共查询到19条相似文献,搜索用时 140 毫秒
1.
周军 《数学物理学报(B辑英文版)》2013,(4):1031-1048
We consider the growth rate and quenching rate of the following problem with singular nonlinearityfor some positive constants b:, b2 (see Theorem 3.3 for the parametersfor some constantsHence, the solution (u, v) quenches at the originx = 0 at the same time '1' (see Theorem 4.3). We also tind various other conditions tor the solution to quench in a finite time and obtain the corresponding decay rate of the solution near the quenching time. 相似文献
2.
In this paper,we consider a localized problem with free boundary for the heat equation in higher space dimensions and heterogeneous environment.For simplicity,we assume that the environment and solution are radially symmetric.First,by using the contraction mapping theorem,we prove that the local solution exists and is unique.Then,some sufficient conditions are given under which the solution will blow up in finite time.Our results indicate that the blowup occurs if the initial data are sufficiently large.Finally,the long time behavior of the global solution is discussed.It is shown that the global fast solution does exist if the initial data are sufficiently small,while the global slow solution is possible if the initial data are suitably large. 相似文献
3.
In this paper, the authors study the existence and nonexistence of multiple positive solutions for problem(*)μwhere h ∈ H-1(RN), N ≥ 3, |f(x,u)| ≤ C1up-1 + C2u with C1 > 0, C2∈ [0,1) being some constants and 2 < p < ∞. Under some assumptions on f and h, they prove that there exists a positive constant μ* <∞ such that problem (*)μ has at least one positive solution uμ if μ,∈ (0,μ*), there are no solutions for (*)μ if μ, > μ* and uμ is increasing with respect to μ∈ (0,μ*); furthermore, problem (*)μ has at least two positive solution for μ ∈ (0,μ*) and a unique positive solution for μ, =μ* if p ≤2N/N-2. 相似文献
4.
Nakao HAYASHI Pavel I. NAUMKIN 《数学学报(英文版)》2006,22(5):1441-1456
We study large time asymptotics of solutions to the Korteweg-de Vries-Burgers equation ut+uux-uxx+uxxx=0,x∈R,t〉0. We are interested in the large time asymptotics for the case when the initial data have an arbitrary size. We prove that if the initial data u0 ∈H^s (R)∩L^1 (R), where s 〉 -1/2, then there exists a unique solution u (t, x) ∈C^∞ ((0,∞);H^∞ (R)) to the Cauchy problem for the Korteweg-de Vries-Burgers equation, which has asymptotics u(t)=t^-1/2fM((·)t^-1/2)+0(t^-1/2) as t →∞, where fM is the self-similar solution for the Burgers equation. Moreover if xu0 (x) ∈ L^1 (R), then the asymptotics are true u(t)=t^-1/2fM((·)t^-1/2)+O(t^-1/2-γ) where γ ∈ (0, 1/2). 相似文献
5.
HYPERBOLIC MEAN CURVATURE FLOW:EVOLUTION OF PLANE CURVES 总被引:2,自引:0,他引:2
In this paper we investigate the one-dimensional hyperbolic mean curvatureflow for closed plane curves. More precisely, we consider a family of closed curves F : S1 × [0, T ) → R^2 which satisfies the following evolution equation δ^2F /δt^2 (u, t) = k(u, t)N(u, t)-▽ρ(u, t), ∨(u, t) ∈ S^1 × [0, T ) with the initial data F (u, 0) = F0(u) and δF/δt (u, 0) = f(u)N0, where k is the mean curvature and N is the unit inner normal vector of the plane curve F (u, t), f(u) and N0 are the initial velocity and the unit inner normal vector of the initial convex closed curve F0, respectively, and ▽ρ is given by
▽ρ Δ=(δ^2F /δsδt ,δF/δt) T , in which T stands for the unit tangent vector. The above problem is an initial value problem for a system of partial differential equations for F , it can be completely reduced to an initial value problem for a single partial differential equation for its support function. The latter equation is a hyperbolic Monge-Ampere equation. Based on this, we show that there exists a class of initial velocities such that the solution of the above initial value problem exists only at a finite time interval [0, Tmax) and when t goes to Tmax, either the solution convergesto a point or shocks and other propagating discontinuities are generated. Furthermore, we also consider the hyperbolic mean curvature flow with the dissipative terms and obtain the similar equations about the support functions and the curvature of the curve. In the end, we discuss the close relationship between the hyperbolic mean curvature flow and the equations for the evolving relativistic string in the Minkowski space-time R^1,1. 相似文献
▽ρ Δ=(δ^2F /δsδt ,δF/δt) T , in which T stands for the unit tangent vector. The above problem is an initial value problem for a system of partial differential equations for F , it can be completely reduced to an initial value problem for a single partial differential equation for its support function. The latter equation is a hyperbolic Monge-Ampere equation. Based on this, we show that there exists a class of initial velocities such that the solution of the above initial value problem exists only at a finite time interval [0, Tmax) and when t goes to Tmax, either the solution convergesto a point or shocks and other propagating discontinuities are generated. Furthermore, we also consider the hyperbolic mean curvature flow with the dissipative terms and obtain the similar equations about the support functions and the curvature of the curve. In the end, we discuss the close relationship between the hyperbolic mean curvature flow and the equations for the evolving relativistic string in the Minkowski space-time R^1,1. 相似文献
6.
