共查询到20条相似文献,搜索用时 15 毫秒
1.
P. I. Naumkin 《Studies in Applied Mathematics》1992,87(1):45-60
In this paper we continue to study large time asymptotic behavior of solutions to the Cauchy problem for a class of nonlinear nonlocal equations with dissipation. When t → ∞ and x → ∞ simultaneously, the asymptotics of solutions for a generalization of the Kolmogorov-Petrovsky-Piscounov equation, a model equation studied by Whitham, and an equation introduced by Ott, Sudan, and Ostrovsky is found. The character of asymptotics obtained is quasilinear. 相似文献
2.
We study asymptotics as t → ∞ of solutions to a linear, parabolic system of equations with time‐dependent coefficients in Ω × (0, ∞), where Ω is a bounded domain. On ? Ω × (0, ∞) we prescribe the homogeneous Dirichlet boundary condition. For large values of t, the coefficients in the elliptic part are close to time‐independent coefficients in an integral sense which is described by a certain function κ (t). This includes in particular situations when the coefficients may take different values on different parts of Ω and the boundaries between them can move with t but stabilize as t → ∞. The main result is an asymptotic representation of solutions for large t. As a corollary, it is proved that if κ ∈ L1(0, ∞), then the solution behaves asymptotically as the solution to a parabolic system with time‐independent coefficients (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
3.
We consider the M/G/1 queue with an arrival rate λ that depends weakly upon time, as λ = λ(εt) where ε is a small parameter. In the asymptotic limit ε → 0, we construct approximations to the probability p
n(t)that η customers are present at time t. We show that the asymptotics are different for several ranges of the (slow) time scale Τ= εt. We employ singular perturbation techniques and relate the various time scales by asymptotic matching.
This revised version was published online in June 2006 with corrections to the Cover Date. 相似文献
4.
We construct small solutions x(t) → 0 as t → 0 of nonlinear operator equations F(x(t), x(α(t)),t) = 0 with a functional perturbation α(t) of the argument. By the Newton diagram method, we reduce the problem to quasilinear operator equations with a functional
perturbation of the argument. We show that the solutions of such equations can have not only algebraic but also logarithmic
branching points and contain free parameters. The number of free parameters and the form of the solution depend on the properties
of the Jordan structure of the operator coefficients of the equation. 相似文献
5.
We consider one-phase (formal) asymptotic solutions in the Kuzmak-Whitham form for the nonlinear Klein-Gordon equation and for the Korteweg-de Vries equation.
In this case, the leading asymptotic expansion term has the form X(S(x, t)/h+Φ(x, t), I(x, t), x, t) +O(h), where h ≪ 1 is a small parameter and the phase S}(x, t) and slowly changing parameters I(x, t) are to be found from the system of “averaged” Whitham equations. We obtain the equations for the phase shift Φ(x, t) by studying the second-order correction to the leading term. The corresponding procedure for finding the phase shift is
then nonuniform with respect to the transition to a linear (and weakly nonlinear) case. Our observation, which essentially
follows from papers by Haberman and collaborators, is that if we incorporate the phase shift Φ into the phase and adjust the
parameter Ĩ by setting $
\tilde S
$
\tilde S
= S +hΦ+O(h
2),Ĩ = I + hI
1 + O(h
2), then the functions $
\tilde S
$
\tilde S
(x, t, h) and Ĩ(x, t, h) become solutions of the Cauchy problem for the same Whitham system but with modified initial conditions. These functions
completely determine the leading asymptotic term, which is X($
\tilde S
$
\tilde S
(x, t, h)/h, Ĩ(x, t, h), x, t) + O(h). 相似文献
6.
When the Hurst coefficient of a fBm B
t
H is greater than 1/2, it is possible to define a stochastic integral with respect to B
t
H as the pathwise limit of Riemann sums. In this article we consider diffusion equations of the type Xt = x0 + 0
T (Xs) dBs
H. We then construct a simple-to-use estimator of the diffusion coefficient (x), based on the number of crossings of level x of the process X
t. We then study consistency in probability of this estimator and calculate convergence rates in probability. 相似文献
7.
An algorithm of the regularization method is developed for singularly perturbed integral equations with a higher-order diagonal degeneration of the kernel. The leading term of the asymptotics is analyzed to solve the problem of initialization (that is, extraction of the class of right-hand sides and kernels of the integral operator for which the exact solution of the original equation tends to some limit function as ε → +0 on the entire time interval, including the boundary-layer zone). 相似文献
8.
