一类奇异半线性反应扩散方程初值问题解的存在性

本文获得了如下的奇异半线性反应扩散方程初值问题{(e)u/(e)t-(1/tσ)△u=up+f(x),t＞0,x∈Rnlim t→0+ u (t,x)=0, x∈Rn广义解(mild solution)在L∞ loe[(0,∞);L∞(Rn)]中的存在性.其中σ＞0,0＜p＜1,f(x)非负且f(x)∈L∞(Rn).     

奇异半线性反应扩散方程组Cauchy问题

本文讨论如下问题其中{(б)u/(б)t-(1/tσ)△u=αvp1+β1vp1+f1(x),t>0,x∈RN,(б)u/(б)t-(1/tσ)△v=α2uq2+β2vp2+f2(x),t>0,∈RN,limt→0+u(t,x)=limt→0+v(t,x)=0,x∈Rn,其中σ＞0,pi＞1,qi＞1(i=1,2),α1≥0,α2＞0,β1＞0,β2≥0,fi(x)(i=1,2)连续有界非负,(f1(x),f2(x))(≡／)(0,0).给出了非负局部解存在的几个充分条件和解的爆破结果.     

Optimal Transportation for Generalized Lagrangian

This paper deals with the optimal transportation for generalized Lagrangian L = L(x, u, t), and considers the following cost function: c(x, y) = inf x(0)=x x(1)=y u∈U∫_0~1 L(x(s), u(x(s), s), s)ds, where U is a control set, and x satisfies the ordinary equation x(s) = f(x(s), u(x(s), s)).It is proved that under the condition that the initial measure μ0 is absolutely continuous w.r.t. the Lebesgue measure, the Monge problem has a solution, and the optimal transport map just walks along the characteristic curves of the corresponding Hamilton-Jacobi equation:V_t(t, x) + sup u∈UV_x(t, x), f(x, u(x(t), t), t)-L(x(t), u(x(t), t), t) = 0,V(0, x) = Φ0(x).     

应用题新编选登

题1　某企业有一条价值a万元的生产流水线,要提高该生产流水线的生产能力,提高产品的增加值,就要对流水线进行技术改造,假设增加值y万元与技改投入x万元之间的关系满足1y与(a- x)x2 成正比例.2当x =a2 时,y=a32 .30≤x2 (a- x)≤t,其中t为常数且t∈[0 ,2 ].1)设y=f(x) ,求出f(x)的表达式,并求其定义域;2 )求出增加值y的最大值,并求出此时的技改投入x值.解　1)设y=f (x) =k(a- x ) x2 ,因当x =a2时,y=a32 .故a32 =k(a- a2 ) (a2 ) 2 ,∴k=4 ,从而有y=4 (a- x) x2 .因0≤x2 (a- x) ≤t,解得0≤x≤2 t1+ 2 ta,于是f(x) =4 (a- x) x2 (0≤x≤2 t…     

Multiple Positive Solutions for a Nonlinear Elliptic Equation Involving Hardy–Sobolev–Maz'ya Term

In this paper, we study the existence and nonexistence of multiple positive solutions for the following problem involving Hardy–Sobolev–Maz'ya term:-Δu- λu/|y|2=|u|pt-1u/|y|t+ μf(x), x ∈Ω,where Ω is a bounded domain in RN(N ≥ 2), 0 ∈Ω, x =(y, z) ∈ Rk× RN-kand pt =N +2-2t N-2(0 ≤ t ≤2). For f(x) ∈ C1(Ω)\{0}, we show that there exists a constant μ* 0 such that the problem possessesat least two positive solutions if μ∈(0, μ*) and at least one positive solution if μ = μ*. Furthermore,there are no positive solutions if μ∈(μ*, +∞).     

一类非线性Schr(o)dinger方程的高精度守恒数值格式

1 引言 本文讨论下面非线性Schr(o)dinger方程(NLS)方程的初边值问题: i(e)u/(e)t (e)2u/(e)x2 2|u2|u=0, (1) u(xl,t)=u(xr,t)=0, t＞0, (2) u(x,0)=u0(x), xl≤x≤xr, (3) 其中u(x,t)是复值函数,u0(x)为已知的复值函数,i2=-1.该问题有着如下的电荷与能量守恒关系: Q=∫xrxl|u(x,t)|2dx=‖u‖2=Q0, (4) E=∫xrxl(|(e)u/(e)x|2-|u|4)dx=E0, (5) 其中Q0,E0为常数,并且称公式(4),(5)分别为电荷和能量守恒.由(4),(5)式可以证明[3] ‖u‖L∞≤C, (6) 其中C为一般正常数.     

