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1.
The paper discusses the existence of positive solutions, dead core solutions and pseudodead core solutions of the singular
Dirichlet problem (ϕ(u′))′ = λf(t, u, u′), u(0) = u(T) = A. Here λ is the positive parameter, A > 0, f is singular at the value 0 of its first phase variable and may be singular at the value A of its first and at the value 0 of its second phase variable.
This work was supported by grant no. A100190703 of the Grant Agency of the Academy of Sciences of the Czech Republic and by
the Council of Czech Government MSM 6198959214. 相似文献
2.
We study the existence and the properties of reduced measures for the parabolic equations ∂
t
u − Δu + g(u) = 0 in Ω × (0, ∞) subject to the conditions (P): u = 0 on ∂Ω × (0, ∞), u(x, 0) = μ and (P′): u = μ′ on ∂Ω × (0, ∞), u(x, 0) = 0, where μ and μ′ are positive Radon measures and g is a continuous nondecreasing function. 相似文献
3.
Shu-Yu Hsu 《Mathematische Annalen》2003,325(4):665-693
We prove that the solution u of the equation u
t
=Δlog u, u>0, in (Ω\{x
0})×(0,T), Ω⊂ℝ2, has removable singularities at {x
0}×(0,T) if and only if for any 0<α<1, 0<a<b<T, there exist constants ρ0, C
1, C
2>0, such that C
1
|x−x
0|α≤u(x,t)≤C
2|x−x
0|−α holds for all 0<|x−x
0|≤ρ0 and a≤t≤b. As a consequence we obtain a sufficient condition for removable singularities at {∞}×(0,T) for solutions of the above equation in ℝ2×(0,T) and we prove the existence of infinitely many finite mass solutions for the equation in ℝ2×(0,T) when 0≤u
0∉L
1
(ℝ2) is radially symmetric and u
0L
loc
1(ℝ2).
Received: 16 December 2001 / Revised version: 20 May 2002 / Published online: 10 February 2003
Mathematics Subject Classification (1991): 35B40, 35B25, 35K55, 35K65 相似文献
4.
V. M. Prokip 《Ukrainian Mathematical Journal》2012,63(8):1314-1320
Polynomial n × n matrices A(x) and B(x) over a field
\mathbbF \mathbb{F} are called semiscalar equivalent if there exist a nonsingular n × n matrix P over
\mathbbF \mathbb{F} and an invertible n × n matrix Q(x) over
\mathbbF \mathbb{F} [x] such that A(x) = PB(x)Q(x). We give a canonical form with respect to semiscalar equivalence for a matrix pencil A(x) = A
0x
- A
1, where A
0 and A
1 are n × n matrices over
\mathbbF \mathbb{F} , and A
0 is nonsingular. 相似文献
5.
Kaouther Ammar 《Central European Journal of Mathematics》2010,8(3):548-568
The paper addresses the existence and uniqueness of entropy solutions for the degenerate triply nonlinear problem: b(v)
t
− div α(v, ▽g(v)) = f on Q:= (0, T) × Ω with the initial condition b(v(0, ·)) = b(v
0) on Ω and the nonhomogeneous boundary condition “v = u” on some part of the boundary (0, T) × ∂Ω”. The function g is continuous locally Lipschitz continuous and has a flat region [A
1, A
2,] with A
1 ≤ 0 ≤ A
2 so that the problem is of parabolic-hyperbolic type. 相似文献
6.
《Mathematical and Computer Modelling》1999,29(2):1-18
This paper is concerned with the construction of accurate continuous numerical solutions for partial self-adjoint differential systems of the type (P(t) ut)t = Q(t)uxx, u(0, t) = u(d, t) = 0, u(x, 0) = f(x), ut(x, 0) = g(x), 0 ≤ x ≤ d, t >- 0, where P(t), Q(t) are positive definite oRr×r-valued functions such that P′(t) and Q′(t) are simultaneously semidefinite (positive or negative) for all t ≥ 0. First, an exact theoretical series solution of the problem is obtained using a separation of variables technique. After appropriate truncation strategy and the numerical solution of certain matrix differential initial value problems the following question is addressed. Given T > 0 and an admissible error ϵ > 0 how to construct a continuous numerical solution whose error with respect to the exact series solution is smaller than ϵ, uniformly in D(T) = {(x, t); 0 ≤ x ≤ d, 0 ≤ t ≤ T}. Uniqueness of solutions is also studied. 相似文献
7.
A. I. Perov 《Functional Analysis and Its Applications》2010,44(1):69-72
Let M be a complete K-metric space with n-dimensional metric ρ(x, y): M × M → R
n
, where K is the cone of nonnegative vectors in R
n
. A mapping F: M → M is called a Q-contraction if ρ (Fx,Fy) ⩽ Qρ (x,y), where Q: K → K is a semi-additive absolutely stable mapping. A Q-contraction always has a unique fixed point x* in M, and ρ(x*,a) ⩽ (I - Q)-1 ρ(Fa, a) for every point a in M. The point x* can be obtained by the successive approximation method x
k
= Fx
k-1, k = 1, 2,..., starting from an arbitrary point x
0 in M, and the following error estimates hold: ρ (x*, x
k
) ⩽ Q
k
(I - Q)-1ρ(x
1, x
0) ⩽ (I - Q)-1
Q
k
ρ(x
1, x
0), k = 1, 2,.... Generally the mappings (I - Q)-1 and Q
k
do not commute. For n = 1, the result is close to M. A. Krasnosel’skii’s generalized contraction principle. 相似文献
8.
