We continue our investigations on pointwise multipliers for Besov spaces of dominating mixed smoothness. This time we study the algebra property of the classes S_(p,q)~rB(R^d) with respect to pointwise multiplication....We continue our investigations on pointwise multipliers for Besov spaces of dominating mixed smoothness. This time we study the algebra property of the classes S_(p,q)~rB(R^d) with respect to pointwise multiplication. In addition, if p≤q, we are able to describe the space of all pointwise multipliers for S_(p,q)~rB(R^d).展开更多
The study of the control and stabilization of the KdV equation began with the work ofRussell and Zhang in late 1980s.Both exact control and stabilization problems have been intensivelystudied since then and significan...The study of the control and stabilization of the KdV equation began with the work ofRussell and Zhang in late 1980s.Both exact control and stabilization problems have been intensivelystudied since then and significant progresses have been made due to many people's hard work andcontributions.In this article,the authors intend to give an overall review of the results obtained so farin the study but with an emphasis on its recent progresses.A list of open problems is also providedfor further investigation.展开更多
In this paper, we study exact controllability and feedback stabilization forthe distributed parameter control system described by high-order KdV equation posedon a periodic domain T with an internal control acting on ...In this paper, we study exact controllability and feedback stabilization forthe distributed parameter control system described by high-order KdV equation posedon a periodic domain T with an internal control acting on an arbitrary small nonemptysubdomain w of T. On one hand, we show that the distributed parameter controlsystem is locally exactly controllable with the help of Bourgain smoothing effect; onthe other hand, we prove that the feedback system is locally exponentially stable withan arbitrarily large decay rate when Slemrod＇s feedback input is chosen.展开更多
文摘We continue our investigations on pointwise multipliers for Besov spaces of dominating mixed smoothness. This time we study the algebra property of the classes S_(p,q)~rB(R^d) with respect to pointwise multiplication. In addition, if p≤q, we are able to describe the space of all pointwise multipliers for S_(p,q)~rB(R^d).
文摘The study of the control and stabilization of the KdV equation began with the work ofRussell and Zhang in late 1980s.Both exact control and stabilization problems have been intensivelystudied since then and significant progresses have been made due to many people's hard work andcontributions.In this article,the authors intend to give an overall review of the results obtained so farin the study but with an emphasis on its recent progresses.A list of open problems is also providedfor further investigation.
基金The first author is financially supported by the Natural Science Foundation of Zhejiang Province （# LY18A010024, # LQ16A010003）, the China National Natural Science Foundation （# 11505154, # 11605156） and the Open Foundation from Marine Sciences in the Most Important Subjects of Zhejiang （# 20160101）. The second author is financially sup- ported by Foundation for Distinguished Young Teacher in Higher Education of Guangdong, China （YQ2015167）, Foundation for Characteristic Innovation in Higher Education of Guangdong, China （Analysis of some kinds of models of cell division and the spread of epidemics）, NSF of Guangdong Province （2015A030313707）. The authors are greatly in debt to the anonymous referee for his/her valuable comments and suggestions on modifying this manuscript.
文摘In this paper, we study exact controllability and feedback stabilization forthe distributed parameter control system described by high-order KdV equation posedon a periodic domain T with an internal control acting on an arbitrary small nonemptysubdomain w of T. On one hand, we show that the distributed parameter controlsystem is locally exactly controllable with the help of Bourgain smoothing effect; onthe other hand, we prove that the feedback system is locally exponentially stable withan arbitrarily large decay rate when Slemrod＇s feedback input is chosen.
