非线性非局部边界条件的半线性耦合抛物型方程组

该文研究具有非负初始数据和非局部边界条件u|αΩ×(0,∞)=∫_Ωψ_i(x,y,t)u_i~(l_i)(y,t)dy的半线性抛物型方程组u_(it)=△u_i+c_i(x,t)u_(i+1)~(pi),(x,t)∈Ω×(0,∞).给出了方程组解的整体存在与爆破准则.这些结果表明,权重函数c_i(x,t),ψ_i(x,y,t)和指数p_i,l_i的大小在确定方程组的解是否爆破中起着关键的作用.     

具非线性非局部边界条件的一类多孔介质方程解的整体存在与爆破

本文研究了具有非线性非局部边界条件的一类退化型多孔介质方程.利用比较原理和上下解的方法,获得了方程的解是否在有限时刻爆破或整体存在的准则,这些结果表明,权重函数g(x,y)及指数l的大小对于问题解的爆破与否起着关键的作用.最后研究了爆破解的爆破率.     

我为高考设计题目

题1已知函数y=kx与.y=x~2+2 (x≥0)的图象相交于不同两点A(x_1,y_1), B(x_2,y_2),l_1,l_2分别是y=x~2+2(x≥0)的图象在A,B两点的切线,M,N分别是l_1,l_2与x轴的交点,P为l_1与l_2的交点. (1)求证:直线l_1、y=kx、l_2的斜率成等差数列;     

问题解决中的轨迹意识

<正>我们先来看一个问题:设直线l_1:y=k_1x+1,l_2:y=k_2x-1,其中实数k_1、k_2满足k_1k_2+2=0.(Ⅰ)证明l1与l2相交;(Ⅱ)证明l_1与l_2的交点在椭圆2x~2+y~2=1上.解(Ⅰ)略.解法一由{y=k_1x+1,y=k_2x-1,(Ⅱ)得交点坐标     

求解函数关系式中字母系数的取值范围两例

<正>求函数关系式中自变量的字母系数的取值范围问题,涉及知识点多,求解方法灵活多变.现举例说明如下,供参考.例1如图1,已知直线l_1:y=-2x+4与直线l_2:y=kx+b(k≠0)在第一象限交于点M.若直线l_2与x轴的交点为A(-2,0),求k的取值范围.分析可将点A的坐标代入到直线l_2关系式中,得出用k表示的b,这样,再解由两条直线组成的方程组,求出用k表示的方程组的解,即为点M的坐标,     

带非局部源的退化奇异半线性抛物方程的爆破

本文研究带齐次Dirichlet边界条件的非局部退化奇异半线性抛物方程ut-(xαux)x=∫0af(u)dx在(0,a)×(0,T)内正解的爆破性质,建立了古典解的局部存在性与唯一性.在适当的假设条件下,得到了正解的整体存在性与有限时刻爆破的结论.本文还证明了爆破点集是整个区域,这与局部源情形不同.进而,对于特殊情形:f(u)=up,p>1及,f(u)=eu,精确地确定了爆破的速率.     

关于隐函数样条的凸性分析

本文讨论由隐函数样条F(x)=αg~h(x)-(1-α)f(x)=0,x∈R~(?),0<α<1定义的函数(Functional spline)的凸性,得到:1)当 g(x)=l_0(x),f(x)=multiply from j to k l_j(x),其中,l_j(x)=sum from i=1 to n a_(ij)x_i+b_j 是线性的,且 (?)(x)≥0围成区域Ω,那么在Ω内,当 h>k 时,F(x)=αg~h(x)-(1-α)f(x)=0是凸的;2)在 R~2内,若 f(x,y)=0,g(x,y)=0定义两条凸曲线,那么隐函数样条不一定是凸的.但可以构造 f_1,g_1,使得 f_1与 f 定义同一条曲线,g_1与 g 也定义同一条曲线,而这时的隐函数样条是凸的.本文还给出了一个凸样条的充分条件.     

漏解分析与补遗

<正>一、题目呈现与其流行的解法题目已知直线l_1:x+3y-7=0,l_2:y=kx+b与x轴、y轴的正半轴围成的四边形有外接圆,求k的值及b的取值范围.这是流行于许多数学教辅资料的一道题目,其解法(以下称为流行解法)如下:解由于已知直线l_1:x+3y-7=0,l_2:y=kx+b的斜率分别为k_1=-1/3,k_2=k,又直线l_1、l_2与x轴、y轴正半轴围成的四边形有外接圆,如图1所示.     

具非线性非局部边界条件的一类多孔介质方程解的整体存在与爆破（英文）

本文研究了具有非线性非局部边界条件的一类退化型多孔介质方程.利用比较原理和上下解的方法,获得了方程的解是否在有限时刻爆破或整体存在的准则,这些结果表明,权重函数g(x,y)及指数l的大小对于问题解的爆破与否起着关键的作用.最后研究了爆破解的爆破率.     

