共查询到20条相似文献,搜索用时 109 毫秒
1.
Bang-He Li 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2007,72(3):959-968
There are lots of results on the solutions of the heat equation
\frac?u?t = \mathop?ni=1\frac?2?x2iu,\frac{\partial u}{\partial t} = {\mathop\sum\limits^{n}_{i=1}}\frac{\partial^2}{\partial x^{2}_{i}}u,
but much less on those of the Hermite heat equation
\frac?U?t = \mathop?ni=1(\frac?2?x2i - x2i) U\frac{\partial U}{\partial t} = {\mathop\sum\limits^{n}_{i=1}}\left(\frac{\partial^2}{\partial x^{2}_{i}} - x^{2}_{i}\right) U
due to that its coefficients are not constant and even not bounded. In this paper, we find an explicit relation between the
solutions of these two equations, thus all known results on the heat equation can be transferred to results on the Hermite
heat equation, which should be a completely new idea to study the Hermite equation. Some examples are given to show that known
results on the Hermite equation are obtained easily by this method, even improved. There is also a new uniqueness theorem
with a very general condition for the Hermite equation, which answers a question in a paper in Proc. Japan Acad. (2005). 相似文献
2.
Let θ > 1 and let ϕ : [0,1] → ℂ be such that the two-sided series
converges for all x ∊ [0,1], (then necessarily φ(0) = φ(1) = 0).Suppose
Define
For different classes of functions φ we show that
À notre ami, Jean-Louis Nicolas2000 Mathematics Subject Classification: Primary—11B83, 11B99 相似文献
3.
E.M.E.ZAYED 《数学学报(英文版)》2004,20(2):209-222
The asymptotic expansion for small |t| of the trace of the wave kernel ∧↑μ(t) =∑v=1^∞exp(-it μv^1/2), where i= √-1 and {μv}v=1^∞ are the eigenvalues of the negative Laplacian -△=-∑β=1^2(δ/δx^β)^2 in the (x^1, x^2)-plane, is studied for a multi-connected vibrating membrane Ω in R^2 surrounded by simply connected bounded domains Ωj with smooth boundaries δΩj(j=1,...,n), where a finite number of piecewise smooth Robin boundary conditions on the piecewise smooth components Гi(i=1 κj-1,...,κj) of the boundaries δΩj are considered, such that δΩj=∪i=1 κj-1^κj Гi and κ0=0. The basic problem is to extract information on the geometry of Ω using the wave equation approach. Some geometric quantities of Ω (e.g. the area of Ω, the total lengths of its boundary, the curvature of its boundary, the number of the holes of Ω, etc.) are determined from the asymptotic expansion of the trace of the wave kernel ∧↑μ(t) for small |t|. 相似文献
4.
This paper exhibits an interesting relationship between arbitrary order Bessel functions and Dirac type equations.
Let
be the Euclidean Dirac operator in the n-dimensional flat space
the radial symmetric Euler operator and α and λ be arbitrary non-zero complex parameters. The goal of this paper is to describe
explicitly the structure of the solutions to the PDE system
in terms of arbitrary complex order Bessel functions and homogeneous monogenic polynomials.
Received: 27 October 2005 相似文献
5.
Precise Asymptotics in the Law of the Iterated Logarithm of Moving-Average Processes 总被引:1,自引:0,他引:1
Yun Xia LI Li Xin ZHANG 《数学学报(英文版)》2006,22(1):143-156
In this paper, we discuss the moving-average process Xk = ∑i=-∞ ^∞ ai+kεi, where {εi;-∞ 〈 i 〈 ∞} is a doubly infinite sequence of identically distributed ψ-mixing or negatively associated random variables with mean zeros and finite variances, {ai;-∞ 〈 i 〈 -∞) is an absolutely solutely summable sequence of real numbers. 相似文献
6.
Adimurthi K. Sandeep 《NoDEA : Nonlinear Differential Equations and Applications》2007,13(5-6):585-603
Let Ω be a bounded domain in
, we prove the singular Moser-Trudinger embedding:
if and only if
where
and
. We will also study the corresponding critical exponent problem. 相似文献
7.
