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
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
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.
