首页 | 本学科首页   官方微博 | 高级检索  
相似文献
 共查询到20条相似文献，搜索用时 359 毫秒
1.
Positive solutions and eigenvalue intervals for nonlinear systems   总被引：1，自引：0，他引：1  
This paper deals with the existence of positive solutions for the nonlinear system
. This system often arises in the study of positive radial solutions of nonlinear elliptic system. Here u = (u 1, …, u n) and f i, i = 1, 2, …, n are continuous and nonnegative functions, p(t), q(t): [0, 1] → (0, ∞) are continuous functions. Moreover, we characterize the eigenvalue intervals for
. The proof is based on a well-known fixed point theorem in cones.  相似文献   

2.
This paper is concerned with nonoscillatory solutions of the fourth order quasilinear differential equation
where α > 0, β > 0 and p(t) and q(t) are continuous functions on an infinite interval [a,∞) satisfying p(t) > 0 and q(t) > 0 (t≥a). The growth bounds near t = ∞ of nonoscillatory solutions are obtained, and necessary and sufficient integral conditions are established for the existence of nonoscillatory solutions having specific asymptotic growths as t→∞. This revised version was published online in June 2006 with corrections to the Cover Date.  相似文献   

3.
ＷＥＩＧＨＴＥＤＡＰＰＲＯＸＩＭＡＴＩＯＮＯＦＲＡＮＤＯＭＦＵＮＣＴＩＯＮＳＹＵＪＩＡＲＯＮＧＡｂｓｔｒａｃｔ：Ｌｅｔ（Ω,Ａ,Ｐ）ｂｅａｐｒｏｂａｂｉｌｉｔｙｓｐａｃｅ,Ｘ（ｔ,ω）ａｒａｎｄｏｍｆｕｎｃｔｉｏｎｃｏｎｔｉｎｕｏｕｓｉｎｐｒｏｂａｂ...  相似文献   

4.
The first part of this review paper is devoted to the simple (undamped, unforced) pendulum with a varying coefficient. If the coefficient is a step function, then small oscillations are described by the equation
Using a probability approach, we assume that (a k ) k=1∞ is given, and {t k } k=1∞ is chosen at random so that t k − t k−1 are independent random variables. The first problem is to guarantee that all solutions tend to zero, as t → ∞, provided that a k ↗ ∞ (k → ∞). In the problem of swinging the coefficient a 2 takes only two different values alternating each others, and t k − t k−1 are identically distributed. One has to find the distributions and their critical expected values such that the amplitudes of the oscillations tend to ∞ in some (probabilistic) sense. In the second part we deal with the damped forced pendulum equation
In 1999 J. Hubbard discovered that some motions of this simple physical model are chaotic. Recently, using also the computer (the method of interval arithmetic), we gave a proof for Hubbard’s assertion. Here we show some tools of the proof. Supported by the Hungarian NFSR (OTKA T49516) and by the Analysis and Stochastics Research Group of the Hungarian Academy of Sciences.  相似文献   

5.
This paper treats the rich mathematical structure of the (dimensionless) equation of motion governing the behavior of an elastically restrained simple pendulum subject to a downward force of magnitude f(t) applied to its bob with $\dot{f}(t)>0$\dot{f}(t)>0 for all t>0 and f(t)→∞ as t→∞:
[(q)\ddot]+2n[(q)\dot] +q = f(t)sinq.\ddot{\theta}+2\nu\dot{\theta} +\theta= f(t)\sin\theta.  相似文献   

6.
We consider the nonlinear Sturm–Liouville problem
(1)
where λ > 0 is an eigenvalue parameter. To understand well the global behavior of the bifurcation branch in R + × L 2(I), we establish the precise asymptotic formula for λ(α), which is associated with eigenfunction u α with ‖ u α ‖2 = α, as α → ∞. It is shown that if for some constant p > 1 the function h(u) ≔ f(u)/u p satisfies adequate assumptions, including a slow growth at ∞, then λ(α) ∼ α p−1 h(α) as α → ∞ and the second term of λ(α) as α → ∞ is determined by lim u → ∞ uh′(u). Mathematics Subject Classification (2000) 34B15  相似文献   

7.
The equations under consideration have the following structure:
where 0 < x n < ∞, (x 1, …, x n−1) ∈ Ω, Ω is a bounded Lipschitz domain, is a function that is continuous and monotonic with respect to u, and all coefficients are bounded measurable functions. Asymptotic formulas are established for solutions of such equations as x n → + ∞; the solutions are assumed to satisfy zero Dirichlet or Neumann boundary conditions on ∂Ω. Previously, such formulas were obtained in the case of a ij, ai depending only on (x 1, …, x n−1). __________ Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 25, pp. 98–111, 2005.  相似文献   

