The aim of this work is to understand better the long time behaviour of asymptotically compact random dynamical systems （RDS）, which can be generated by solutions of some stochastic partial differential equations on...The aim of this work is to understand better the long time behaviour of asymptotically compact random dynamical systems （RDS）, which can be generated by solutions of some stochastic partial differential equations on unbounded domains. The conceptual analysis for the long time behavior of RDS will be done through some examples. An application of those analysis will be demonstrated through the proof of the existence of random attractors for asymptotically compact dissipative RDS.展开更多
This is the first part of a work on second order nonlinear, nonmonotone evolution inclusions defined in the framework of an evolution triple of spaces and with a multivalued nonlinearity depending on both x（t） and x...This is the first part of a work on second order nonlinear, nonmonotone evolution inclusions defined in the framework of an evolution triple of spaces and with a multivalued nonlinearity depending on both x（t） and x（t）. In this first part we prove existence and relaxation theorems. We consider the case of an usc, convex valued nonlinearity and we show that for this problem the solution set is nonempty and compact in C^1 （T, H）. Also we examine the Isc, nonconvex case and again we prove the existence of solutions. In addition we establish the existence of extremal solutions and by strengthening our hypotheses, we show that the extremal solutions are dense in C^1 （T, H） to the solutions of the original convex problem （strong relaxation）. An example of a nonlinear hyperbolic optimal control problem is also discussed.展开更多
This paper gives characterizations for diffusion processes on the line and birth-death processes whose generators admit the empty essential spectra. Some equivalent conditions for empty essential spectra for general M...This paper gives characterizations for diffusion processes on the line and birth-death processes whose generators admit the empty essential spectra. Some equivalent conditions for empty essential spectra for general Markov generators are also discussed.展开更多
We contimle the work initiated in [1] （Second order nonlinear evolution inclusions I： Existence and relaxation results. Acta Mathematics Science, English Series, 21（5）, 977-996 （2005）） and study the structural ...We contimle the work initiated in [1] （Second order nonlinear evolution inclusions I： Existence and relaxation results. Acta Mathematics Science, English Series, 21（5）, 977-996 （2005）） and study the structural properties of the solution set of second order evolution inclusions which are defined in the analytic framework of the evolution triple. For the convex problem we show that the solution set is compact Rs, while for the nonconvex problem we show that it is path connected, Also we show that the solution set is closed only if the multivalued nonlinearity is convex valued. Finally we illustrate the results by considering a nonlinear hyperbolic problem with discontinuities.展开更多
Lp Poincare inequalities for general symmetric forms are established by new Cheeger＇s isoperimetric constants. Lp super-Poincare inequalities are introduced to describe the equivalent conditions for the Lp compact em...Lp Poincare inequalities for general symmetric forms are established by new Cheeger＇s isoperimetric constants. Lp super-Poincare inequalities are introduced to describe the equivalent conditions for the Lp compact embedding, and the criteria via the new Cheeger＇s constants for those inequalities are presented. Finally, the concentration or the volume growth of measures for these inequalities are studied.展开更多
In this paper, we study the supercontractivity for Maxkov semigroups and obtain some sufficient and necessary conditions; especially explicit formulae are obtained for birth-death process and diffusion on the line. Su...In this paper, we study the supercontractivity for Maxkov semigroups and obtain some sufficient and necessary conditions; especially explicit formulae are obtained for birth-death process and diffusion on the line. Sufficient conditions and necessary conditions in terms of isoperimetric inequalities are also presented. Moreover, we prove that the supercontractivity is equivalent to the compact embedding of Sobolev space into an Orlicz space.展开更多
In this paper, we consider the long time behaviors for the partly dissipative stochastic reaction diffusion equations. The existence of a bounded random absorbing set is firstly discussed for the systems and then an e...In this paper, we consider the long time behaviors for the partly dissipative stochastic reaction diffusion equations. The existence of a bounded random absorbing set is firstly discussed for the systems and then an estimate on the solution is derived when the time is sufficiently large. Then, we establish the asymptotic compactness of the solution operator by giving uniform a priori estimates on the tails of solutions when time is large enough. In the last, we finish the proof of existence a pullback ran- dom attractor in L^2 （R^n） × L^2 （R^n）. We also prove the upper semicontinuity of random attractors when the intensity of noise approaches zero. The long time behaviors are discussed to explain the corresponding physical phenomenon.展开更多
Let G be a locally compact Abelian group with Haar measure.The authors discuss some basic properties of L_(w_1)~r(G) ∩ L(p,q,w_2dμ)(G) spaces.Then the necessary conditions for compact embeddings of the spaces L_(w_1...Let G be a locally compact Abelian group with Haar measure.The authors discuss some basic properties of L_(w_1)~r(G) ∩ L(p,q,w_2dμ)(G) spaces.Then the necessary conditions for compact embeddings of the spaces L_(w_1)~r(R^d)∩L(p,q,w_2dμ)(R^d) are showed.展开更多
基金the National NSFC under grant No.50579022the Foundation of Pre-973 Program of China under grant No.2004CCA02500+1 种基金the SRF for the ROCS,SEMthe Talent Recruitment Foundation of HUST
文摘The aim of this work is to understand better the long time behaviour of asymptotically compact random dynamical systems （RDS）, which can be generated by solutions of some stochastic partial differential equations on unbounded domains. The conceptual analysis for the long time behavior of RDS will be done through some examples. An application of those analysis will be demonstrated through the proof of the existence of random attractors for asymptotically compact dissipative RDS.
