EXISTENCE OF SOLUTIONS TO THE PARABOLIC EQUATION WITH A SINGULAR POTENTIAL OF THE SOBOLEV-HARDY TYPE

We study the existence of solutions to the following parabolic equation{ut-△pu=λ/|x|s|u|q-2u,(x,t)∈Ω×(0,∞),u(x,0)=f(x),x∈Ω,u(x,t)=0,(x,t)∈Ω×(0,∞),(P)}where-△pu ≡-div(|▽u|p-2▽u),1相似文献   

On approximation of smooth functions from null spaces of optimal linear differential operators with constant coefficients

For a real valued function f defined on a finite interval I we consider the problem of approximating f from null spaces of differential operators of the form Ln(ψ) = n ∑ k=0 akψ(k), where the constant coefficients ak ∈ R may be adapted to f . We prove that for each f ∈ C(n)(I), there is a selection of coefficients {a1, ,an} and a corresponding linear combination Sn( f ,t) = n ∑ k=1 bkeλkt of functions ψk(t) = eλkt in the nullity of L which satisfies the following Jackson’s type inequality: f (m) Sn(m )( f ,t) ∞≤ |an|2n|Im|1/1q/ep|λ|λn|n|I||nm1 Ln( f ) p, where |λn| = mka x|λk|, 0 ≤ m ≤ n 1, p,q ≥ 1, and 1p + q1 = 1. For the particular operator Mn(f) = f + 1/(2n) f(2n) the rate of approximation by the eigenvalues of Mn for non-periodic analytic functions on intervals of restricted length is established to be exponential. Applications in algorithms and numerical examples are discussed.     

Nonexistence of time-periodic solutions of the Dirac equation in non-extreme Kerr-Newman-AdS spacetime

In non-extreme Kerr-Newman-Ad S spacetime, we prove that there is no nontrivial Dirac particle which is Lpfor 0 p≤ 4/3 with arbitrary eigenvalue λ, and for 4/3 p≤ 4/(3-2q), 0 q 3/2 with eigenvalue|λ| |Q| + qκ, outside and away from the event horizon. By taking q =1/2, we show that there is no normalizable massive Dirac particle with mass greater than |Q| +κ/2 outside and away from the event horizon in non-extreme Kerr-Newman-Ad S spacetime, and they must either disappear into the black hole or escape to infinity, and this recovers the same result of Belgiorno and Cacciatori in the case of Q = 0 obtained by using spectral methods.Furthermore, we prove that any Dirac particle with eigenvalue |λ| κ/2 must be L~2 outside and away from the event horizon.     

主理想整环上n阶矩阵环中的Goldbach问题

Abstract: In this paper, we consider the Goldbach's problem for matrix rings, namely, we decompose an n ×n (n > 1) matrix over a principal ideal domain R into a sum of two matrices in Mn(R) with given determinants. We prove the following result: Let n > 1 be a natural number and A = (αij) be a matrix in Mn(R). Define d(A) := g.c.d{αij}. Suppose that p and q are two elements in R. Then (1) If n > 1 is even, then A can be written as a sum of two matrices X, Y in Mn(R) with det(X) = p and det(Y) = q if and only if d(A) |p-q; (2) If n > 1 is odd, then A can be written as a sum of two matrices X, Y in Mn(R) with det(X) = p and det(Y) = q if and only if d(A) |p + q. We apply the result to the matrices in Mn(Z) and Mn(Q[x]) and prove that if R = Z or Q[x], then any nonzero matrix A in Mn(R) can be written as a sum of two matrices in Mn(R) with prime determinants.     