8.
Let M n be an n-dimensional compact C ∞-differentiable manifold, n ≥ 2, and let S be a C 1-differential system on M n . The system induces a one-parameter C 1 transformation group φ t (−∞ < t < ∞) over M n and, thus, naturally induces a one-parameter transformation group of the tangent bundle of M n . The aim of this paper, in essence, is to study certain ergodic properties of this latter transformation group. Among various results established in the paper, we mention here only the following, which might describe quite well the nature of our study. (A) Let M be the set of regular points in M n of the differential system S. With respect to a given C ∞ Riemannian metric of M n , we consider the bundle of all (n−2) spheres Q x n−2, x∈M, where Q x n−2 for each x consists of all unit tangent vectors of M n orthogonal to the trajectory through x. Then, the differential system S gives rise naturally to a one-parameter transformation group ψ t # (−∞<t<∞) of . For an l-frame α = (u 1, u 2,⋯, u l ) of M n at a point x in M, 1 ≥ l ≥ n−1, each u i being in , we shall denote the volume of the parallelotope in the tangent space of M n at x with edges u 1, u 2,⋯, u l by υ(α), and let . This is a continuous real function of t. Let
α is said to be positively linearly independent of the mean if I + *(α) > 0. Similarly, α is said to be negatively linearly independent of the mean if I − *(α) > 0. A point x of M is said to possess positive generic index κ = κ + *(x) if, at x, there is a κ-frame , , of M n having the property of being positively linearly independent in the mean, but at x, every l-frame , of M n with l > κ does not have the same property. Similarly, we define the negative generic index κ − *(x) of x. For a nonempty closed subset F of M n consisting of regular points of S, invariant under φ t (−∞ < t < ∞), let the (positive and negative) generic indices of F be defined by
Theorem κ + *(F)=κ − *(F). (B) We consider a nonempty compact metric space x and a one-parameter transformation group ϕ t (−∞ < t < ∞) over X. For a given positive integer l ≥ 2, we assume that, to each x∈X, there are associated l-positive real continuous functions
of −∞ < t < ∞. Assume further that these functions possess the following properties, namely, for each of k = 1, 2,⋯, l,
(i*)  h k (x, t) = h xk (t) is a continuous function of the Cartesian product X×(−∞, ∞).
(ii*) 
for each x∈X, each −∞ < s < ∞, and each −∞ < t < ∞. Theorem With X, etc., given above, let μ be a normal measure of X that is ergodic and invariant under ϕ t (−∞ < t < ∞). Then, for a certain permutation k→p(k) of k= 1, 2,⋯, l, the set W of points x of X such that all the inequalities (I k )
(II k )
(k=2, 3,⋯, l) hold is invariant under ϕ t (−∞ < t < ∞) and is μ-measurable with μ-measure1. In practice, the functions h xk (t) will be taken as length functions of certain tangent vectors of M n . This theory, established such as in this paper, is expected to be used in the study of structurally stable differential systems on M n . Translated from Qualitative Theory of Differentiable Dynamical Systems, Beijing, China: Science Press, 1996, by Dr. SUN Wen-xiang, School of Mathematical Sciences, Peking University, Beijing 100871, China. The Chinese version of this paper was published in Acta Scientiarum Naturalium Universitatis Pekinensis, 1963, 9: 241–265, 309–326  相似文献   

9.
We investigate the large time behavior of positive solutions with finite mass for the viscous Hamilton-Jacobi equationu t = Δu + |Δu| p ,t>0,x ∈ ℝ N , wherep≥1 andu(0,.)=u 0≥0,u 0≢0,u 0∈L 1. DenotingI ∞=lim t→∞‖u(t)‖1≤∞, we show that the asymptotic behavior of the mass can be classified along three cases as follows:
–  • ifp≤(N+2)/(N+1), thenI ∞=∞ for allu 0;
–  • if (N+2)/(N+1)<p<2, then bothI ∞=∞ andI ∞<∞ occur;
–  • ifp≥2, thenI ∞<∞ for allu 0.
We also consider a similar question for the equationu t=Δu+u p .  相似文献   

10.
We study the nonlinear Sturm-Liouville problem
where λ > 0 is an eigenvalue parameter and f(u) is a rapidly increasing function. For better understanding of the global behavior of the bifurcation branch in R+ × L 2(I), we establish precise asymptotic formulas up to the third term for the eigenvalue λ(α) associated with the eigenfunction u α with ‖u α‖2 = α, as α → ∞. We show that there exists a new type of asymptotic formula for λ (α) as α → ∞.  相似文献   

