Hyperbolic Phenomena in a Degenerate Parabolic Equation

M. Bertsch and R. Dal Passo [1] considered the equation u_t = (φ(u)ψ(u_z))x., where φ > 0 and ψ is a strictly increasing function with lim_{s → ∞} ψ(s) = ψ_∞ < ∞. They have solved the associated Cauchy problem for an increasing initial function. Furthermore, they discussed to what extent the solution behaves like the solution of the first order conservation law u_t = ψ_∞(φ(u))_x. The condition φ > 0 is essential in their paper. In the present paper, we study the above equation under the degenerate condition φ(0) = 0. The solution also possesses some hyperbolic phenomena like those pointed out in [1].     

On the Cauchy Problem and Initial Trace for Nonlinear Filtration Type with Singularity

In this paper, we consider the Cauchy problem \frac{∂u}{∂t} = Δφ(u) in R^N × (0, T] u(x,0} = u_0(x) in R^N where φ ∈ C[0,∞) ∩ C¹(0,∞), φ(0 ) = 0 and (1 - \frac{2}{N})^+ < a ≤ \frac{φ'(s)s}{φ(s)} ≤ m for some a ∈ ((1 - \frac{2}{n})^+, 1), s > 0. The initial value u_0 (z) satisfies u_0(x) ≥ 0 and u_0(x) ∈ L¹_{loc}(R^N). We prove that, under some further conditions, there exists a weak solution u for the above problem, and moreover u ∈ C^{α, \frac{α}{2}}_{x,t_{loc}} (R^N × (0, T]) for some α > 0.     

非交换Lipschitz-φ算子代数

本文引入由紧距离空间(K,d)到给定Banach代数A中的Lipschitz-φ算子构成的非交换Banach代数L~φ(K,A)与l~φ(K,A),证明了它们都是由K到A的全体连续算子构成的非交换Banach代数C(K,A)的子代数,并且关于范数||f||φ=L_φ(f)+||f||∞是Banach代数,研究了不同 Lipschitz尺度函数φ对应的大(小)Lipschitz代数之间的关系。特别当φ(t)=t~α时,引入了极限代数lim_(α→0+)l~α(K,A),lim_(α→+∞)l~α(K,A),lim_(α→0+)L~α(K,A)与lim_(α→+∞)L~α(K,A)以及距离空间的Lipschitz连通性,得到了lim_(α→+∞)l~α(K,A)=A的充要条件,也给出了lim_(α→0+)L~α(K,A)=C(K,A)的条件。     

Generalized Solution of the First Boundary Value Problem for Parabolic Monge-Ampere Equation

The existence and uniqueness of generalized solution to the first boundary value problem for parabolic Monge-Ampère equation - ut det D²_xu = f in Q = Ω × (0, T], u = φ on ∂_pQ are proved if there exists a strict generalized supersolution u_φ, where Ω ⊂ R^n is a bounded convex set, f is a nonnegative bounded measurable function defined on Q, φ ∈ C(∂_pQ), φ(x, 0) is a convex function in \overline{\Omega}, ∀x_0 ∈ ∂Ω, φ(x_0, t) ∈ C^α([0, T]).     

查甫雷金方程的唯一性定理(Ⅰ)

<正>考虑下列混合型议程的唯一性问题 K(y)u_(xx)+u_(yy)=0 (K(0)=0;当y≠0时,■(1) 所考慮的區域D由三條曲綾圍成.其一是雙曲區域(y<0)中由原點引出的特徵线Γ_1,它滿足下面條件     

Vector subdivision schemes in (Lp(Rs))r(1 ≤ p ≤∞) spaces

The purpose of this paper is to investigate the refinement equations of the form ψ(x) = ∑α∈Zs a(α)ψ(Mx - α), x ∈ Rs,where the vector of functions ψ=(ψ1,…,ψr)T is in (Lp(Rs))r, 1≤p≤∞,a(α),α∈Zs,is a finitely supported sequence of r × r matrices called the refinement mask, and M is an s × s integer matrix suchthat lim n→∞ M-n = 0. In order to solve the refinement equation mentioned above, we start with a vectorof compactly supported functions ψ0 ∈ (Lp(Rs))r and use the iteration schemes fn := Qnaψ0,n = 1,2,…,where Qa is the linear operator defined on (Lp(Rs))r given by Qaψ:= ∑α∈Zs a(α)ψ(M·- α),ψ∈ (Lp(Rs))r. This iteration scheme is called a subdivision scheme or cascade algorithm. In this paper, we characterize the Lp-convergence of subdivision schemes in terms of the p-norm joint spectral radius of a finite collection of somelinear operators determined by the sequence a and the set B restricted to a certain invariant subspace, wherethe set B is a complete set of representatives of the distinct cosets of the quotient group Zs/MZs containing 0.     