Let X be a complex Banach space with norm ‖ · ‖, B be the unit ball in X, D
n
be the unit polydisc in ℂ
n
. In this paper, we introduce a class of holomorphic mappings on B or D
n
. Let f(x) be a normalized locally biholomorphic mapping on B such that (Df(x))−1
f(x) ∈ and f(x) − x has a zero of order k + 1 at x = 0. We obtain coefficient estimates for f(x). These results unify and generalize many known results.
This work was supported by National Natural Science Foundation of China (Grant No. 10571164), Specialized Research Fund for
the Doctoral Program of Higher Education (Grant No. 20050358052), the Jiangxi Provincial Natural Science Foundation of China
(Grant No. 2007GZS0177) and Specialized Research Fund for the Doctoral Program of Jiangxi Normal University. 相似文献
9.
Joanna Janczewska 《Central European Journal of Mathematics》2004,2(4):561-572
In this work we study the problem of the existence of bifurcation in the solution set of the equation F(x, λ)=0, where F: X×R
k
→Y is a C
2-smooth operator, X and Y are Banach spaces such that X⊂Y. Moreover, there is given a scalar product 〈·,·〉: Y×Y→R
1 that is continuous with respect to the norms in X and Y. We show that under some conditions there is bifurcation at a point (0, λ0)∈X×R
k
and we describe the solution set of the studied equation in a small neighbourhood of this point. 相似文献
10.
Sergio Vessella 《数学学报(英文版)》2005,21(2):351-380
Let Γ be a portion of a C
1,α boundary of an n-dimensional domain D. Let u be a solution
to a second order parabolic equation in D × (–T, T) and assume that u = 0 on Γ × (–T, T), 0 ∈ Γ. We
prove that u satis.es a three cylinder inequality near Γ × (–T, T) . As a consequence of the previous
result we prove that if u (x, t) = O (|x|k) for every t ∈ (–T, T) and every k ∈ ℕ, then u is identically
equal to zero.
This work is partially supported by MURST, Grant No. MM01111258 相似文献
11.
Zhu Ning 《高校应用数学学报(英文版)》1998,13(3):241-250
ANOTEONTHEBEHAVIOROFBLOW┐UPSOLUTIONSFORONE┐PHASESTEFANPROBLEMSZHUNINGAbstract.Inthispaper,thefolowingone-phaseStefanproblemis... 相似文献
12.
Abstract
The singular second-order m-point boundary value problem
, is considered under some conditions concerning the first eigenvalue of the relevant linear operators, where (Lϕ)(x) = (p(x)ϕ′(x))′ + q(x)ϕ(x) and ξ
i
∈ (0, 1) with 0 < ξ1 < ξ2 < · · · < ξ
m−2 < 1, a
i
∈ [0, ∞). h(x) is allowed to be singular at x = 0 and x = 1. The existence of positive solutions is obtained by means of fixed point index theory. Similar conclusions hold for some
other m-point boundary value conditions.
Supported by the National Natural Science Foundation of China (No.10371066, No.10371013) 相似文献
13.
We consider the parametric programming problem (Q
p
) of minimizing the quadratic function f(x,p):=x
T
Ax+b
T
x subject to the constraint Cx≤d, where x∈ℝ
n
, A∈ℝ
n×n
, b∈ℝ
n
, C∈ℝ
m×n
, d∈ℝ
m
, and p:=(A,b,C,d) is the parameter. Here, the matrix A is not assumed to be positive semidefinite. The set of the global minimizers and the set of the local minimizers to (Q
p
) are denoted by M(p) and M
loc
(p), respectively. It is proved that if the point-to-set mapping M
loc
(·) is lower semicontinuous at p then M
loc
(p) is a nonempty set which consists of at most ?
m,n
points, where ?
m,n
= is the maximal cardinality of the antichains of distinct subsets of {1,2,...,m} which have at most n elements. It is proved also that the lower semicontinuity of M(·) at p implies that M(p) is a singleton. Under some regularity assumption, these necessary conditions become the sufficient ones.
Received: November 5, 1997 / Accepted: September 12, 2000?Published online November 17, 2000 相似文献
14.
《偏微分方程通讯》2013,38(7-8):1409-1425
15.