11.
The solvability of the boundary-value problem
in the space H 0 2 (0, 1) is proved under the following assumptions: p0(t)t3(1 − t)3 ∈ L(0, 1), p1(t)t(1 − t) ∈ L(0, 1), f(t)t3/2(1 − t)3/2 ∈ L(0, 1), 0 ≤ p2(t)[t(1 − t)]k+1 ∈ L(0, 1), 0 ≤ f0(t)[t(1 − t)]3/2 ∈ L(0, 1), 0 ≤ f1(t)[t(1 − t)]3m+3 ∈ L(0, 1), ϕ(u)u ≥ −c|u|, c > 0,
. Bibliography: 6 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 334, 2006, pp. 233–245.  相似文献   

12.
Ibαf ( x) =∫R ∏mj=1( bj( x) - bj( y) ) 1| x - y| n-αf ( y) dyare considered.The following priori estimates are proved.For 1 01Φ1t| {y∈Rn:| Ibαf( y) | >t}| 1q ≤csupt>01Φ1t| {y∈Rn:ML( log L) 1r ,α(‖b‖f ) ( y) >t}| 1q,where‖b‖=∏mj=1‖bj‖Oscexp Lrj,Φ( t) =t( 1 + log+t) 1r,1r =1r1+ ...+ 1rm,ML(…  相似文献   

13.
This paper summarized recent achievements obtained by the authors about the box dimensions of the Besicovitch functions given byB(t) := ∞∑k=1 λs-2k sin(λkt),where 1 ＜ s ＜ 2, λk ＞ 0 tends to infinity as k →∞ and λk satisfies λk 1/λk ≥λ＞ 1. The results show thatlimk→∞ log λk 1/log λk = 1is a necessary and sufficient condition for Graph(B(t)) to have same upper and lower box dimensions.For the fractional Riemann-Liouville differential operator Du and the fractional integral operator D-v,the results show that if λ is sufficiently large, then a necessary and sufficient condition for box dimension of Graph(D-v(B)),0 ＜ v ＜ s - 1, to be s - v and box dimension of Graph(Du(B)),0 ＜ u ＜ 2 - s, to be s uis also lim k→∞logλk 1/log λk = 1.  相似文献   

14.
In this paper the forced neutral difterential equation with positive and negative coefficients d/dt [x(t)-R(t)x(t-r)] P(t)x(t-x)-Q(t)x(t-σ)=f(t),t≥t0,is considered,where f∈L^1(t0,∞)交集C([t0,∞],R^ )and r,x,σ∈(0,∞),The sufficient conditions to oscillate for all solutions of this equation are studied.  相似文献   

15.
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   

16.
This paper studies the existence of solutions to the singular boundary value problem
, where g: (0, 1) × (0, ∞) → ℝ and h: (0, 1) × [0, ∞) → [0, ∞) are continuous. So our nonlinearity may be singular at t = 0, 1 and u = 0 and, moreover, may change sign. The approach is based on an approximation method together with the theory of upper and lower solutions. The research is supported by NNSF of China(10301033).  相似文献   

17.
By using a specially constructed cone and the fixed point index theory, this paper investigates the existence of multiple positive solutions for the third-order threepoint singular semipositone BVP:
where 1/2 < η < 1, the non-linear term ƒ(t, x): (0, 1) × (0, + ∞) → (-∞, + ∞) is continuous and may be singular att = 0,t = 1, andx = 0, also may be negative for some values oft andx, λ is a positive parameter.  相似文献   

18.
In this paper, necessary and sufficient conditions for the oscillation and asymptotic behaviour of solutions of the second order neutral delay differential equation (NDDE)
are obtained, where q, h ∈ C([0, ∞), ℝ) such that q(t) ≥ 0, r ∈ C (1) ([0, ∞), (0, ∞)), p ∈ C ([0, ∞), ℝ), G ∈ C (ℝ, ℝ) and τ ∈ ℝ+. Since the results of this paper hold when r(t) ≡ 1 and G(u) ≡ u, therefore it extends, generalizes and improves some known results.   相似文献   

19.
This paper is concerned with the large time behavior of traveling wave solutions to the Cauchy problem of generalized Benjamin–Bona–Mahony–Burgers equations
with prescribed initial data
Here v( > 0), β are constants, u  ±  are two given constants satisfying u + ≠ u − and the nonlinear function f(u) ∈C 2(R) is assumed to be either convex or concave. An algebraic time decay rate to traveling waves of the solutions of the Cauchy problem of generalized Benjamin-Bona-Mahony-Burgers equation is obtained by employing the weighted energy method developed by Kawashima and Matsumura in [6] to discuss the asymptotic behavior of traveling wave solutions to the Burgers equation. revised: May 23 and August 8, 2007  相似文献   

20.
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   

设为首页 | 免责声明 | 关于勤云 | 加入收藏

Copyright©北京勤云科技发展有限公司  京ICP备09084417号