Let A be a Banach algebra which does not contain any nonzero idempotent element, let γ > 0, and let
. We show that if
then
. We also show, assuming a suitable spectral condition on x, that if
, then
Received: 12 July 2006 Revised: 31 January 2007 相似文献
8.
Junjie Li 《Mathematische Annalen》2007,339(2):251-285
We are concerned with existence, positivity property and long-time behavior of solutions to the following initial boundary
value problem of a fourth order degenerate parabolic equation in higher space dimensions 相似文献
9.
E.M.E.ZAYED 《数学学报(英文版)》2003,19(4):679-694
The asymptotic expansions of the trace of the heat kernel θ(t)=∑^∞v=1^exp(-tλv) for small positive t,where {λv} are the eigenvalues of the negative Laplacian -△n=-∑^ni=1(D/Dx^1)^2 in R^2(n=2 or 3),are studied for a general annular bounded domain Ω with a smooth inner boundary DΩ1 and a smooth outer boundary DΩ2,where a finite number of piecewise smooth Robin boundary conditions(D/Dnj γh)Ф=0 on the components Гj(j= 1,...,m) of (DΩ1 and on the components Гj (j=k 1,…,m) of of DΩ2 are considered such that DΩl=U^kj=lГj and DΩ2= U^m=k 1Гj and where the coefficients γj(j=1,...,m) are piecewise smooth positive functions. Some applications of θ(t) for an ideal gas enclosed in the general annular bounded domain Ω are given. Further results are also obtained. 相似文献
10.
We study holomorphic flows on Stein manifolds. We prove that a holomorphic flow with isolated singularities and a dicritical
singularity of the form
on a Stein manifold
with
, is globally analytically linearizable; in particular M is biholomorphic to
. A complete stability result for periodic orbits is also obtained.
Bruno Scárdua: Partially supported by ICTP-Trieste-Italy.
Received: 27 September 2006 相似文献
11.
Let
and
. We are interested in the lower bounds of the integral:
where h > 0 and
. Using the lower bounds for these integrals we obtain in particular for the so-called Fejér operator
of
the following asymptotic expression
which essentially improves the results concerning the approximation behavior of this operator.
Received: 10 January 2006 相似文献
12.
Bin Heng SONG Huai Yu JIAN 《数学学报(英文版)》2005,21(5):1183-1190
We establish the existence of fundamental solutions for the anisotropic porous medium equation, ut = ∑n i=1(u^mi)xixi in R^n × (O,∞), where m1,m2,..., and mn, are positive constants satisfying min1≤i≤n{mi}≤ 1, ∑i^n=1 mi 〉 n - 2, and max1≤i≤n{mi} ≤1/n(2 + ∑i^n=1 mi). 相似文献
13.
In this paper, we have considered the generalized bi-axially symmetric Schr\"{o}dinger equation $$\frac{\partial^2\varphi}{\partial x^2}+\frac{\partial^2\varphi}{\partial y^2} + \frac{2\nu} {x}\frac{\partial \varphi} {\partial x} + \frac{2\mu} {y}\frac{\partial \varphi} {\partial y} + \{K^2-V(r)\} \varphi=0,$$ where $\mu,\nu\ge 0$, and $rV(r)$ is an entire function of $r=+(x^2+y^2)^{1/2}$ corresponding to a scattering potential $V(r)$. Growth parameters of entire function solutions in terms of their expansion coefficients, which are analogous to the formulas for order and type occurring in classical function theory, have been obtained. Our results are applicable for the scattering of particles in quantum mechanics. 相似文献
14.
