In this paper,we study Sobolev spaces in infinite dimensions and the corresponding embedding theorems.Our underlying spaces areℓ^(r)for r ∈[1,∞),equipped with corresponding probability measures.For the weighted Sobo...In this paper,we study Sobolev spaces in infinite dimensions and the corresponding embedding theorems.Our underlying spaces areℓ^(r)for r ∈[1,∞),equipped with corresponding probability measures.For the weighted Sobolev space W_(b)^(1,p)(ℓ^(r),γa)with a weight a∈ℓ^(r)of the Gaussian measureγa and a gradient weight b∈l^(∞),we characterize the relation between the weights(a and b)and the continuous(resp.compact)log-Sobolev embedding for p∈[1,∞)(resp.p∈(1,∞)).Several counterexamples are also constructed,which are of independent interest.展开更多
Let X be a rearrangement invariant space is R^n and Wxr1…,rn be an anisotropicSobolev space which is a generalization of Wpri,…,rn,The main subject of this paper is to prove the embedding theorem fot Wpri,…,rn.
Based on the methods introduced by Klainerman and Ponce, and Cohn, a lower bounded estimate of the existence time for a kind of semilinear Schrdinger equation is obtained in this paper. The implementation of this meth...Based on the methods introduced by Klainerman and Ponce, and Cohn, a lower bounded estimate of the existence time for a kind of semilinear Schrdinger equation is obtained in this paper. The implementation of this method depends on the L p-L q estimate and the energy estimate.展开更多
New Sobolev type embeddings for some weighted Banach spaces are established. Using such embeddings and the singular positive radial entire solutions, we construct singular positive weak solutions with a prescribed sin...New Sobolev type embeddings for some weighted Banach spaces are established. Using such embeddings and the singular positive radial entire solutions, we construct singular positive weak solutions with a prescribed singular set for a weighted elliptic equation. Our main results in this paper also provide positive weak solutions with a prescribed singular set to an equation with Hardy potential.展开更多
New embeddings of some weighted Sobolev spaces with weights a(x)and b(x)are established.The weights a(x)and b(x)can be singular.Some applications of these embeddings to a class of degenerate elliptic problems of the f...New embeddings of some weighted Sobolev spaces with weights a(x)and b(x)are established.The weights a(x)and b(x)can be singular.Some applications of these embeddings to a class of degenerate elliptic problems of the form-div(a(x)?u)=b(x)f(x,u)in?,u=0 on??,where?is a bounded or unbounded domain in RN,N 2,are presented.The main results of this paper also give some generalizations of the well-known Caffarelli-Kohn-Nirenberg inequality.展开更多
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
In this paper, we obtain some existence results for a class of singular semilinear elliptic problems where we improve some earlier results of Zhijun Zhang. We show the existence of entire positive solutions without th...In this paper, we obtain some existence results for a class of singular semilinear elliptic problems where we improve some earlier results of Zhijun Zhang. We show the existence of entire positive solutions without the monotonic condition imposed in Zhang’s paper. The main point of our technique is to choose an approximating sequence and prove its convergence. The desired compactness can be obtained by the Sobolev embedding theorems.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
基金supported by National Natural Science Foundation of China(Grant Nos.11931011 and 11901407)New Cornerstone Investigator Program and the Science Development Project of Sichuan University(Grant No 2020SCUNL201).
文摘In this paper,we study Sobolev spaces in infinite dimensions and the corresponding embedding theorems.Our underlying spaces areℓ^(r)for r ∈[1,∞),equipped with corresponding probability measures.For the weighted Sobolev space W_(b)^(1,p)(ℓ^(r),γa)with a weight a∈ℓ^(r)of the Gaussian measureγa and a gradient weight b∈l^(∞),we characterize the relation between the weights(a and b)and the continuous(resp.compact)log-Sobolev embedding for p∈[1,∞)(resp.p∈(1,∞)).Several counterexamples are also constructed,which are of independent interest.
基金Supported by National Natural Science Foundation of China(10871173,10931001,11101372)
文摘Let X be a rearrangement invariant space is R^n and Wxr1…,rn be an anisotropicSobolev space which is a generalization of Wpri,…,rn,The main subject of this paper is to prove the embedding theorem fot Wpri,…,rn.
文摘Based on the methods introduced by Klainerman and Ponce, and Cohn, a lower bounded estimate of the existence time for a kind of semilinear Schrdinger equation is obtained in this paper. The implementation of this method depends on the L p-L q estimate and the energy estimate.
基金supported by National Natural Science Foundation of China (Grant Nos. 11171092 and 11571093)
文摘New Sobolev type embeddings for some weighted Banach spaces are established. Using such embeddings and the singular positive radial entire solutions, we construct singular positive weak solutions with a prescribed singular set for a weighted elliptic equation. Our main results in this paper also provide positive weak solutions with a prescribed singular set to an equation with Hardy potential.
基金supported by National Natural Science Foundation of China (Grant Nos. 11171092, 11571093 and 11371117)
文摘New embeddings of some weighted Sobolev spaces with weights a(x)and b(x)are established.The weights a(x)and b(x)can be singular.Some applications of these embeddings to a class of degenerate elliptic problems of the form-div(a(x)?u)=b(x)f(x,u)in?,u=0 on??,where?is a bounded or unbounded domain in RN,N 2,are presented.The main results of this paper also give some generalizations of the well-known Caffarelli-Kohn-Nirenberg inequality.
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.
基金supported in part by NSF(Youth) of Shandong Province and NNSF of China
文摘In this paper, we obtain some existence results for a class of singular semilinear elliptic problems where we improve some earlier results of Zhijun Zhang. We show the existence of entire positive solutions without the monotonic condition imposed in Zhang’s paper. The main point of our technique is to choose an approximating sequence and prove its convergence. The desired compactness can be obtained by the Sobolev embedding theorems.
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.
