共查询到20条相似文献,搜索用时 484 毫秒
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2.
Linghai ZHANG 《数学年刊B辑(英文版)》2008,29(2):179-198
Let u=u(x,t,uo)represent the global solution of the initial value problem for the one-dimensional fluid dynamics equation ut-εuxxt+δux+γHuxx+βuxxx+f(u)x=αuxx,u(x,0)=uo(x), whereα〉0,β〉0,γ〉0,δ〉0 andε〉0 are constants.This equation may be viewed as a one-dimensional reduction of n-dimensional incompressible Navier-Stokes equations. The nonlinear function satisfies the conditions f(0)=0,|f(u)|→∞as |u|→∞,and f∈C^1(R),and there exist the following limits Lo=lim sup/u→o f(u)/u^3 and L∞=lim sup/u→∞ f(u)/u^5 Suppose that the initial function u0∈L^I(R)∩H^2(R).By using energy estimates,Fourier transform,Plancherel's identity,upper limit estimate,lower limit estimate and the results of the linear problem vt-εv(xxt)+δvx+γHv(xx)+βv(xxx)=αv(xx),v(x,0)=vo(x), the author justifies the following limits(with sharp rates of decay) lim t→∞[(1+t)^(m+1/2)∫|uxm(x,t)|^2dx]=1/2π(π/2α)^(1/2)m!!/(4α)^m[∫R uo(x)dx]^2, if∫R uo(x)dx≠0, where 0!!=1,1!!=1 and m!!=1·3…(2m-3)…(2m-1).Moreover lim t→∞[(1+t)^(m+3/2)∫R|uxm(x,t)|^2dx]=1/2π(x/2α)^(1/2)(m+1)!!/(4α)^(m+1)[∫Rρo(x)dx]^2, if the initial function uo(x)=ρo′(x),for some functionρo∈C^1(R)∩L^1(R)and∫Rρo(x)dx≠0. 相似文献
3.
We establish the existence of positive bound state solutions for the singular quasilinear Schrodinger equation iψ/t=-div(ρ(|ψ|^2) ψ)+ω(|ψ|62)ψ-λρ(|ψ|^2)ψ,x∈Ω,t〉0,where ω(τ^2)τ→∞ as τ→ 0 and, λ 〉 0is a parameter and Ω is a ball in ^RN. This problem is studied in connection with the following quasilinear eigenvalue problem with Dirichlet boundary condition -div(ρ(| ψ|^2) ψ)=λ1ρ(|ψ|^2)ψ=λ1ρ(|ψ|^2)ψ,x∈Ω.Indeed, we establish the existence of solutions for the above Schrodinger equation when A belongs to a certain neighborhood of the first eigenvahie λ1 of this eigenvalue problem. The main feature of this paper is that the nonlinearity ω|ψ|^2 is unbounded around the origin and also the presence of the second order nonlinear term. Our analysis shows the importance of the role played by the parameter A combined with the nonlinear nonhomogeneous term div (ρ(| ψ|^2) ψ) which leads us to treat this prob- lem in an appropriate Orlicz space. The proofs are based on various techniques related to variational methods and implicit function theorem. 相似文献
4.
研究了下列Henon方程解的渐近性态:-△u=|x|^αu^p-1,u〉0,x∈B1(0)∪→R(n≥3),u=0,x∈δB1(0).这里α〉0,p从左边趋近于p(α)=2(n+α)/n-1〉2n/n-2(n≥3). 相似文献
5.
本文研究了如下问题:-div(|x|β△u)=|x|^a|u|^2(α,β)-2u+λ|x|σ|u|^q-2,x∈Ω,u=0,x∈δΩ,这里Ω∪→R^N是有界光滑区域且0∈Ω,2(α,β)=2(N+α)/N+β-2,运用Sobolev-Hardy不等式和山路几何,证明了在一定的条件下方程至少存在一个非平凡解。 相似文献
6.
Let Ω be a bounded domain with a smooth C2 boundary in RN(N ≥ 3), 0 ∈Ω, and n denote the unit outward normal to ЭΩ.We are concerned with the Neumann boundary problems: -div(|x|α|△u|p-2△u)=|x|βup(α,β)-1-λ|x|γup-1,u(x)〉0,x∈Ω,Эu/Эn=0 on ЭΩ,where 1〈p〈N and α〈0,β〈0 such that p(α,β)△=p(N+β)/N-p+α〉p,y〉α-p.For various parameters α,βorγ,we establish certain existence results of the solutions in the case 0∈Ω or 0∈ЭΩ. 相似文献
7.