Local and Global Existence of Solutions to Initial Value Problems of Nonlinear Kaup-Kupershmidt Equations 总被引:6,自引:0,他引:6
Shuang Ping TAO Shang Bin CUI 《数学学报(英文版)》2005,21(4):881-892
This paper is devoted to studying the initial value problems of the nonlinear Kaup Kupershmidt equations δu/δt + α1 uδ^2u/δx^2 + βδ^3u/δx^3 + γδ^5u/δx^5 = 0, (x,t)∈ E R^2, and δu/δt + α2 δu/δx δ^2u/δx^2 + βδ^3u/δx^3 + γδ^5u/δx^5 = 0, (x, t) ∈R^2. Several important Strichartz type estimates for the fundamental solution of the corresponding linear problem are established. Then we apply such estimates to prove the local and global existence of solutions for the initial value problems of the nonlinear Kaup- Kupershmidt equations. The results show that a local solution exists if the initial function u0(x) ∈ H^s(R), and s ≥ 5/4 for the first equation and s≥301/108 for the second equation. 相似文献
15.
E.M.E.ZAYED 《应用数学学报(英文版)》2004,20(2):215-230
The asymptotic expansion of the heat kernel Θ(t)=sum from ∞to j=1 exp(-tλ_j) where {λ_j}_(j=1)~∞are the eigen-values of the negative Laplacian -Δ_n=-sum from n to k=1((?))~2 in R~n(n=2 or 3) is studied for short-time t for a generalbounded domain Ωwith a smooth boundary (?)Ω.In this paper,we consider the case of a finite number of theDirichlet conditions φ=0 on Γ_i (i=1,...,J) and the Neumann conditions (?)=0 on Γ_i (i=J 1,...,k) andthe Robin conditions ((?) γ_i)φ=0 on Γ_i (i=k 1,...,m) where γ_i are piecewise smooth positive impedancefunctions,such that (?)Ωconsists of a finite number of piecewise smooth components Γ_i(i=1,...,m) where(?)Ω=(?)Γ_i.We construct the required asymptotics in the form of a power series over t.The senior coefficients inthis series are specified as functionals of the geometric shape of the domain Ω.This result is applied to calculatethe one-particle partition function of a“special ideal gas”,i.e.,the set of non-interacting particles set up in abox with Dirichlet,Neumann and Robin boundary conditions for the appropriate wave function.Calculationof the thermodynamic quantities for the ideal gas such as the internal energy,pressure and specific heat revealsthat these quantities alone are incapable of distinguishing between two different shapes of the domain.Thisconclusion seems to be intuitively clear because it is based on a limited information given by a one-particlepartition function;nevertheless,its formal theoretical motivation is of some interest. 相似文献
16.
E. M. E. Zayed 《数学学报(英文版)》2000,16(4):627-636
Abstract
Small-time asymptotics of the trace of the heat semigroup
where {μ
ν
} are the eigenvalues of the negative Laplacian
in the (x
1, x
2)-plane, is studied for a general bunded domain Ω with a smooth boundary ∂Ω, where a finite number of Dirichlet, Neumann and
Robin boundary conditions, on the piecewise smooth parts Γ
i
(i = 1, ..., n) of ∂Ω such that
, are considered. Some geometrical properties associated with Ω are determined. 相似文献
17.
In this note we prove that the Laplacian with generalized Wentzell boundary
conditions on an open bounded regular domain in
defined by
generates an analytic semigroup of angle
on
for every > 0 and
(for the definition of
cf. (1.3)).Received: 13 July 2002 相似文献
18.
Guang Lin RANG Zhi Yuan HUANG 《数学学报(英文版)》2007,23(5):895-904
In this paper a complete proof for the existence of generalized operators satisfying abstract 02 dynamical equations of quantum motions δ^2/δt^2Ф(t, x) + ( △- m^2)Ф(t, x) = -λ :Ф^3(t, x), subject to a suitable initial condition, is given under the framework of white noise analysis. Also some important commutation relations related to Ф44 quantum fields are discussed and proved in detail. 相似文献
19.
We consider the Allen–Cahn equation
where Ω is a smooth and bounded domain in
such that the mean curvature is positive at each boundary point. We show that there exists a sequence ε j → 0 such that the Allen–Cahn equation has a solution
with an interface which approaches the boundary as j → + ∞. 相似文献