Li MA 《数学年刊B辑(英文版)》2013,34(4):587-592
The author studies the boundary value problem of the classical semilinear parabolic equations ut-△u = |u|p-1u inΩ×(0, T), and u = 0 on the boundary × [0, T) and u = φ at t = 0, where Rnis a compact C1domain, 1 < p ≤ p S is a fixed constant, and φ∈ C1 0(Ω) is a given smooth function. Introducing a new idea, it is shown that there are two sets W and Z, such that for φ∈ W, there is a global positive solution u(t) ∈ W with H1omega limit 0 and for φ∈ Z, the solution blows up at finite time. 相似文献
7.
Zhou Fujun Cui Shangbin 《高校应用数学学报(英文版)》2005,20(4):441-447
The solvability of the fifth-order nonlinear dispersive equation δtu+au (δxu)^2+βδx^3u+γδx^5u = 0 is studied. By using the approach of Kenig, Ponce and Vega and some Strichartz estimates for the corresponding linear problem,it is proved that if the initial function u0 belongs to H^5(R) and s〉1/4,then the Cauchy problem has a unique solution in C([-T,T],H^5(R)) for some T〉0. 相似文献
8.
The Self-similar Solution to Some Nonlinear Integro-differential Equations Corresponding to Fractional Order Time Derivative 总被引:3,自引:0,他引:3
Chang Xing MIAO Han YANG 《数学学报(英文版)》2005,21(6):1337-1350
In this paper we study the self-similar solution to a class of nonlinear integro-differential equations which correspond to fractional order time derivative and interpolate nonlinear heat and wave equation. Using the space-time estimates which were established by Hirata and Miao in [1] we prove the global existence of self-similar solution of Cauchy problem for the nonlinear integro-differential equation in C*([0,∞];B^8pp,∞(R^n). 相似文献
9.
Fan Lili Yin Hui Zhao Huijiang 《偏微分方程(英文版)》2008,21(2):141-172
Abstract This paper is concerned with the initial-boundary value problem for damped wave equations with a nonlinear convection term in the half space R+{utt-txx+ut+f(u)x=0,t〉0,x∈R+,u(0,x)=u0(x)→u+,asx→+∞,ut(0,x)=u1(x),u(t,0)=ub.For the non-degenerate case f](u+) 〈 0, it is shown in [1] that the above initialboundary value problem admits a unique global solution u(t,x) which converges to the stationary wave φ(x) uniformly in x ∈ R+ as time tends to infinity provided that the initial perturbation and/or the strength of the stationary wave are sufficiently small. Moreover, by using the space-time weighted energy method initiated by Kawashima and Matsumura [2], the convergence rates (including the algebraic convergence rate and the exponential convergence rate) of u(t, x) toward φ(x) are also obtained in [1]. We note, however, that the analysis in [1] relies heavily on the assumption that f'(ub) 〈 0. The main purpose of this paper is devoted to discussing the case of f'(ub)= 0 and we show that similar results still hold for such a case. Our analysis is based on some delicate energy estimates. 相似文献
10.
Yong-ping Sun 《高校应用数学学报(英文版)》2008,23(3):279-285
In this paper, the existence of monotone positive solution for the following secondorder three-point boundary value problem is studied:
x″(t)+f(t,x(t))=0,0〈t〈1,
x′(0)=0,x(1)+δx′(η)=0,
where η ∈ (0, 1), δ∈ [0, ∞), f ∈ C([0, 1] × [0, ∞), [0, ∞)). Under certain growth conditions on the nonlinear term f and by using a fixed point theorem of cone expansion and compression of functional type due to Avery, Anderson and Krueger, sufficient conditions for the existence of monotone positive solution are obtained and the bounds of solution are given. At last, an example is given to illustrate the result of the paper. 相似文献
x″(t)+f(t,x(t))=0,0〈t〈1,
x′(0)=0,x(1)+δx′(η)=0,
where η ∈ (0, 1), δ∈ [0, ∞), f ∈ C([0, 1] × [0, ∞), [0, ∞)). Under certain growth conditions on the nonlinear term f and by using a fixed point theorem of cone expansion and compression of functional type due to Avery, Anderson and Krueger, sufficient conditions for the existence of monotone positive solution are obtained and the bounds of solution are given. At last, an example is given to illustrate the result of the paper. 相似文献
11.