Convergence of Iterative Difference Method with Nonuniform Meshes for Quasilinear Parabolic Systems

In this paper, we study the general difference schemes with nonuniform meshes for the following problem: u_t = A(x,t,u,u_x)u_{xx}, + f(x,t,u,u_x), 0 < x < l, 0 < t ≤ T \qquad (1) u(0,t) = u(l ,t) = 0, 0 < t ≤ T \qquad\qquad (2) u(x,0) = φ(x), 0 ≤ x ≤ l \qquad\qquad (3) where u, φ, and f are m-dimensional vector valued functions, u_t = \frac{∂u}{∂t}, u_x = \frac{∂u}{∂x}, u_{xx} = \frac{∂²u}{∂_x²}. In the practical computation, we usually use the method of iteration to calculate the approximate solutions for the nonlinear difference schemes. Here the estimates of the iterative sequence constructed from the iterative difference schemes for the problem (1)-(3) is proved. Moreover, when the coefficient matrix A = A(x, t, u) is independent of u_x, t he convergence of the approximate difference solution for the iterative difference schemes to the unique solution of the problem (1)-(3) is proved without imposing the assumption of heuristic character concerning the existence of the unique smooth solution for the original problem (1)-(3).     

Cauchy Problem for Semilinear Wave Equations in Four Space Dimensions with Small Initial Data

In this paper, we consider the Cauchy problem ◻u(t,x) = |u(t,x)|^p, (t,x) ∈ R^+ × R^4 t = 0 : u = φ(x), u_t = ψ(x), x ∈ R^4 where ◻ = ∂²_t - Σ^4_{i=1}∂²_x_i, is the wave operator, φ, ψ ∈ C^∞_0 (R^4). We prove that for p > 2 the problem has a global solution provided tile initial data is sufficiently small.     

Factorization of mappings of topological spaces and homomorphisms of topological groups in accordance with weight and dimension ind

Symbols w(X), nw(X), and hl(X) denote the weight, the network weight, and the hereditary Lindelöf number of a space X, respectively. We prove the following factorization theorems.

1. Let X and Y be Tychonoff spaces, φ: X→Y a continuous mapping, hl(X)≤τ, and w(Y)≤τ. Then there exist a Tychonoff space Z and continuous mappings ψ: X→Z, χ: Z→Y such that φ=χ o ψ, Z=ψ(X), w(Z)≤τ andind Z≤ind X. Moreover, if nw(X)≤τ, then mapping ψ is one-to-one.
2. Let π: G→H be a continuous homomorphism of a Hausdorff topological group G to a Hausdorff topological group H, hl(G)≤τ and w(H)≤τ. Then there are a Hausdorff topological group G^* and continuous homomorphisms g: G→G^*, h: G^*→H so that π=h o g, G^*=g(G), w(G^*)≤τ andind G^*≤ind G. If nw(G)≤τ, then g is one-to-one.
3. For every continuous mapping φ: X→Y of a regular Lindelöf space X to a Tychonoff space Y one can find a Tychonoff space Z and continuous mappings ψ: X→Z, χ: Z→Y such that φ=χ o ψ, Z=ψ(X), w(Z)≤w(Y),dim Z≤dim X, andind ₀ Z≤ind ₀ X, whereind ₀ is the dimension function defined by V.V.Filippov with the help of G_δ-partitions. If we additionally suppose that X has a countable network, then ψ can be chosen to be one-to-one. The analogous result also holds for topological groups.
4. For each continuous homomorphism π: G→H of a Hausdorff Lindelöf Σ-group G (in particular, of a σ-compact group G) to a Hausdorff group H there exist a Hausdorff group G^* and continuous homomorphisms g: G→G^*, h:G^*→H so that π=h o g, G^*=g(G), w(G^*)≤w(H),dimG^*≤dimG, andind G^*≤ind G. Bibliography: 25 titles.