Guizhen LIU 《Frontiers of Mathematics in China》2009,4(2):311-323
Let G be a digraph with vertex set V(G) and arc set E(G) and let g = (g
−, g
+) and ƒ = (ƒ
−, ƒ
+) be pairs of positive integer-valued functions defined on V(G) such that g
−(x) ⩽ ƒ
−(x) and g
+(x) ⩽ ƒ
+(x) for each x ∈ V(G). A (g, ƒ)-factor of G is a spanning subdigraph H of G such that g
−(x) ⩽ id
H
(x) ⩽ ƒ
−(x) and g
+(x) ⩽ od
H
(x) ⩽ ƒ
+(x) for each x ∈ V(H); a (g, ƒ)-factorization of G is a partition of E(G) into arc-disjoint (g, ƒ)-factors. Let
= {F
1, F
2,…, F
m} and H be a factorization and a subdigraph of G, respectively.
is called k-orthogonal to H if each F
i
, 1 ⩽ i ⩽ m, has exactly k arcs in common with H. In this paper it is proved that every (mg+m−1,mƒ−m+1)-digraph has a (g, f)-factorization k-orthogonal to any given subdigraph with km arcs if k ⩽ min{g
−(x), g
+(x)} for any x ∈ V(G) and that every (mg, mf)-digraph has a (g, f)-factorization orthogonal to any given directed m-star if 0 ⩽ g(x) ⩽ f(x) for any x ∈ V(G). The results in this paper are in some sense best possible.
相似文献
16.
Yu. K. Sabitova 《Russian Mathematics (Iz VUZ)》2009,53(12):41-49
We consider the equation y
m
u
xx
− u
yy
− b
2
y
m
u = 0 in the rectangular area {(x, y) | 0 < x < 1, 0 < y < T}, where m < 0, b ≥ 0, T > 0 are given real numbers. For this equation we study problems with initial conditions u(x, 0) = τ(x), u
y
(x, 0) = ν(x), 0 ≤ x ≤ 1, and nonlocal boundary conditions u(0, y) = u(1, y), u
x
(0, y) = 0 or u
x
(0, y) = u
x
(1, y), u(1, y) = 0 with 0≤y≤T. Using the method of spectral analysis, we prove the uniqueness and existence theorems for solutions to these problems 相似文献
17.
Xiao-nlng Lin Hong Wang Da-qing Jiang 《应用数学学报(英文版)》2007,23(2):289-302
This paper is devoted to the study ofthe existence of single and multiple positive solutions forthe first order boundary value problem x′= f(t,x),x(0)=x(T),where f ∈ C([0,T]×R).In addition,weapply our existence theorems to a class of nonlinear periodic boundary value problems with a singularity at theorigin.Our proofs are based on a fixed point theorem in cones.Our results improve some recent results in theliteratures. 相似文献
18.
Xiaojing Yang 《Archiv der Mathematik》2005,85(5):460-469
In this paper, the existence of unbounded solutions for the following nonlinear asymmetric oscillator
is discussed, where α, β are positive constants satisfying
for some ω ∈R+ /Q, h(t) ∈L∞ [0, 2π ] is 2π-periodic, x±=max {±x, 0 }.
Received: 23 September 2004 相似文献
19.
Extremal probabilities for Gaussian quadratic forms 总被引:1,自引:0,他引:1
Denote by Q an arbitrary positive semidefinite quadratic form in centered Gaussian random variables such that E(Q)=1. We prove that for an arbitrary x>0, inf
Q
P(Q≤x)=P(χ2
n
/n≤x), where χ
n
2
is a chi-square distributed rv with n=n(x) degrees of freedom, n(x) is a non-increasing function of x, n=1 iff x>x(1)=1.5364…, n=2 iff x[x(2),x(1)], where x(2)=1.2989…, etc., n(x)≤rank(Q). A similar statement is not true for the supremum: if 1<x<2 and Z
1
,Z
2
are independent standard Gaussian rv's, then sup0≤λ≤1/2
P{λZ
1
2
+(1−λ)Z
2
2
≤x} is taken not at λ=0 or at λ=1/2 but at 0<λ=λ(x)<1/2, where λ(x) is a continuous, increasing function from λ(1)=0 to λ(2)=1/2, e.g. λ(1.5)=.15…. Applications of our theorems include asymptotic
quantiles of U and V-statistics, signal detection, and stochastic orderings of integrals of squared Gaussian processes.
Received: 24 June 2002 / Revised version: 26 January 2003
Published online: 15 April 2003
Research supported by NSA Grant MDA904-02-1-0091
Mathematics Subject Classification (2000): Primary 60E15, 60G15; Secondary 62G10 相似文献
20.
Wolfgang Müller 《Monatshefte für Mathematik》2008,171(2):233-250
Let Q
1,…,Q
r
be quadratic forms with real coefficients. We prove that the set
{(Q1(x),?,Qr(x)) | x ? \Bbb Zs}\{(Q_1(x),\ldots ,Q_r(x))\,\vert\, x\in{\Bbb Z}^s\}
is dense in
\Bbb Rr{\Bbb R}^r
, provided that the system Q
1(x) = 0,…,Q
r
(x) = 0 has a nonsingular real solution and all forms in the real pencil generated by Q
1,…,Q
r
are irrational and have rank larger than 8r. Moreover, we give a quantitative version of the above assertion. As an application we study higher correlation functions
of the value distribution of a positive definite irrational quadratic form. 相似文献