光滑度量测度空间上的加权扩散方程的梯度估计

本文主要考虑了一类加权非线性扩散方程正解的梯度估计.在m-维Bakry-(E)mery Ricci曲率下有界的假设下,得到加权多孔介质方程(γ＞1)正解的Li-Yau型梯度估计,此外对于加权快速扩散方程(0＜γ＜1),证明了Hamilton型椭圆梯度估计,结论分别推广了Lu,Ni,Vázquez and Villani在文[1]和Zhu在文[2]中的结果.     

计算一类反应扩散方程分歧问题的Fourier配点法

1引言 关于反应扩散方程的研究由来已久,特别是对一些含参数的非线性反应扩散方程,由于其多解性和丰富的分歧现象,经常受到人们的关注.本文考虑如下非线性反应扩散方程组 {ut=γf(u,v)+uxx, vt=γg(u,v)+dvxx, (1) 相应的边界条件为 ux(t,0):ux(t,π)=vx(t,0)=vx(t,π)=0. (2) 我们选取Gierer-Meinhardt模型[1,2]为研究对象,即 {f(u,V)=a-bu+u2/v, g(u,v)=u2-v, 其中a、b和γ是正常数,d为参数.     

一个反应扩散方程整体解的存在性

对一个由反应扩散方程推出的抛物型自由边界问题进行了研究,该问题含有参数γ和τ,已证明了当γ充分大时,该问题存在唯一稳定解.现在主要证明γ,τ均充分大时,该问题存在整体解,即当0＜t＜+∞时,该问题的解存在.     

一类反应扩散方程解的熄灭现象

利用能量估计方法讨论了下述反应扩散方程的初边值问题解的渐近性态,分别给出解熄灭的充分条件和必要条件。这里λ>0,γ>0,β>0为常数,Ω?RN为有界域。文末给出说明文中方法处理高阶方程的例子。     

非线性奇异边值问题的高效数值算法

正1引言本文考虑下列非线性奇异边值问题■α≥1,a≥0,b0.此类非线性奇异边值问题出现在许多物理学领域,例如,电流体动力学、核物理、原子结构和原子计算.当α=2并且f(x,y)=ny/y+k,n0,k0时,方程(1)表示一个带有Michaelis-Menten吸收动力学的稳态氧扩散模型[1].对于α=1,f(c,y)=y~γ,这里γ是一个物理常数,上述公式用于研究热爆炸[2],α=2,f(x,y)=γe~y,这里γ是一个表示     

一个反应扩散方程整体解的存在性

对一个由反应扩散方程推出的抛物型自由边界问题进行了研究,该问题含有参数γ和τ,已证明了当γ充分大时,该问题存在唯一稳定解．现在主要证明γ,τ均充分大时,该问题存在整体解,即当0＜t＜ ∞时,该问题的解存在．     

求解扩散—对流方程的CAYЛbEB型CE方法

1　引　　言扩散—对流方程是描述粘性流体运动的非线性方程—Burgers方程的线性化模型,并且它本身也描述了许多自然现象,例如在水中和大气中污染物质浓度的扩散,沿海盐度、温度扩散等.因此求解扩散—对流方程的计算方法引起了充分的重视.考虑扩散—对流方程的初边值问题如下:ut=aux+ε2ux2　(00)(1.3)其中a为常数,ε>0为小参数.对网格区域R{0≤x≤1,t>0}进行均匀剖分,其网格点xj=jh,j=0,1,…,J,h=1J;tn=nτ,n=0,1,….h和τ分别为空间步长和时间步长.关于问题(1.1)—…     

具有正负系数的连续变量的差分方程解的零点分布

考虑具有正负系数的连续变量的差分方程x(t)-x(t-γ)+P(t)x(t-т)-Q(t)x(t-σ)=0,其中P,Q∈C([t0,∞),R+),т,σ,γ∈(0,∞).本文给出了上述方程解的零点分布及方程所有解振动的充分条件并改进和推广了已有的结果.     

二阶非线性时滞差分方程解的振动性

研究了二阶非线性时滞差分方程△(α_n(△(x_n+p_nx_(n-r)))~γ)+bn(△(x_n+p_nx_(n-T)))~γ+f(n,x_(n-σ))=0给出方程振动的充分条件,推广和改进了中立时滞差分方程的许多结果.     

第三类Painlevé方程(δ=0或γ=0)解的渐近性态

对带有一般实参数第三类Painlevé方程,已有γ<0,δ>0时,解的有界性以及振荡渐近解的表达形式的结论.在本文中,我们给出当δ=0或γ=0时其振荡渐近解的表达形式.     

Determining a Rotation of a Tetrahedron from a Projection

The following problem, arising from medical imaging, is addressed: Suppose that T is a known tetrahedron in ?³ with centroid at the origin. Also known is the orthogonal projection U of the vertices of the image ?T of T under an unknown rotation ? about the origin. Under what circumstances can ? be determined from T and U?     

Symmetry-based solution of a model for a combination of a risky investment and a riskless investment

Benth and Karlsen [F.E. Benth, K.H. Karlsen, A note on Merton's portfolio selection problem for the Schwartz mean-reversion model, Stoch. Anal. Appl. 23 (2005) 687-704] treated a problem of the optimisation of the selection of a portfolio based upon the Schwartz mean-reversion model. The resulting Hamilton-Jacobi-Bellman equation in 1+2 dimensions is quite nonlinear. The solution obtained by Benth and Karlsen was very ingenious. We provide a solution of the problem based on the application of the Lie theory of continuous groups to the partial differential equation and its associated boundary and terminal conditions.     

On a mapping of a projective space into a sphere

We obtain an exact estimate for the minimum multiplicity of a continuous finite-to-one mapping of a projective space into a sphere for all dimensions. For finite-to-one mappings of a projective space into a Euclidean space, we obtain an exact estimate for this multiplicity for n = 2, 3. For n ≥ 4, we prove that this estimate does not exceed 4. Several open questions are formulated.     

Calculating a function of a matrix with a real spectrum

Let T be a square matrix with a real spectrum, and let f be an analytic function. The problem of the approximate calculation of f(T) is discussed. Applying the Schur triangular decomposition and the reordering, one can assume that T is triangular and its diagonal entries t_ii are arranged in increasing order. To avoid calculations using the differences t_ii ? t_jj with close (including equal) t_ii and t_jj, it is proposed to represent T in a block form and calculate the two main block diagonals using interpolating polynomials. The rest of the f(T) entries can be calculated using the Parlett recurrence algorithm. It is also proposed to perform some scalar operations (such as the building of interpolating polynomials) with an enlarged number of significant decimal digits.

     

Folding a surface to a polygon

A concept of folding for compact connected surfaces, involving the partition of the surface into combinatorially identical n-sided topological polygons, is defined. The existence of such foldings for given n and given surfaces is explored, with definitive results for the sphere and the torus. We obtain necessary conditions for the existence of such foldings in all other cases.Supported by Kuwait University Grant SM 043.     

Approaching a vertex in a shrinking domain under a nonlinear flow

We consider here the homogeneous Dirichlet problem for the equation , in a noncylindrical domain in space-time given by . By means of matched asymptotic expansion techniques we describe the asymptotics of the maximal solution approaching the vertex x=0, t=T, in the three different cases p>1/2, p=1/2(vertex regular), p<1/2 (vertex irregular).