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1.
When the Laplace transform F(p) of a function f(x) has no poles but is singular only on the real negative semiaxis because of a cut required to make it single-valued, the inverse transform f(x) can easily be computed by means of the integral of a real-valued function. This result is applied to the calculation of a class of exact eternal solutions of the Boltzmann equation, recently found by the authors. The new approach makes it easier to prove that these solutions are positive, as well as to study their asymptotics.  相似文献   

2.
Let {α12,…} be a sequence of real numbers outside the interval [−1,1] and μ a positive bounded Borel measure on this interval satisfying the Erd s–Turán condition μ′>0 a.e., where μ′ is the Radon–Nikodym derivative of the measure μ with respect to the Lebesgue measure. We introduce rational functions n(x) with poles {α1,…,αn} orthogonal on [−1,1] and establish some ratio asymptotics for these orthogonal rational functions, i.e. we discuss the convergence of n+1(x)/n(x) as n tends to infinity under certain assumptions on the location of the poles. From this we derive asymptotic formulas for the recurrence coefficients in the three-term recurrence relation satisfied by the orthonormal functions.  相似文献   

3.
We prove that the averaged scattering solutions to the Schrödinger equation with short-range electromagnetic potentials (V, A) where V(x) = O(|x|), A(x) = O(|x|), |x| → ∞, ρ > 1, are dense in the set of all solutions to the Schrödinger equation that are in L 2(K) where K is any connected bounded open set in ? n ,n ≥ 2, with smooth boundary. We use this result to prove that if two short-range electromagnetic potentials (V 1, A 1) and (V 2, A 2) in ? n , n ≥ 3, have the same scattering matrix at a fixed positive energy and if the electric potentials V j and the magnetic fields F j : = curl A j , j = 1, 2, coincide outside of some ball they necessarily coincide everywhere. In a previous paper of Weder and Yafaev the case of electric potentials and magnetic fields that are asymptotic sums of homogeneous terms at infinity was studied. It was proven that all these terms can be uniquely reconstructed from the singularities in the forward direction of the scattering amplitude at a fixed positive energy. The combination of the new uniqueness result of this paper and the result of Weder and Yafaev implies that the scattering matrix at a fixed positive energy uniquely determines electric potentials and magnetic fields that are a finite sum of homogeneous terms at infinity, or more generally, that are asymptotic sums of homogeneous terms that actually converge, respectively, to the electric potential and to the magnetic field.  相似文献   

4.
We study the differential-delay equation x′(t) = ?αf(x(t–1)), where α is a positive parameter and f is an odd function which decays like x?r at infinity. In particular, we consider the case r ? 2, and prove the existence of periodic solutions with special symmetries which are different from previously known periodic solutions of minimal period 4. For r = 2 we prove sharp asymptotic estimates for the minimal periods of these solutions. Our results disprove a conjecture of R. D. Nussbaum.  相似文献   

5.
In this article, we study the asymptotic behaviour of solutions to the nonlocal operator u t (x, t) = (?1) n?1 (J * Id ? 1) n (u(x, t)), x ∈ ? N , which is the nonlocal analogous to the higher order local evolution equation v t = (?1) n?1(Δ) n v. We prove that the solutions of the nonlocal problem converge to the solution of the higher order problem with the right-hand side given by powers of the Laplacian when the kernel J is rescaled in an appropriate way. Moreover, we prove that solutions to both equations have the same asymptotic decay rate as t goes to infinity.  相似文献   

6.
This work is devoted to the analysis of the asymptotic behavior of positive solutions to some problems of variable exponent reaction-diffusion equations, when the boundary condition goes to infinity (large solutions). Specifically, we deal with the equations ??u = u p(x), ??u = ?m(x)u?+?a(x)u p(x) where a(x)??? a 0 >?0, p(x)??? 1 in ??, and ??u = e p(x) where p(x)??? 0 in ??. In the first two cases p is allowed to take the value 1 in a whole subdomain ${\Omega_c\subset \Omega}$ , while in the last case p can vanish in a whole subdomain ${\Omega_c\subset \Omega}$ . Special emphasis is put in the layer behavior of solutions on the interphase ?? i :?= ??? c ???. A similar study of the development of singularities in the solutions of several logistic equations is also performed. For example, we consider ???u = ?? m(x)u?a(x) u p(x) in ??, u = 0 on ???, being a(x) and p(x) as in the first problem. Positive solutions are shown to exist only when the parameter ?? lies in certain intervals: bifurcation from zero and from infinity arises when ?? approaches the boundary of those intervals. Such bifurcations together with the associated limit profiles are analyzed in detail. For the study of the layer behavior of solutions the introduction of a suitable variant of the well-known maximum principle is crucial.  相似文献   

