I
10.1098/rspa.1998.0253
中图分类号
O [数理科学和化学]; P [天文学、地球科学]; Q [生物科学]; N [自然科学总论];
学科分类号
07 ; 0710 ; 09 ;
摘要
The critical forms of the minimal closed global attractor (M) over cap and the minimal closed global B-attractor M for operator semigroups are given, i.e. if the set [U(x is an element of X)omega(x)](X) attracts each point x in the phase space X, then it must be the unique minimal closed global attractor (M) over cap, and if the set [U(B is an element of B)w(B)](x) attracts each bounded set in X, then it must be the unique minimal closed global B-attractor M. On this basis, the sufficient conditions for the existence of (M) over cap and M, interrelation between (M) over cap and M are discussed. We introduce some classes of semigroups and study a variety of relations of them and the different classes of semigroups considered by Ladyzhenskaya, Hale and Temam. When a semigroup's continuity is not required, it is proved that the semigroups with B-ACP introduced by us coincide with the asymptotically smooth semigroups, the semigroups of class AK or B-AK, respectively, and that if these semigroups are B-bounded and U(x is an element of x)w(x)is an element of B, then each of them possesses the unique non-empty minimal invariant compact global attractor (M) over cap = [U-x is an element of x w(x)](x) and the unique non-empty minimal invariant compact global B-attractor M = [U(B is an element of B)w(B)](x). M = w(O-delta((M) over cap)), where 6 > O is a small number. Furthermore, the sufficient conditions for M = M and (M) over cap subset of(not equal)M are given. It is further proved that these conditions, which guarantee the existence of the compact invariant global B-attractor M, are not only sufficient, but also necessary. So, in order to guarantee a semigroup to have such attractor M, the conditions which are put on the semigroup cannot be weakened again. These results will be useful in studying the existence and construction of attractors.
引用
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页码:2157 / 2171
页数:15
相关论文
共 7 条
[1]  
[Anonymous], USP MAT NAUK
[2]  
Chepyzhov V., 1992, ADV SOV MATH, V10, P85, DOI DOI 10.1090/ADVSOV/010/02
[3]  
Hale J. K., 1988, ASYMPTOTIC BEHAV DIS
[4]  
LADYZHENSKAYA O, 1991, ATTRACTORS SEMIGROUP, P3
[5]  
Ladyzhenskaya O.A, 1987, USPEKHI MAT NAUK, V42, P25
[6]  
Ladyzhenskaya OA., 1972, ZAP NAUCHN SEM LOMI, V27, P91
[7]  
TEMAM R, 1988, APPL MATH SCI SERIES, V68