We study the large-time asymptotics for solutions u( x , t) of the wave equation with Dirichlet boundary data, generated by a time-harmonic force distribution of frequency ω, in a class of domains with non-compact boundaries and show that the results obtained in [11] for a special class of local perturbations of Ω0 ? ?2 × (0,1) can be extended to arbitrary smooth local perturbations Ω of Ω0. In particular, we prove that u is bounded as t → ∞ if Ω does not allow admissible standing waves of frequency ω in the sense of [8]. This implies in connection with [8]. Theorem 3.1 that the logarithmic resonances of the unperturbed domain Ω0 at the frequencies ω = πk (k = 1, 2,…) observed in [14] can be simultaneously removed by small perturbations of the boundary. As a main step of our analysis, the determination of admissible solutions of the boundary value problem ΔU + κ2U = ? f in Ω, U = 0 on ?Ω is reduced to a compact operator equation. 相似文献
9.
S. M. Frolovitchev 《Archiv der Mathematik》2002,79(5):385-391
In this paper we deal with an asymptotics of the fundamental solution to parabolic equations of second order with a small parameter. We obtain a uniform multiplicative estimate of the difference between the fundamental solution and its asymptotics. We point out the asymptotics of the logarithmic limit in a special case when the matrix defining the principle term of the parabolic equation depends on the spatial variables and time. 相似文献
10.
M. R. Gonzalez-Dorrego 《Mathematische Nachrichten》1994,165(1):133-158
In this paper the asymptotic behaviour of the solutions of x' = A(t)x + h(t,x) under the assumptions of instability is studied, A(t) and h(t,x) being a square matrix and a vector function, respectively. The conditions for the existence of bounded solutions or solutions tending to the origin as t → ∞ are obtained. The method: the system is recasted to an equation with complex conjugate coordinates and this equation is studied by means of a suitable Lyapunov function and by virtue of the Wazevski topological method. Applications to a nonlinear differential equation of the second order are given. 相似文献
11.
《偏微分方程通讯》2013,38(7-8):1385-1408
The purpose of this paper is to study the limit in L 1(Ω), as t → ∞, of solutions of initial-boundary-value problems of the form ut ? Δw = 0 and u ∈ β(w) in a bounded domain Ω with general boundary conditions ?w/?η + γ(w) ? 0. We prove that a solution stabilizes by converging as t → ∞ to a solution of the associated stationary problem. On the other hand, since in general these solutions are not unique, we characterize the true value of the limit and comment the results on the related concrete situations like the Stefan problem and the filtration equation. 相似文献
12.
N. V. Krylov 《偏微分方程通讯》2016,41(4):644-664
Several negative results are presented concerning the solvability in Sobolev classes of the Cauchy problem for the inhomogeneous second-order uniformly parabolic equations without lower order terms in one space dimension. The main coefficient is assumed to be a bounded measurable function of (t, x) bounded away from 0. We also discuss upper and lower estimates of certain kind on the fundamental solutions of such equations. 相似文献
13.
We develop a method for constructing exact cosmological solutions of the Einstein equations based on representing them as
a second-order linear differential equation. In particular, the method allows using an arbitrary known solution to construct
a more general solution parameterized by a set of 3N constants, where N is an arbitrary natural number. The large number of free parameters may prove useful for constructing
a theoretical model that agrees satisfactorily with the results of astronomical observations. Cosmological solutions on the
Randall-Sundrum brane have similar properties. We show that three-parameter solutions in the general case already exhibit
inflationary regimes. In contrast to previously studied two-parameter solutions, these three-parameter solutions can describe
an exit from inflation without a fine tuning of the parameters and also several consecutive inflationary regimes.
__________
Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 158, No. 2, pp. 312–320, February, 2009. 相似文献
14.
We investigate the spreading of thin liquid films of power-law rheology. We construct an explicit travelling wave solution and source-type similarity solutions. We show that when the nonlinearity exponent λ for the rheology is larger than one, the governing dimensionless equation ht + (hλ+2|hxxx|λ−1hxxx)x=0 admits solutions with compact support and moving fronts. We also show that the solutions have bounded energy dissipation rate. 相似文献
15.