7.
Given a nonlinear analytic difference equation of level 1 with a formal power series solution ? 0 we associate with it a stable manifold of solutions with asymptotic expansion ? 0. This manifold can be represented by means of Borel summable series. All solutions with asymptotic expansion ? 0 in some sector can be written as certain exponential series which are called transseries. Some of their properties are investigated: are resurgence properties and Stokes transition. Analogous problems for differential equations have been studied by Costin in [7]  相似文献   

8.
We consider asymptotics for orthogonal polynomials with respect to varying exponential weights wn(x)dx = enV(x) dx on the line as n → ∞. The potentials V are assumed to be real analytic, with sufficient growth at infinity. The principle results concern Plancherel‐Rotach‐type asymptotics for the orthogonal polynomials down to the axis. Using these asymptotics, we then prove universality for a variety of statistical quantities arising in the theory of random matrix models, some of which have been considered recently in [31] and also in [4]. An additional application concerns the asymptotics of the recurrence coefficients and leading coefficients for the orthonormal polynomials (see also [4]). The orthogonal polynomial problem is formulated as a Riemann‐Hilbert problem following [19, 20]. The Riemann‐Hilbert problem is analyzed in turn using the steepest‐descent method introduced in [12] and further developed in [11, 13]. A critical role in our method is played by the equilibrium measure V for V as analyzed in [8]. © 1999 John Wiley & Sons, Inc.  相似文献   

9.
The equationx (n)(t)=(−1) n x(t) k withk>1 is considered. In the casen≦4 it is proved that solutions defined in a neighbourhood of infinity coincide withC(t−t0)−n/(k−1), whereC is a constant depending only onn andk. In the general case such solutions are Kneser solutions and can be estimated from above and below by a constant times (t−t 0)−n/(k−1). It is shown that they do not necessarily coincide withC(t−t0)−n/(k−1). This gives a negative answer to two conjectures posed by Kiguradze that Kneser solutions are determined by their value in a point and that blow-up solutions have prescribed asymptotics. Dedicated to Professor Vladimir Maz'ya on the occasion of his 60th birthday. The author was supported by the Swedish Natural Science Research Council (NFR) grant M-AA/MA 10879-304.  相似文献   

10.
We investigate the exponential decay of the tail probability P(X?>?x) of a continuous type random variable X. Let ?(s) be the Laplace–Stieltjes transform of the probability distribution function F(x)?=?P(X?≤?x) of X, and σ0 be the abscissa of convergence of ?(s). We will prove that if ?∞?0?s) on the axis of convergence are only a finite number of poles, then the tail probability decays exponentially. For the proof of our theorem, Ikehara's Tauberian theorem will be extended and applied.  相似文献   

11.
We establish some assertions of Tauberian and Abelian types which enable us to find connections between the asymptotic properties of the Laplace transform at infinity and the asymptotics of the corresponding densities of rapidly decaying distributions (at infinity or in some neighborhood of zero). As applications of our Tauberian type theorems we present asymptotics for the density f (α,ρ)(x) of “extreme” stable laws with parameters (α, ρ) for ρ = ±1 and x lying in the domain of rapid decay of f (α,ρ)(x). This asymptotics had been found in [1–5] by a more complicated method.  相似文献   

12.
In this paper we develop a time-independent approach for the study of the spectral shift function (SSF for short). We apply this method for the perturbed Stark Hamiltonian. We obtain a weak and a Weyl-type asymptotics with optimal remainder estimate of the SSF of the operator pair (P = P0 + V(x), P0 = ? h2Δ +x1), x = (x1,…, xn) where V(x) ∈ 𝒞(?n, ?) decays sufficiently fast at infinity, and h is a small positive parameter. Near a non-trapping energy λ, we give a pointwise asymptotic expansions in powers of h of the derivative of the SSF, and we compute explicitly the two leading terms.  相似文献   