Estimates for wave and Klein-Gordon equations on modulation spaces

We prove that the fundamental semi-group eit(m 2I+|Δ|)1/2(m = 0) of the Klein-Gordon equation is bounded on the modulation space M ps,q(Rn) for all 0 < p,q ∞ and s ∈ R.Similarly,we prove that the wave semi-group eit|Δ|1/2 is bounded on the Hardy type modulation spaces μsp,q(Rn) for all 0 < p,q ∞,and s ∈ R.All the bounds have an asymptotic factor tn|1/p 1/2| as t goes to the infinity.These results extend some known results for the case of p 1.Also,some applications for the Cauchy problems related to the semi-group eit(m2I+|Δ|)1/2 are obtained.Finally we discuss the optimum of the factor tn|1/p 1/2| and raise some unsolved problems.     

Anisotropic Hardy-Lorentz spaces and their applications

Let p ∈(0, 1], q ∈(0, ∞] and A be a general expansive matrix on Rn. We introduce the anisotropic Hardy-Lorentz space H~(p,q)_A(R~n) associated with A via the non-tangential grand maximal function and then establish its various real-variable characterizations in terms of the atomic and the molecular decompositions, the radial and the non-tangential maximal functions, and the finite atomic decompositions. All these characterizations except the ∞-atomic characterization are new even for the classical isotropic Hardy-Lorentz spaces on Rn.As applications, we first prove that Hp,q A(Rn) is an intermediate space between H~(p1,q1)_A(Rn) and H~(p2,q2)_A(R~n) with 0 p1 p p2 ∞ and q1, q, q2 ∈(0, ∞], and also between H~(p,q1)_A(Rn) and H~(p,q2)_A(R~n) with p ∈(0, ∞)and 0 q1 q q2 ∞ in the real method of interpolation. We then establish a criterion on the boundedness of sublinear operators from H~(p,q)_A(R~n) into a quasi-Banach space; moreover, we obtain the boundedness of δ-type Calder′on-Zygmund operators from H~(p,∞)_A(R~n) to the weak Lebesgue space L~(p,∞)(R~n)(or to H~p_A(R~n)) in the ln λcritical case, from H~(p,q)_A(R~n) to L~(p,q)(R~n)(or to H~(p,q)_A(R~n)) with δ∈(0,(lnλ)/(ln b)], p ∈(1/(1+,δ),1] and q ∈(0, ∞], as well as the boundedness of some Calderon-Zygmund operators from H~(p,q)_A(R~n) to L~(p,∞)(R~n), where b := | det A|,λ_:= min{|λ| : λ∈σ(A)} and σ(A) denotes the set of all eigenvalues of A.     

Existence of Nonnegative Solutions for a Class of Systems Involving Fractional(p,q)-Laplacian Operators

The authors study the following Dirichlet problem of a system involving fractional(p, q)-Laplacian operators:{(-△)_p~su=λa(x)|u|+~(p-2)u+λb(x)|u|~(α-2)|u|~βu+μ(x)/αδ|u|~(γ-2)|v|~δu in Ω,(-△)_p~su=λc(x)|v|+~(q-2)v+λb(x)|u|~α|u|~(β-2)v+μ(x)/βγ|u|~γ|v|~(δ-2)v in Ω,u=v=0 on R~N\Ω where λ 0 is a real parameter, ? is a bounded domain in RN, with boundary ?? Lipschitz continuous, s ∈(0, 1), 1 p ≤ q ∞, sq N, while(-?)s pu is the fractional p-Laplacian operator of u and, similarly,(-?)s qv is the fractional q-Laplacian operator of v. Since possibly p = q, the classical definitions of the Nehari manifold for systems and of the Fibering mapping are not suitable. In this paper, the authors modify these definitions to solve the Dirichlet problem above. Then, by virtue of the properties of the first eigenvalueλ_1 for a related system, they prove that there exists a positive solution for the problem when λ λ_1 by the modified definitions. Moreover, the authors obtain the bifurcation property when λ→λ_1~-. Finally, thanks to the Picone identity, a nonexistence result is also obtained when λ≥λ_1.     