In this paper, we investigate problems of approximation for the solution of a system of coupled semi-linear parabolic partial differential equations that model diffusion-reaction problems in chemical engineering. Given that the solutions belong to Hs (0, ∞), we consider finite-element approximations on bounded domains (0, R(h)) such that limh→0[R(h)] = ∞. Optimal convergence estimates are found to depend on the asymptotic behaviour of the solution and its regularity near t = 0. In the L2-norm, they cannot exceed an order of O((;h2/t3/4) + h2[In h]2). For that purpose, a Wheeler-type argument is also generalized to non-coercive elliptic forms. Fully discrete schemes that preserve the positivity of the solutions are also considered. Due to the singularity at t = 0, they lead to estimates of the order O(τ1/4 + h2/t3/4). 相似文献
16.
V. B. Levenshtam 《Differential Equations》2008,44(1):54-70
We consider the problem on the periodic solutions of a system of ordinary differential equations of arbitrary order n containing terms oscillating at a frequency ω ? 1 with coefficients of the order of ω n/2. For this problem, we construct the averaged (limit) problem and justify the averaging method as well as another efficient algorithm for constructing the complete asymptotics of the solution. 相似文献
17.
Hans Ringstrm 《纯数学与应用数学通讯》2004,57(5):657-703
We consider a class of space‐times for which the essential part of Einstein's equations can be written as a wave map equation. The domain is not the standard one, but the target is hyperbolic space. One ends up with a 1 + 1 nonlinear wave equation, where the space variable belongs to the circle and the time variable belongs to the positive real numbers. The main objective of this paper is to analyze the asymptotics of solutions to these equations as t → ∞. For each point in time, the solution defines a loop in hyperbolic space, and the first result is that the length of this loop tends to 0 as t?1/2 as t → ∞. In other words, the solution in some sense becomes spatially homogeneous. However, the asymptotic behavior need not be similar to that of spatially homogeneous solutions to the equations. The orbits of such solutions are either a point or a geodesic in the hyperbolic plane. In the nonhomogeneous case, one gets the following asymptotic behavior in the upper half‐plane (after applying an isometry of hyperbolic space if necessary):
- 1 The solution converges to a point.
- 2 The solution converges to the origin on the boundary along a straight line (which need not be perpendicular to the boundary).
- 3 The solution goes to infinity along a curve y = const.
- 4 The solution oscillates around a circle inside the upper half‐plane.
18.
We show that the solutions of nonlinear diffusion equations of the form u
t
= ΔΦ(u) appearing in filtration theory may present complicated asymptotics as t → ∞ whenever we alternate infinitely many times
in a suitable manner the behavior of the nonlinearity Φ. Oscillatory behaviour is demonstrated for finite-mass solutions defined
in the whole space when they are renormalized at each time t > 0 with respect to their own second moment, as proposed in [Tos05, CDT05]; they are measured in the L
1 norm and also in the Euclidean Wasserstein distance W
2. This complicated asymptotic pattern formation can be constructed in such a way that even a chaotic behavior may arise depending
on the form of Φ.
In the opposite direction, we prove that the assumption that the asymptotic normalized profile does not depend on time implies
that Φ must be a power-law function on the appropriate range of values. In other words, the simplest asymptotic behavior implies
a homogeneous nonlinearity. 相似文献
19.
In this paper we study two transient characteristics of a Markov-fluid-driven queue, viz., the busy period and the covariance
function of the workload process. Both metrics are captured in terms of their Laplace transforms. Relying on sample-path large
deviations, we also identify the logarithmic asymptotics of the probability that the busy period lasts longer than t, as t→∞. Examples illustrating the theory are included. 相似文献
20.
We prove the existence and uniqueness of a continuous solution F = φ + w of the initial-value problem for vehicular traffic according to the nonlinear Prigogine-Herman model, where φ is a suitable t- and x-independent car distribution.We then show that the perturbation w is strongly continuous and strongly differentiable any number of times with respect to the probability of not passing q. Moreover, the derivatives (in the strong sense) satisfy linear systems.We finally investigate the behavior of w(t) as t → + ∞ and, under the assumption that the probability of not passing remains unchanged after the instant t = 0, we prove that lim ∥w(t)∥ = 0 as t → + ∞. 相似文献