     

关于伽辽金方法收敛性的判别准则

<正> 伽辽金方法的重要性已为工程数学界所公认.有关它的收敛性的讨论,亦有大量文献与专著.但从算子方程的角度来看,所加的条件还很苛刻.本文则在较一般的条件下给出了伽辽金方法收敛性的一系列判别准则.我们相信,这些结果对于实际应用将是有益的.     

Time periodic solutions of porous medium equation

In this article, we study the time periodic solutions to the following porous medium equation under the homogeneous Dirichlet boundary condition: The existence of nontrivial nonnegative solution is established provided that 0≤α<m. The existence is also proved in the case α=m but with an additional assumption $\mathop{\min}\nolimits_{\overline{\Omega}\times[0,T]}a(x,t){>}{\lambda}_1In this article, we study the time periodic solutions to the following porous medium equation under the homogeneous Dirichlet boundary condition: The existence of nontrivial nonnegative solution is established provided that 0≤α<m. The existence is also proved in the case α=m but with an additional assumption $\mathop{\min}\nolimits_{\overline{\Omega}\times[0,T]}a(x,t){>}{\lambda}_1$, where λ₁ is the first eigenvalue of the operator ?Δ under the homogeneous Dirichlet boundary condition. We also show that the support of these solutions is independent of time by providing a priori estimates for their upper bounds using Moser iteration. Further, we establish the attractivity of maximal periodic solution using the monotonicity method. Copyright © 2010 John Wiley & Sons, Ltd.     

<Emphasis Type="Italic">q</Emphasis>-Deformed Barut–Girardello <Emphasis Type="Italic">su</Emphasis>(1, 1) coherent states and Schrödinger cat states

We define Schrödinger cat states as superpositions of q-deformed Barut–Girardello su(1, 1) coherent states with an adjustable angle φ in a q-deformed Fock space. We study the statistical properties of the q-deformed Barut–Girardello su(1, 1) coherent states and Schrödinger cat states. The statistical properties of photons are always sub-Poissonian for q-deformed Barut–Girardello su(1, 1) coherent states. For Schrödinger cat states in the cases φ = 0, π/2, π, the statistical properties of photons are always sub-Poissonian if φ = π/2, and the other cases are hard to determine because they depend on the parameters q and k. Moreover, we find some interesting properties of Schrödinger cat states in the limit |z| → 0, where z is the parameter of those states. We also derive that the statistical properties of photons are sub-Poissonian in the undeformed case where π/2 ≤ φ ≤ 3π/2.     

2维球面到复Grassmann流形的调和映照

在这篇文章中我们证明了：设φ：Ｓ２→Ｇｔ（ＣＮ）（ｔ＞１）是调和映照，它的迷向阶Ｉ（φ）≥ ｋ． ｋ为整数．若φ的能量密度 ｅ（φ） ≤ ２πｋ（关于 Ｓ２上标准度量ｄＳ２＝ｄｚｄｚ），则φ是迷向的．作为推论，我们获得了φ：Ｓ２→Ｇｔ（ＣＮ）是调和映照（ｔ＞１）．若ｅ（φ）≤２π（关于Ｓ２上标准度量ｄＳ２＝ｄｚｄｚ），则φ是迷向的．     

广义Mellin变换Ⅰ

<正> 推广至广义函数.古典 Mellin 变换作用于半直线(0,∞).因此我们将以半直线上的广义函数类(定义1)为 Mellin 变换的定义域.古典理论中 Mellin 像函数一般为解析函数,因此将以某种“解析”的广义函数类(定义2)为像域.作指数变换后,Mellin 变     

集值映射的不动点指数与带间断非线性项的椭圆型方程的多重解

<正> 本文研究二阶半线性椭圆边值问题■的多重解(符号详见§3),其中φ(x,t)允许对t是不连续的.一些自由边界问题可以化归这类问题.为了统一处理φ(x,t)对t连续与不连续两种情形,我们采用集值映射的观点.为此推广了经典的算子与Hammerstein算子到集值映射,并发展了集值映射的Leray-Schauder度理论;与已有的集值映射理论不同,现在处理的是映射串(定     

Solving dynamical equations in general homogeneous isotropic cosmologies with a scalaron