CLASSIFICATION OF POSITIVE SOLUTIONS TO A SYSTEM OF HARDY-SOBOLEV TYPE EQUATIONS

In this paper, we are concerned with the following Hardy-Sobolev type system{(-?)~(α/2) u(x) =v~q(x)/|y|~(t_2) (-?)α/2 v(x) =u~p(x)/|y|~(t_1),x =(y, z) ∈(R ~k\{0}) × R~(n-k),(0.1)where 0 α n, 0 t_1, t_2 min{α, k}, and 1 p ≤τ_1 :=(n+α-2t_1)/( n-α), 1 q ≤τ_2 :=(n+α-2 t_2)/( n-α).We first establish the equivalence of classical and weak solutions between PDE system(0.1)and the following integral equations(IE) system{u(x) =∫_( R~n) G_α(x, ξ)v~q(ξ)/|η|t~2 dξ v(x) =∫_(R~n) G_α(x, ξ)(u~p(ξ))/|η|~(t_1) dξ,(0.2)where Gα(x, ξ) =(c n,α)/(|x-ξ|~(n-α))is the Green's function of(-?)~(α/2) in R~n. Then, by the method of moving planes in the integral forms, in the critical case p = τ_1 and q = τ_2, we prove that each pair of nonnegative solutions(u, v) of(0.1) is radially symmetric and monotone decreasing about the origin in R~k and some point z0 in R~(n-k). In the subcritical case (n-t_1)/(p+1)+(n-t_2)/(q+1) n-α,1 p ≤τ_1 and 1 q ≤τ_2, we derive the nonexistence of nontrivial nonnegative solutions for(0.1).     

A CONJECTURE CONCERNING THE HADAMARD PRODUCT OF INVERSE M-MATRICES

1 IntroductionFor an n×n matrix A which is an inverse M-matrix,M.Neumann in [1]conjecturedthat the Hadamard product A·A is an inverse of an M-matrix.They have checked hisconjecture without failure on Ultrametric matrices and inverse of MMA-matrices,Uni-pathicmatrices and the Willongby inverse M-matrices.Bo-Ying Wang et al.in[2]haveinvestigated Triangular inverse M-matrices which are closed under the Hadamard multipli-cation.Lu Linzheng,Sun Weiwei and Li Wen in[3]presented a more general conjecture     

Blow-up of p-Laplacian evolution equations with variable source power

We study the blow-up and/or global existence of the following p-Laplacian evolution equation with variable source power ut(x,t)=div(|?u|~(p-2)?u)+u~(q(x)) in?×(0,T),where ? is either a bounded domain or the whole space R~N,and q(x) is a positive and continuous function defined in ? with 0q_-=inf q(x)=q(x)=sup q(x)=q_+∞.It is demonstrated that the equation with variable source power has much richer dynamics with interesting phenomena which depends on the interplay of q(x) and the structure of spatial domain ?,compared with the case of constant source power.For the case that ? is a bounded domain,the exponent p-1 plays a crucial role.If q_+p-1,there exist blow-up solutions,while if q_+p-1,all the solutions are global.If q_-p-1,there exist global solutions,while for given q_-p-1q_+,there exist some function q(x) and ? such that all nontrivial solutions will blow up,which is called the Fujita phenomenon.For the case ?=R~N,the Fujita phenomenon occurs if 1q_-=q_+=p-1+p/N,while if q_-p-1+p/N,there exist global solutions.     

M-P逆和加权M-P逆的有限迭代计算公式

本文，对于任意给定的矩阵A，我们给出了计算其M—P逆和加权M—P逆的有限迭代计算公式．根据这一迭代公式，当我们选取初始矩阵为X0=A^#，则矩阵A的加权M—P逆A^＋MN在不考虑舍入误差的情况下，可以在有限迭代的情况得到，同样当我们选取初始矩阵X0=A^＊，其M—P逆A^＋亦可以在有限迭代下获得．最后我们用数值例子检验了我们算法的正确性。     

A Recursive Algorithm for Computing the Weighted Moore-Penrose Inverse $A^+_{MN}$

In this paper, we give a recursive algorithm for computing the weighted Moore-Penrose inverse $A^+_{MN}$. This method is a generalization of Greville's method for computing Moore-Penrose inverse $A^+$, and the technique of its proof is new. This method suits the weighted least-squares problem.     