文摘This is the first part of a work on second order nonlinear, nonmonotone evolution inclusions defined in the framework of an evolution triple of spaces and with a multivalued nonlinearity depending on both x（t） and x（t）. In this first part we prove existence and relaxation theorems. We consider the case of an usc, convex valued nonlinearity and we show that for this problem the solution set is nonempty and compact in C^1 （T, H）. Also we examine the Isc, nonconvex case and again we prove the existence of solutions. In addition we establish the existence of extremal solutions and by strengthening our hypotheses, we show that the extremal solutions are dense in C^1 （T, H） to the solutions of the original convex problem （strong relaxation）. An example of a nonlinear hyperbolic optimal control problem is also discussed.
基金Research supported in part by RFDP(No.2001002707)973 ProjectNSFC(No.10121101)
文摘This paper gives characterizations for diffusion processes on the line and birth-death processes whose generators admit the empty essential spectra. Some equivalent conditions for empty essential spectra for general Markov generators are also discussed.
文摘We contimle the work initiated in [1] （Second order nonlinear evolution inclusions I： Existence and relaxation results. Acta Mathematics Science, English Series, 21（5）, 977-996 （2005）） and study the structural properties of the solution set of second order evolution inclusions which are defined in the analytic framework of the evolution triple. For the convex problem we show that the solution set is compact Rs, while for the nonconvex problem we show that it is path connected, Also we show that the solution set is closed only if the multivalued nonlinearity is convex valued. Finally we illustrate the results by considering a nonlinear hyperbolic problem with discontinuities.
基金Supported in part by Program for New Century Excellent Talents in University (NCET)973 Project (Grant No.2006CB805901)National Natural Science Foundation of China (Grant No.10721091)
文摘Lp Poincare inequalities for general symmetric forms are established by new Cheeger＇s isoperimetric constants. Lp super-Poincare inequalities are introduced to describe the equivalent conditions for the Lp compact embedding, and the criteria via the new Cheeger＇s constants for those inequalities are presented. Finally, the concentration or the volume growth of measures for these inequalities are studied.
基金Research supported in part by RFDP(No 20010027007)973 ProjectNSFC(No 10121101,No 10025105 and No 10301007)
文摘In this paper, we study the supercontractivity for Maxkov semigroups and obtain some sufficient and necessary conditions; especially explicit formulae are obtained for birth-death process and diffusion on the line. Sufficient conditions and necessary conditions in terms of isoperimetric inequalities are also presented. Moreover, we prove that the supercontractivity is equivalent to the compact embedding of Sobolev space into an Orlicz space.
基金This work is supported by The National Natural Science Foundation of China （Grant No： 11301043）. We also express our thanks to the referee for helpful comments and suggestions.
文摘In this paper, we consider the long time behaviors for the partly dissipative stochastic reaction diffusion equations. The existence of a bounded random absorbing set is firstly discussed for the systems and then an estimate on the solution is derived when the time is sufficiently large. Then, we establish the asymptotic compactness of the solution operator by giving uniform a priori estimates on the tails of solutions when time is large enough. In the last, we finish the proof of existence a pullback ran- dom attractor in L^2 （R^n） × L^2 （R^n）. We also prove the upper semicontinuity of random attractors when the intensity of noise approaches zero. The long time behaviors are discussed to explain the corresponding physical phenomenon.
文摘Let G be a locally compact Abelian group with Haar measure.The authors discuss some basic properties of L_(w_1)~r(G) ∩ L(p,q,w_2dμ)(G) spaces.Then the necessary conditions for compact embeddings of the spaces L_(w_1)~r(R^d)∩L(p,q,w_2dμ)(R^d) are showed.