Xiao Hong CAO Mao Zheng GUO Bin MENG 《数学学报(英文版)》2006,22(1):169-178
When A ∈ B(H) and B ∈ B(K) are given, we denote by Mc an operator acting on the Hilbert space HΘ K of the form Me = ( A0 CB). In this paper, first we give the necessary and sufficient condition for Mc to be an upper semi-Fredholm (lower semi-Fredholm, or Fredholm) operator for some C ∈B(K,H). In addition, let σSF+(A) = {λ ∈ C : A-λI is not an upper semi-Fredholm operator} bc the upper semi-Fredholm spectrum of A ∈ B(H) and let σrsF- (A) = {λ∈ C : A-λI is not a lower semi-Fredholm operator} be the lower semi Fredholm spectrum of A. We show that the passage from σSF±(A) U σSF±(B) to σSF±(Mc) is accomplished by removing certain open subsets of σSF-(A) ∩σSF+ (B) from the former, that is, there is an equality σSF±(A) ∪σSF± (B) = σSF± (Mc) ∪& where L is the union of certain of the holes in σSF±(Mc) which ilappen to be subsets of σSF- (A) A σSF+ (B). Weyl's theorem and Browder's theorem are liable to fail for 2 × 2 operator matrices. In this paper, we also explore how Weyl's theorem, Browder's theorem, a-Weyl's theorem and a-Browder's theorem survive for 2 × 2 upper triangular operator matrices on the Hilbert space. 相似文献
12.
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
13.
14.
Eberhard MALKOWSKY M. MURSALEEN Suthep SUANTAI 《数学学报(英文版)》2007,23(3):521-532
Let p = (pk)k=0^∞ be a bounded sequence of positive reals, m C N and u be s sequence of nonzero terms. If x = (xk)k=0^∞ is any sequence of complex numbers we write Δ(m)x for the sequence of the m th order differences of x and Δu^(m)X = {x=(x)k=0^∞ uΔ(m)x ∈ X} for any set X of sequences. We determine the α-, β- and γ-duals of the sets Δμ^(m)X for X=co(p),c(p),l∞(p) and characterize some matrix transformations between these spaces Δ^(m)X. 相似文献
15.
GuiQiaoXU YongPingLIU 《数学学报(英文版)》2004,20(1):81-92
This paper concerns the problem of average σ-width of Sobolev-Wiener classes W^rpq(R^d),W^rpq(M,R^d),and Besov-Wiener classes S^rpqθb(R^d).S^rpqθB(R^d),S^rpqθb(M,R^d),S^rpqθB(R^d)in the metric Lq(R^d) for 1≤q≤p≤∞.The weak asymptotic results concerning the average linear widths,the average Bernstein widths and the infinite-dimensional Gel‘fand widths are obtained,respectively. 相似文献
16.
Positive Solutions for Semipositone
<Emphasis Type="Italic">m</Emphasis>-point Boundary-value
Problems 总被引:7,自引:0,他引:7
Abstract
Let ξ
i
∈ (0, 1) with 0 <
ξ1 < ξ2 <
··· < ξ
m−2 < 1,
a
i
, b
i
∈ [0,∞) with
and
. We consider the
m-point boundary-value
problem
where f(x, y) ≥ −M, and M is a positive constant. We show the
existence and multiplicity of positive solutions by applying the
fixed point theorem in cones.
*Supported by the NSFC (10271095).
GG-110-10736-1003, NWNU-KJCXGC-212 and the Foundation of Major
Project of Science and Technology of Chinese Education
Ministry 相似文献
17.
Choonkil BAAK 《数学学报(英文版)》2006,22(6):1789-1796
Let X, Y be vector spaces. It is shown that if a mapping f : X → Y satisfies f((x+y)/2+z)+f((x-y)/2+z=f(x)+2f(z),(0.1) f((x+y)/2+z)-f((x-y)/2+z)f(y),(0.2) or 2f((x+y)/2+x)=f(x)+f(y)+2f(z)(0.3)for all x, y, z ∈ X, then the mapping f : X →Y is Cauchy additive.
Furthermore, we prove the Cauchy-Rassias stability of the functional equations (0.1), (0.2) and (0.3) in Banach spaces. The results are applied to investigate isomorphisms between unital Banach algebras. 相似文献
18.
Herz-type Triebel-Lizorkin Spaces, Ⅰ 总被引:1,自引:0,他引:1
Jing Shi XU Da Chun YANG 《数学学报(英文版)》2005,21(3):643-654
Let s ∈R,0〈β≤∞, 0〈 q, p〈 ∞ and-n/q〈α. In this paper the authors introduce the Herz-type Triebel-Lizorkin spaces,Kq^α,pFβ^s(R^n)andKq^α,pFβ^s(R^n)which are the generalizations of the well-known Herz-type spaces and the inhomogeneous Triebel-Lizorkin spaces, Some properties on these Herz-type Triebel Lizorkin spaces are also given. 相似文献
19.
Weak Hopf Algebras Corresponding to Borcherds-Cartan Matrices 总被引:1,自引:0,他引:1
Li Xia YE Zhi Xiang WU Xue Feng MEI 《数学学报(英文版)》2007,23(10):1729-1744
Let y be a generalized Kac-Moody algebra with an integral Borcherds-Cartan matrix. In this paper, we define a d-type weak quantum generalized Kac-Moody algebra wUq^d(y), which is a weak Hopf algebra. We also study the highest weight module over the weak quantum algebra wUdq^d(y) and weak A-forms of wUq^d(y). 相似文献