半线性抛物型方程的双移动边界问题

本文运用Fourier方法和压缩映像不动点原理,证明了半线性抛物型方程的双移动边界问题 u_t=a~2u_(xx) F(x,t,u,u_x),(x,t)∈D_∞, u(l_1(t),t)=0,l_1(0)=0,t∈(0, ∞), u(l_2(t),t)=0,l_2(0)=l_0,t∈(0, ∞), u(x,0)=φ(x),0≤x≤l_0,φ(0)=φ(l_0)=0.解的存在唯一性,其中D_∞={(x,t)|l_1(t)相似文献   

A degenerate parabolic equation with a nonlocal source and an absorption term

This article deals with a class of nonlocal and degenerate quasilinear parabolic equation u _t = f(u)(Δu + a∫_Ω u(x, t)dx ? u) with homogeneous Dirichlet boundary conditions. The local existence of positive classical solutions is proved by using the method of regularization. The global existence of positive solutions and blow-up criteria are also obtained. Furthermore, it is shown that, under certain conditions, the solutions have global blow-up property. When f(s) = s ^p , 0 < p ≤ 1, the blow-up rate estimates are also obtained.     

Zur Existenz von Lösungen gewisser Randwertaufgaben

Summary With the aid of some known results about integral equations of the Hammerstein type there is proofed an existence theorem for the following class of boundary value problems–y–l ₂ y=f(x,y),y(a)=y(b)=0,l ₂>0 mit|f(x, y)|<=l ₁ |y|+l ₃ (x),l ₁ >=0,l ₃ (x)>0. The existence range is determined by the greatest eigenvalue of some linear problem.     

非局部退化拟线性抛物型方程组解的爆破和整体存在性

该文采用弱上下解方法和正则化技巧,研究了一类非局部退化抛物型方程组解的爆破和整体存在性,给出了爆破指标,并对非退化情形m=n=1,p_1=q_1=0,p_2q_21给出了一致爆破速率.     

带非局部源的退化半线性抛物方程的解的爆破性质

This paper deals with the blow-up properties of the positive solutions to the nonlocal degenerate semilinear parabolic equation u _t − (x ^a u _x ) _x =∫ ₀ ^a f(u)dx in (0,a) × (0,T) under homogeneous Dirichlet conditions. The local existence and uniqueness of classical solution are established. Under appropriate hypotheses, the global existence and blow-up in finite time of positve solutions are obtained. It is also proved that the blow-up set is almost the whole domain. This differs from the local case. Furthermore, the blow-up rate is precisely determined for the special case: f(u)=u ^p , p>1.     

Uniform blow-up rate for parabolic equations with a weighted nonlocal nonlinear source

In this paper, the blow-up rate of solutions of semi-linear reaction-diffusion equations with a more complicated source term, which is a product of nonlocal (or localized) source and weight function a(x), is investigated. It is proved that the solutions have global blow-up, and that the rates of blow-up are uniform in all compact subsets of the domain. Furthermore, the blow-up rate of _∞|u(t)| is precisely determined.     

Global boundedness and blow-up for a parabolic system with positive Dirichlet boundary value

This paper deals with a parabolic system coupled via nonlocal sources, subjecting to positive Dirichlet boundary value conditions. By using the super-, sub-solution methods and techniques, and piecewise functions, the blow-up criteria and global boundedness of nonnegative solutions are determined. The results show the positive boundary value ε ₀ plays an important role in the case of blow-up.     

A note on the behavior of blow-up solutions for one-phase Stefan problems

ＡＮＯＴＥＯＮＴＨＥＢＥＨＡＶＩＯＲＯＦＢＬＯＷ┐ＵＰＳＯＬＵＴＩＯＮＳＦＯＲＯＮＥ┐ＰＨＡＳＥＳＴＥＦＡＮＰＲＯＢＬＥＭＳＺＨＵＮＩＮＧＡｂｓｔｒａｃｔ．Ｉｎｔｈｉｓｐａｐｅｒ,ｔｈｅｆｏｌｏｗｉｎｇｏｎｅ－ｐｈａｓｅＳｔｅｆａｎｐｒｏｂｌｅｍｉｓ...     

Relationship between solutions of families of two-point boundary value problems and Cauchy problems

We carry out a qualitative analysis and suggest a method for the solution of the two-point boundary value problems A _l υ = g, x ? [0, l], l ? (0, L) = B ? R ₊; $$ \alpha v'_x \left| {_{x = + 0} = F_1 (v,l)} \right|_{x = 0} , \beta v'_x \left| {_{x = l - 0} = F_2 (v,l)} \right|_{x = l} , $$ , where α, β ? R, υ is the unknown function, g = g(x) is a given real function, F ₁(y, l) and F ₂(y, l) are known real functions defined on the sets Y ₁ × B and Y ₂ × B, respectively, where Y ₁ ∪ Y ₂ ? R, and A _l is the restriction of A corresponding to the embedding parameter l. (Here A is an operator taking an arbitrary function in the set of function classes defined in the paper to C(B ₁), where B ₁ = [0, L).) The study takes into account the dependence of solutions of various versions of these two-point boundary value problems on the parameter l. We construct algorithms for the reduction of these families of two-point boundary value problems to Cauchy problems for ordinary differential equations and integro-differential equations that contain only first derivatives of the unknown functions with respect to the parameter l.