13.
 We prove that the solution u of the equation u t =Δlog u, u>0, in (Ω\{x 0})×(0,T), Ω⊂ℝ2, has removable singularities at {x 0}×(0,T) if and only if for any 0<α<1, 0<a<b<T, there exist constants ρ0, C 1, C 2>0, such that C 1 |xx 0|αu(x,t)≤C 2|xx 0|−α holds for all 0<|xx 0|≤ρ0 and atb. As a consequence we obtain a sufficient condition for removable singularities at {∞}×(0,T) for solutions of the above equation in ℝ2×(0,T) and we prove the existence of infinitely many finite mass solutions for the equation in ℝ2×(0,T) when 0≤u 0L 1 (ℝ2) is radially symmetric and u 0L loc 1(ℝ2). Received: 16 December 2001 / Revised version: 20 May 2002 / Published online: 10 February 2003 Mathematics Subject Classification (1991): 35B40, 35B25, 35K55, 35K65  相似文献   

14.
We study the long time behavior of viscosity solutions of the Cauchy problem for Hamilton–Jacobi equations in ? n . We prove that if the Hamiltonian H(x, p) is coercive and strictly convex in a mild sense in p and upper semi-periodic in x, then any solution of the Cauchy problem “converges” to an asymptotic solution for any lower semi-almost periodic initial function.  相似文献   

15.
Kiefer considered the asymptotics of q-sample Cramer-Von Mises statistics for a fixed q and sample sizes tending to infinity. For univariate observations, McDonald proved the asymptotic normality of these statistics when q goes to infinity while the sample sizes stay fixed. Here we define a class of multivariate randomness statistics that generalizes the class considered by McDonald. We also prove the asymptotic normality of such statistics when the sample sizes stay fixed while q tends to infinity.  相似文献   

16.
We prove a removability result for nonlinear elliptic equations withp (x)‐type nonstandard growth and estimate the growth of solutions near a nonremovable isolated singularity. To accomplish this, we employ a Harnack estimate for possibly unbounded solutions and the fact that solutions with nonremovable isolated singularities are p (x)‐superharmonic functions (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)  相似文献   

17.
We analyze the Krawtchouk polynomials K n (x,N,p,q) asymptotically. We use singular perturbation methods to analyze them for N→∞, with appropriate scalings of the two variables x and n. In particular, the WKB method and asymptotic matching are used. We obtain asymptotic approximations valid in the whole domain [0,N]×[0,N], involving some special functions. We give numerical examples showing the accuracy of our formulas.   相似文献   

18.
We prove the global existence of weak solutions to the Navier-Stokes equations for compressible, heat-conducting flow in one space dimension with large, discontinuous initial data, and we obtain apriori estimates for these solutions which are independent of time, sufficient to determine their asymptotic behavior. In particular, we show that, as time goes to infinity, the solution tends to a constant state determined by the initial mass and the initial energy. and that the magnitudes of singularities in the solution decay to zero.  相似文献   

19.
Unbounded stochastic control problems may lead to Hamilton-Jacobi-Bellman equations whose Hamiltonians are not always defined, especially when the diffusion term is unbounded with respect to the control. We obtain existence and uniqueness of viscosity solutions growing at most like o(1+|x| p ) at infinity for such HJB equations and more generally for degenerate parabolic equations with a superlinear convex gradient nonlinearity. If the corresponding control problem has a bounded diffusion with respect to the control, then our results apply to a larger class of solutions, namely those growing like O(1+|x| p ) at infinity. This latter case encompasses some equations related to backward stochastic differential equations.  相似文献   

20.
In this paper, we study the asymptotic behavior as x1→+∞ of solutions of semilinear elliptic equations in quarter- or half-spaces, for which the value at x1=0 is given. We prove the uniqueness and characterize the one-dimensional or constant profile of the solutions at infinity. To do so, we use two different approaches. The first one is a pure PDE approach and it is based on the maximum principle, the sliding method and some new Liouville type results for elliptic equations in the half-space or in the whole space RN. The second one is based on the theory of dynamical systems.  相似文献   

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