In this paper we consider the Cauchy problem for the singular semilinear parabolic equation u t-Δu+V 1(x)u=V 2(x)u p,x∈R n\{0},t>0, where V 1(x),V 2(x) may have singularities at the origin. Using functions...In this paper we consider the Cauchy problem for the singular semilinear parabolic equation u t-Δu+V 1(x)u=V 2(x)u p,x∈R n\{0},t>0, where V 1(x),V 2(x) may have singularities at the origin. Using functions of the Kato class and the Green tight functions we got the existence of the positive solution being singular at the origin.展开更多
This paper is motivated by looking for a loop solution of the Hamiltonian systems such that (0.1) q'(t)+V′(q(t))=0 for t∈ with some T>0 and (0.2) 12|q′(t)| 2+V(q(t))=h ...This paper is motivated by looking for a loop solution of the Hamiltonian systems such that (0.1) q'(t)+V′(q(t))=0 for t∈ with some T>0 and (0.2) 12|q′(t)| 2+V(q(t))=h for t∈ with q(0)=q(T)=x 0 where q∈C 2(, R n 0}), n≥2, x 0∈R n 0} is a fixed point, h∈R is a given number, V∈C 2(R n 0}), R is a potential with a singularity and V′ denotes its gradient. Our main existence results are obtained by a appropriately defined lengthdecreasing (or rather energy decreasing) deformation and a min max procedure which is a combined version of Bahri Rabinowitz and Klingenberg . Our main assumptions are geodesic convex conditions found by the author and the strong force condition of Gordon . As a direct application, for the relativistic gravitational potential V(x)=|x| -1 +|x| -2 or its large scale perturbation, there always exists an almost periodic solution of (0.1)-(0.2) for any h∈R and any x 0∈R n 0} with | x 0 | small enough. This is an interesting phenomenon because we know that there exists no periodic solution of prescribed nonnegative energy for such a Hamiltonian system.展开更多
We employ the invariant eigen-operator （lEO） method to find the invariant eigen-operators of N-body singular oscillators＇ Hamiltonians and then derive their energy gaps. The Hamiltonians of parametric amplifiers wi...We employ the invariant eigen-operator （lEO） method to find the invariant eigen-operators of N-body singular oscillators＇ Hamiltonians and then derive their energy gaps. The Hamiltonians of parametric amplifiers with singular potential are also discussed in this way.展开更多
The precipitation in Shandong in July, August as well as the whole summer (JJA) and the corresponding 500 hPa geopotential height fields are analyzed by means of the SVD (singular value decomposition) methodology. It ...The precipitation in Shandong in July, August as well as the whole summer (JJA) and the corresponding 500 hPa geopotential height fields are analyzed by means of the SVD (singular value decomposition) methodology. It is found that the general circulations in East Asia and the Western Pacific underwent decadal changes around 1979. The geopotential height, in particular over key areas like the South China Sea and the Philippines, increased after 1979. Corresponding to the changes in the geopotential height, the rainfall in Shandong started to decrease around 1979. The synthesized analysis shows that when the geopotential height at 500hPa level decreases in the key areas, the Western Pacific subtropical high shifts northward and an anticyclonic anomalous cell enforces the southerly flow over Shandong-Korea-Japan, Shandong could experience a wet period. A dry period is likely to occur when the geopotential height increases in these key areas, the subtropical high moves southward or expands westward to a great distance, and a cyclonic anomalous cell controls Shandong. Respective conceptual models for the causative mechanism are obtained for the cases of July, August and the whole summer (JJA) .展开更多
We show that the eigenfunctions of Kohn-Sham equations can be decomposed as ■ = F ψ, where F depends on the Coulomb potential only and is locally Lipschitz, while ψ has better regularity than ■.
文摘In this paper we consider the Cauchy problem for the singular semilinear parabolic equation u t-Δu+V 1(x)u=V 2(x)u p,x∈R n\{0},t>0, where V 1(x),V 2(x) may have singularities at the origin. Using functions of the Kato class and the Green tight functions we got the existence of the positive solution being singular at the origin.
文摘This paper is motivated by looking for a loop solution of the Hamiltonian systems such that (0.1) q'(t)+V′(q(t))=0 for t∈ with some T>0 and (0.2) 12|q′(t)| 2+V(q(t))=h for t∈ with q(0)=q(T)=x 0 where q∈C 2(, R n 0}), n≥2, x 0∈R n 0} is a fixed point, h∈R is a given number, V∈C 2(R n 0}), R is a potential with a singularity and V′ denotes its gradient. Our main existence results are obtained by a appropriately defined lengthdecreasing (or rather energy decreasing) deformation and a min max procedure which is a combined version of Bahri Rabinowitz and Klingenberg . Our main assumptions are geodesic convex conditions found by the author and the strong force condition of Gordon . As a direct application, for the relativistic gravitational potential V(x)=|x| -1 +|x| -2 or its large scale perturbation, there always exists an almost periodic solution of (0.1)-(0.2) for any h∈R and any x 0∈R n 0} with | x 0 | small enough. This is an interesting phenomenon because we know that there exists no periodic solution of prescribed nonnegative energy for such a Hamiltonian system.
基金The project supported by the Specialized Research Fund for the Doctorial Program of Higher Education of China
文摘We employ the invariant eigen-operator （lEO） method to find the invariant eigen-operators of N-body singular oscillators＇ Hamiltonians and then derive their energy gaps. The Hamiltonians of parametric amplifiers with singular potential are also discussed in this way.
文摘The precipitation in Shandong in July, August as well as the whole summer (JJA) and the corresponding 500 hPa geopotential height fields are analyzed by means of the SVD (singular value decomposition) methodology. It is found that the general circulations in East Asia and the Western Pacific underwent decadal changes around 1979. The geopotential height, in particular over key areas like the South China Sea and the Philippines, increased after 1979. Corresponding to the changes in the geopotential height, the rainfall in Shandong started to decrease around 1979. The synthesized analysis shows that when the geopotential height at 500hPa level decreases in the key areas, the Western Pacific subtropical high shifts northward and an anticyclonic anomalous cell enforces the southerly flow over Shandong-Korea-Japan, Shandong could experience a wet period. A dry period is likely to occur when the geopotential height increases in these key areas, the subtropical high moves southward or expands westward to a great distance, and a cyclonic anomalous cell controls Shandong. Respective conceptual models for the causative mechanism are obtained for the cases of July, August and the whole summer (JJA) .
基金supported by National Natural Science Foundation of China (Grant No. 91330202)the Funds for Creative Research Groups of China (Grant No. 11321061)+1 种基金National Basic Research Program of China (Grant No. 2011CB309703)the National Center for Mathematics and Interdisciplinary Sciences of the Chinese Academy of Sciences
文摘We show that the eigenfunctions of Kohn-Sham equations can be decomposed as ■ = F ψ, where F depends on the Coulomb potential only and is locally Lipschitz, while ψ has better regularity than ■.