Joseph A. Iaia 《偏微分方程(英文版)》2010,(1):33-67
We study capillary spreadings of thin films of liquids of power-law rheology. These satisfy ut+(u^λ+2|uxxx|^λ-1uxxx)x=0,where u (x, t) represents the thickness of the one-dimensional liquid and λ 〉 1. We look for traveling wave solutions so that u(x,t) =g(x+ct) and thus g satisfies g'''=|g-ε|^1/λ/g^1+2/λ sgn(g-ε) We show that for each ε 〉 0 there is an infinitely oscillating solution, gε, such that limt→∞ gε=ε and that gε→ g0 as ε → O, where g0≡ 0 for t ≥ 0 and g0=cλ|t|3λ/2λ+1 for t〈0 for some constant cλ. 相似文献
8.
类似于Campanato空间理论,本文对具有以下连续模的函数类给出了一局部积分刻画:|u(x)-u(y)|≤C|x—y|^α(log|x—y|^-1)β,z,y∈Ω,其中C〉0,Ω为R^d中一正则区域,α∈(0,1],β∈[0,1]. 相似文献
9.
The hyper Hilbert transform Tnf(x) =∫-1^1 f(x - Γ(t))e^-i|t|-β|t|^-1-αdt along an appropriate curve Γ(t) on R^n is investigated,where β 〉 α 〉 0.An L^p boundedness theorem of T4 is obtained,which is an extension of some earlier results of n = 2 and n = 3. 相似文献
10.
1 问题的提出
题绘出以下函数的图像.
①y=||x|-2|-1
②y=|x+2|-|x|+|x-2|-3
利用几何画板软件,画出两个函数的图像(如图1)是一样的: 相似文献
11.
Shiquan WU 《应用数学学报(英文版)》1996,12(4):377-383
Letn, s
1,s
2, ... ands
n
be positive integers. Assume
is an integer for eachi}. For
,
, and
, denotes
p
(a)={j|1jn,a
j
p},
, and
.
is called anI
t
p
-intersecting family if, for any a,b
,a
i
b
i
=min(a
i
,b
i
)p for at leastt i's.
is called a greedyI
t
P
-intersecting family if
is anI
t
p
-intersecting family andW
p
(A)W
p
(B+A
c
) for anyAS
p
(
) and any
with |B|=t–1.In this paper, we obtain a sharp upper bound of |
| for greedyI
t
p
-intersecting families in
for the case 2ps
i
(1in) ands
1>s
2>...>s
n
.This project is partially supported by the National Natural Science Foundation of China (No.19401008) and by Postdoctoral Science Foundation of China. 相似文献
12.
13.
Mihai Mihăilescu 《Czechoslovak Mathematical Journal》2008,58(1):155-172
We study the boundary value problem in Ω, u = 0 on ∂Ω, where Ω is a smooth bounded domain in ℝ
N
. Our attention is focused on two cases when , where m(x) = max{p
1(x), p
2(x)} for any x ∈ or m(x) < q(x) < N · m(x)/(N − m(x)) for any x ∈ . In the former case we show the existence of infinitely many weak solutions for any λ > 0. In the latter we prove that if λ is large enough then there exists a nontrivial weak solution. Our approach relies on the variable exponent theory of generalized
Lebesgue-Sobolev spaces, combined with a ℤ2-symmetric version for even functionals of the Mountain Pass Theorem and some adequate variational methods. 相似文献
14.
In this paper we give Lp-bound edness for the operator Tu defined bywhere P(x,y) is a real nontrivial polynomial on Rn×Rn,Ωis homogeneous of degree zero,Ω∈Lq(Sn-1),q>1/(1-μ) and b(r)∈BV(R+),The result can be regarded as an improvement of F.Ricci and E.M.Stein's result for fractional oscillatory integral operator with smoothness kernel. 相似文献
15.
L
p
approximation capability of radial basis function (RBF) neural networks is investigated. If g: R
+1 → R
1 and ∈ L
loc
p
(R
n
) with 1 ≤ p < ∞, then the RBF neural networks with g as the activation function can approximate any given function in L
p
(K) with any accuracy for any compact set K in R
n
, if and only if g(x) is not an even polynomial.