环上矩阵广义Moore-Penrose逆的存在性

讨论带有对合反自同构*有单位元的结合环R上矩阵的广义Moore-Penrose 逆,给出了环R上矩阵的广义Moore-Penrose逆存在的几个充要条件．特别,得到了环 R上矩阵A的关于M和N的广义Moore-Penrose逆存在的充要条件是A有分解A= GDH,其中D2=D,(MD)*=MD,(GD)*MGD+M(I-D)和DHN-1(DH)*+ (I-D)M-1均可逆．     

AN APPROXIMATION THEOREM OF A M-P INVERSE BY OUTER INVERSES

Let H1 and H2 be separable Hilbert spaces, and B(H1,H2) all of boundedlinear operators from H1 into H2. In this note, we prove the following theorem: for any positive integer N and T ε B(H1, H2) with a closed range, there exists an outerinverse TN^# with finite rank N such that T y = lim TN^#y for any y ε H2, where T N→∞ denotes the Moore-Penrose inverse of T. Thus computing T is reduced to computingouter inverses TN^# with finite rank N. Moreover, because of the stability of boundedouter inverse of a T ε B(H1,H2), this is very useful.     

Nonnegative matrices having same nonnegative moore-penrose and group inverses

Nonnegative matrices A whose Moore-Penrose generalized inverse A⁺ is nonnegative and has any one of the three equivalent properties (i) AA⁺ = A⁺A (ii) A⁺ = A, the group inverse, (iii) A⁺ = p(A), some polynomial in A with scalar coefficients, are characterized. This characterization generalizes known results on nonnegative matrices Awhose Moore-Penrose generalized inverse is equal to some power of A.     

整环上Moore-Penrose逆的刻画

若A为整环上的n阶可逆矩阵,则X=A-1是满足方程rank■=rank(A)的唯一矩阵.把它推广到满足Rao条件的整环上得到关于矩阵A的Moore-Penrose逆A+的刻画.     

关于分块矩阵〔ABCD〕的g-逆的块独立性

本文证明了分块阵M=〔ABCD〕的g-逆有块独立性的充要条件是M适合秩可加性条件,M~+有一特定的表达式的充要条件也是M适合秩可加性条件.本文还给出了包含M的g-逆的子块的不变矩阵以及这些子块的定义方程.     

分块矩阵的Moore-Penrose逆

该文研究了两类3×3分块矩阵M1=AOOBCODEF,M2=ABCDEFGHK的Moore-Penrose逆的表达式,并给出了表达式成立时的条件.     

Fuzzy矩阵的加权Moore-Penrose逆

给出了Fuzzy矩阵加权Moore-Penrose逆AM+N的定义,研究了Fuzzy矩阵加权Moore-Penrose逆AM+N的存在性问题,证明了当权矩阵M,N满足一定条件时,AM+N存在且A+MN=AT的充要条件是ANATMA≤A,推广了Fuzzy矩阵和Boolean矩阵的相应结果.     

Convergence of Rump’s method for computing the Moore-Penrose inverse

We extend Rump’s verified method (S.Oishi, K.Tanabe, T.Ogita, S.M.Rump (2007)) for computing the inverse of extremely ill-conditioned square matrices to computing the Moore-Penrose inverse of extremely ill-conditioned rectangular matrices with full column (row) rank. We establish the convergence of our numerical verified method for computing the Moore-Penrose inverse. We also discuss the rank-deficient case and test some ill-conditioned examples. We provide our Matlab codes for computing the Moore-Penrose inverse.