Boundedness of Marcinkiewicz integral related to multilinear commutators with variable kernels on Hardy spaces

Let→b=(b1,b2,…,bm),bi∈∧βi(Rn),1≤I≤m,βi＞0,m∑I=1βi=β,0＜β＜1,μΩ→b(f)(x)=(∫∞0|F→b,t(f)(x)|2dt/t3)1/2,F→b,t(f)(x)=∫|x-y|≤t Ω(x,x-y)/|x-y|n-1 mΠi=1[bi(x)-bi(y)dy.We consider the boundedness of μΩ,→b on Hardy type space Hp→b(Rn).     

Biharmonic equations with asymptotically linear nonlinearities

This article considers the equation △2u = f(x, u)with boundary conditions either u|(a)Ω = (a)u/(a)n|(a)Ω = 0 or u|(a)Ω = △u|(a)Ω = 0, where f(x,t) is asymptotically linear with respect to t at infinity, and Ω is a smooth bounded domain in RN, N ＞ 4. By a variant version of Mountain Pass Theorem, it is proved that the above problems have a nontrivial solution under suitable assumptions of f(x, t).     

关于双曲型偏微分方程u_xy=f(x,y,u,u_x,u_y)的唯一性问题

关于双曲型偏微分方程 u_(xy)=f(x,y,u,u_x,u_y),0≤x≤a,0≤y≤b,-∞相似文献   

一个局部波动方程组的解的全局存在性与爆破

我们研究初始值问题(e)u1/(e)t2=(e)2u1/(e)x2+‖u2(·,t)‖p, (e)2u2/(e)t2=(e)2u2/(e)x2+‖u1(·,t)‖q,-∞<x<∞,t>0,u1(x,0)=f1(x), (e)u1/(e)t(x,0)=g1(x),u2(x,0)=f2(x), (e)u2/(e)t(x,0)=g2(x),- ∞<x<∞,where‖ui(·,t)‖=∫∞-∞(4)i(x)|ui(x,t)|dx with (4)i(x)≥0 and ∫∞-∞(4)i(x)dx=1,i=1,2.然后建立解的全局存在和爆破的标准,提出爆破增长率.     

求解Riesz空间分数阶扩散方程的一种新的数值方法

本文利用Diethelm方法构造了一种逼近Riesz空间分数阶导数的O（△x^3-α）格式,其中1 < α < 2,△x是空间步长.进一步对一阶时间导数采用Crank-Nicolson方法离散,得到了求解Riesz空间分数阶扩散方程的一种新的有限差分格式,并用矩阵方法证明了稳定性和收敛性,其误差估计为O（△t²+△x^3-α）,其中△t为时间步长.最后,数值算例验证了差分格式的正确性和有效性.     

与双调和方程相关的连续模和Heinz-Schwarz型不等式

对于给定的两个正整数n ≥ 2和m ≥ 1，假设函数f满足如下条件：（1）在Bⁿ内满足非齐次双调和方程△（△f）=g（g ∈ C（Bⁿ，R^m））；（2）在S^n-1上满足f=ψ₁（ψ₁ ∈ C（S^n-1，R^m）），以及∂f/∂n=ψ₂（ψ₂ ∈ C（S^n-1，R^m）），其中∂/∂n表示内法线方向导数，Bⁿ表示Rⁿ中的单位球以及S^n-1表示Bⁿ的边界.本文主要研究f的连续模和Heinz-Schwarz型不等式.     

二维线性双曲型方程Neumann边值问题的紧交替方向隐格式

对二维Neumann边界条件的线性双曲型方程建立了紧交替方向的隐格式.利用方程和边界条件得到在空间上的三阶与五阶导数的边界值,进而在内点、边界内点和边界角点分别建立9点、6点和4点紧差分格式;通过引进新的范数和L₂范数估计L_∞范数;借助能量估计、Gronwall不等式和Schwarz不等式等技巧,详细分析了差分格式在无穷范数下关于时间和空间分别为二阶和四阶收敛性,并给出了稳定性结果;通过数值算例,验证了理论分析结果.     