Partly supported by the National Natural Science Foundation of China (10471017) 相似文献
16.
The notion of p-adic multiresolution analysis (MRA) is introduced. We discuss a “natural” refinement equation whose solution (a refinable function) is the characteristic function
of the unit disc. This equation reflects the fact that the characteristic function of the unit disc is a sum of p characteristic functions of mutually disjoint discs of radius p
−1. This refinement equation generates a MRA. The case p=2 is studied in detail. Our MRA is a 2-adic analog of the real Haar MRA. But in contrast to the real setting, the refinable
function generating our Haar MRA is 1-periodic, which never holds for real refinable functions. This fact implies that there
exist infinity many different 2-adic orthonormal wavelet bases in ℒ2(ℚ2) generated by the same Haar MRA. All of these new bases are described. We also constructed infinity many different multidimensional 2-adic Haar orthonormal wavelet bases for ℒ2(ℚ2
n
) by means of the tensor product of one-dimensional MRAs. We also study connections between wavelet analysis and spectral
analysis of pseudo-differential operators. A criterion for multidimensional p-adic wavelets to be eigenfunctions for a pseudo-differential operator (in the Lizorkin space) is derived. We proved also
that these wavelets are eigenfunctions of the Taibleson multidimensional fractional operator. These facts create the necessary
prerequisites for intensive using our wavelet bases in applications. Our results related to the pseudo-differential operators
develop the investigations started in Albeverio et al. (J. Fourier Anal. Appl. 12(4):393–425, 2006).
相似文献
17.
Jean-Christophe Bourgoin 《Calculus of Variations and Partial Differential Equations》2006,25(4):469-489
In this paper, we investigate the minimality of the map
from the Euclidean unit ball Bn to its boundary 핊n−1 for weighted energy functionals of the type Ep,f = ∫Bn f(r)‖∇ u‖p dx, where f is a non-negative function. We prove that in each of the two following cases:
Mathematics Subject Classification (2000) 58E20; 53C43 相似文献
i) | p = 1 and f is non-decreasing, |
ii) | p is integer, p ≤ n−1 and f = rα with α ≥ 0, the map minimizes Ep,f among the maps in W1,p(Bn, 핊n−1) which coincide with on ∂ Bn. We also study the case where f(r) = rα with −n+2 < α < 0 and prove that does not minimize Ep,f for α close to −n+2 and when n ≥ 6, for α close to 4−n. |
18.
Approximation by Reciprocals of Polynomials with Positive Coefficients in L
p
Spaces 总被引:5,自引:0,他引:5
We prove that, if f(x) L
p
[0,1], 1 < p < , f(x) 0, x [0,1], f 0, then there is a polynomial p(x) +
n
such that f - 1/p
LP
C(p)(f,n
-1/2)
LP
where +
n
indicates the set of all polynomials of degree n with positive coeficients (see the definition (1) in the text). 相似文献
19.
Let be a continuous function such that H(p)→H0∈R as |p|→+∞. Fixing a domain Ω in R2 we study the behaviour of a sequence (un) of approximate solutions to the H-system Δu=2H(u)ux∧uy in Ω. Assuming that supp∈R3|(H(p)−H0)p|<1, we show that the weak limit of the sequence (un) solves the H-system and un→u strongly in H1 apart from a countable set S made by isolated points. Moreover, if in addition H(p)=H0+o(1/|p|) as |p|→+∞, then in correspondence of each point of S we prove that the sequence (un) blows either an H-bubble or an H0-sphere. 相似文献
20.
MUCHUNLAI TANGXIANJIANG 《高校应用数学学报(英文版)》1998,13(1):31-38
In this paper L^p-L^q estimates for the solution u(x,t) to the following perturbed high-er order hyperbolic equation are considered, (ρπ--a△)(ρπ--b△)u V(x)u=O, x∈R^n,n≥6, ρ1eu(x,O) = O, ρ^3eu(x,O) = f(x), (j = O,1,2).We assume that the otential V(x) and the initial data f(x) are compactly supported, andV(x) is sufficiently small, then the solution u (x,t) of the above problem satisfies the same L^p-L^q estimates as that of the unperturbed problem. 相似文献