We consider gauge-dependent dynamical equations describing homogeneous isotropic cosmologies coupled to a scalar field ψ (scalaron). For flat cosmologies (k = 0), we analyze the gauge-independent equation describing the differential χ(α) ≡ ψ (a) of the map of the metric a to the scalaron field ψ, which is the main mathematical characteristic of a cosmology and locally defines its portrait in the so-called a version. In the more customary ψ version, the similar equation for the differential of the inverse map \(\bar \chi (\psi ) \equiv \chi ^{ - 1} (\alpha )\) is solved in an asymptotic approximation for arbitrary potentials v(ψ). In the flat case, \(\bar \chi (\psi )\) and χ^?1(α) satisfy first-order differential equations depending only on the logarithmic derivative of the potential, v(ψ)/v(ψ). If an analytic solution for one of the χ functions is known, then we can find all characteristics of the cosmological model. In the α version, the full dynamical system is explicitly integrable for k ≠ 0 with any potential v(α) ≡ v[ψ(α)] replacing v(ψ). Until one of the maps, which themselves depend on the potentials, is calculated, no sort of analytic relation between these potentials can be found. Nevertheless, such relations can be found in asymptotic regions or by perturbation theory. If instead of a potential we specify a cosmological portrait, then we can reconstruct the corresponding potential. The main subject here is the mathematical structure of isotropic cosmologies. We also briefly present basic applications to a more rigorous treatment of inflation models in the framework of the α version of the isotropic scalaron cosmology. In particular, we construct an inflationary perturbation expansion for χ. If the conditions for inflation to arise are satisfied, i.e., if v > 0, k = 0, χ² < 6, and χ(α) satisfies a certain boundary condition as α→-∞, then the expansion is invariant under scaling the potential, and its first terms give the standard inflationary parameters with higher-order corrections.     

Approximation Solutions of Nonlinear Strongly Accretive Operator Equations by Ishikawa Iteration Procedure with Errors

Let 1＜ρ≤2,E be a real ρ-uniformly smooth Banach space and T:E→E be a continuous and strongly accretive operator.The purpose of this paper is to investigate the problem of approximating solutions to the equation Tx=f by the Ishikawa iteration procedure with errors (?) where x_0 ∈ E,{u_n},{υ_n}are bounded sequences in E and{α_n},{b_n},{c_n},{a_n~'},{b_n~'},{c_n~'} are real sequences in[0,1].Under the assumption of the condition 0＜α≤b_n c_n,An≥0, it is shown that the iterative sequence{x_n}converges strongly to the unique solution of the equation Tx=f.Furthermore,under no assumption of the condition(?)(b_n~' c_n~')=0,it is also shown that{x_n}converges strongly to the unique solution of Tx=f.     

Time periodic solutions of a class of degenerate parabolic equations

1.IntroductionManypapershavebeendevotedtotheexistenceoftimeperiodicsolutionsforsemilinearparabolicequations,see[1--8].Atthesametime,thestudyofquasilinearperiodic-parabolicequationsalsoattractedmanyauthors,seealso[9--141.Inparticular,recentlyHess,PozioandTesei[13]usedthemonotonicitymethodstodealwiththeequationsonot=aam a(x,t)u,wherem>1andaisafunctionperiodicint,andMizoguchi[lllappliedtheLeray-Schauderdegreetheorytoinvestigatetheequationswithsuperlinearforcingtermwherem>1,hisapositiveperiodicf…     

Regularity of the obstacle problem for a fractional power of the laplace operator

Given a function φ and s ∈ (0, 1), we will study the solutions of the following obstacle problem:

• u ≥ φ in ?ⁿ,
• (??)^su ≥ 0 in ?ⁿ,
• (??)^su(x) = 0 for those x such that u(x) > φ(x),
• lim_{|x| → + ∞} u(x) = 0.

We show that when φ is C^{1, s} or smoother, the solution u is in the space C^{1, α} for every α < s. In the case where the contact set {u = φ} is convex, we prove the optimal regularity result u ∈ C^{1, s}. When φ is only C^{1, β} for a β < s, we prove that our solution u is C^{1, α} for every α < β. © 2006 Wiley Periodicals, Inc.     

重乘积的计算

<正> §1.对弧连通的拓扑空间 X J.H.C.Whitehead 介绍了一种乘积,它使α∈πm(X),β∈πn(X)对应(?),这种乘积使我们有可能去了解低维同伦群的元素对高维同伦群的影响.