带有临界增长的分数阶Kirchhoff方程的半经典解

本文研究如下带有临界增长的分数阶Kirchhoff方程ε^2s(ε^2s-3∫∫R³×R³|u(x)-u(y)/|²|x-y|^3+2s),x∈R³,其中M是一个连续正的Kirchhoff函数,λ>0是一个参数,3/40充分小和λ足够大时,我们首先证明了上述问题正基态解的存在性.其次,证明了基态解集中在一个由位势函数所刻画的特定集合中.最后,研究了基态解的衰减估计.     

时间分数阶扩散方程的一个新的高阶有限差分/谱逼近

研究时间分数阶扩散方程,结合时间方向的有限差分格式和空间方向的Legendre Collocation谱方法,构造了一个高阶稳定数值格式.数值算例表明该格式是无条件稳定和长时间稳定的,其收敛阶为O(Δt^3-α+N^-m),其中Δt,N和m分别是时间步长,空间多项式阶数以及精确解的正则度.     

临界或超临界增长分数阶Schrodinger-Poisson方程正解的存在性

本文研究如下分数阶Schrodinger-Poisson方程{(-△)^su+Vx(u)+K(x)φu=f(u)+λ|u|^q-2ux∈R³,(-△)^tφ=K(x)u²,x∈R³其中S∈(3/4,1),t∈(0,1),f是在原点超线性无穷远次临界的连续非线性项,指数q≥2_s^*=6/3-2x.当λ>0充分小时,我们利用变分方法证明上述问题正解的存在性.本文的主要贡献是处理了超临界情形.     

最大度不小于3的图的星全色数的一个上界

对圈、扇和轮作了简单的剖分,得到了其剖分图的星全色数,并运用Lovasz局部引理证明了若G(V,E)是一个最大度为△≥3的简单无向图,则Χ_(st)(G)≤22Δ~2.     

An application of a boundary element method to natural convection

The direct boundary element method is applied to the numerical modelling of thermal fluid flow in a transient state. The Navier-Stokes equations are considered under the Boussinesq approximation and the viscous thermal flow equations are expressed in terms of stream function, vorticity, and temperature in two dimensions. Boundary integral equations are derived using logarithmic potential and time-dependent heat potential as fundamental solutions. Boundary unknowns are discretized by linear boundary elements and flow domains are divided into a series of triangular cells. Charged points are translated upstream in the numerical evaluation of convective terms. Unknown stream function, vorticity, and temperature are staggered in the computational scheme.

Simple iteration is found to converge to the quasi steady-state flow. Boundary solutions for two-dimensional examples at a Reynolds number 100 and Grashoff number 10⁷ are obtained.     

带非奇扰动项的(2,p)-Laplace方程无穷多解的存在性

本文研究带非奇扰动项的(2,p)-Laplace方程{u=0,-△u-△pu=a(x)|u|^q-2u+f(x,u)x∈ЭΩ,x∈Ω,其中ΩСR^N是有界光滑区域,1相似文献   

An improved Chebyshev collocation algorithm for direct simulation of 2D turbulent convection in differentially heated cavities

We present a modified Chebyshev collocation algorithm for direct numerical simulations of 2D turbulent convection in differentially heated cavities. The numerical algorithm integrates the Navier-Stokes equations in velocity-pressure formulation with a Chebyshev spatial approximation and a second order finite difference time-stepping scheme. A coordinate stretching is introduced which allows one to redistribute the collocation points where needed in order to resolve more economically the small scales that appear at cavity mid-height. This algorithm is used to perform simulations in a square differentially heated cavity with adiabatic top and bottom walls filled with a fluid of Prandtl number equal to 0.71 for Rayleigh number values up to 10¹⁰ which is almost two orders of magnitude higher than the onset of unsteadiness. The time-dependent dynamics of the solutions are investigated and the time-averaged flow structures are displayed. The influence of unsteadiness on the local and global heat transfer coefficients is examined.