共查询到20条相似文献,搜索用时 312 毫秒
1.
N. G. Khoma 《Ukrainian Mathematical Journal》1998,50(12):1917-1923
In three spaces, we find exact classical solutions of the boundary-value periodic problem utt - a2uxx = g(x, t) u(0, t) = u(π, t) = 0, u(x, t + T) = u(x, t), x ∈ ℝ, t ∈ ℝ. We study the periodic boundary-value problem for a quasilinear equation whose left-hand side is the d’Alembert operator
and whose right-hand side is a nonlinear operator.
Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 12, pp. 1680–1685, December, 1998. 相似文献
2.
Yu. M. Semenov 《Differential Equations》2011,47(11):1668-1674
We describe the controllability sets of linear nonautonomous systems ẋ = A(t)x + B(t)u, x ∈ ℝ
n
, u ∈ U ⊆ ℝ
m
, with entire matrix functions A(t) and B(t) and with a linear set U of control constraints. We derive a criterion for the complete controllability of these linear systems in terms of derivatives
of the entire matrix functions A(t) and B(t) at zero. This complete controllability criterion is compared with the Kalman and Krasovskii criteria. 相似文献
3.
Consider the Cauchy problem ∂u(x, t)/∂t = ℋu(x, t) (x∈ℤd, t≥ 0) with initial condition u(x, 0) ≡ 1 and with ℋ the Anderson Hamiltonian ℋ = κΔ + ξ. Here Δ is the discrete Laplacian, κ∈ (0, ∞) is a diffusion constant,
and ξ = {ξ(x): x∈ℤ
d
} is an i.i.d.random field taking values in ℝ. G?rtner and Molchanov (1990) have shown that if the law of ξ(0) is nondegenerate,
then the solution u is asymptotically intermittent.
In the present paper we study the structure of the intermittent peaks for the special case where the law of ξ(0) is (in the
vicinity of) the double exponential Prob(ξ(0) > s) = exp[−e
s
/θ] (s∈ℝ). Here θ∈ (0, ∞) is a parameter that can be thought of as measuring the degree of disorder in the ξ-field. Our main result
is that, for fixed x, y∈ℤ
d
and t→∈, the correlation coefficient of u(x, t) and u(y, t) converges to ∥w
ρ∥−2
ℓ2Σz ∈ℤd
w
ρ(x+z)w
ρ(y+z). In this expression, ρ = θ/κ while w
ρ:ℤd→ℝ+ is given by w
ρ = (v
ρ)⊗
d
with v
ρ: ℤ→ℝ+ the unique centered ground state (i.e., the solution in ℓ2(ℤ) with minimal l
2-norm) of the 1-dimensional nonlinear equation Δv + 2ρv log v = 0. The uniqueness of the ground state is actually proved only for large ρ, but is conjectured to hold for any ρ∈ (0, ∞).
empty
It turns out that if the right tail of the law of ξ(0) is thicker (or thinner) than the double exponential, then the correlation
coefficient of u(x, t) and u(y, t) converges to δ
x, y
(resp.the constant function 1). Thus, the double exponential family is the critical class exhibiting a nondegenerate correlation
structure.
Received: 5 March 1997 / Revised version: 21 September 1998 相似文献
4.
In this work we prove that the initial value problem of the Benney-Lin equation ut + uxxx + β(uxx + u xxxx) + ηuxxxxx + uux = 0 (x ∈ R, t ≥0 0), where β 〉 0 and η∈R, is locally well-posed in Sobolev spaces HS(R) for s ≥ -7/5. The method we use to prove this result is the bilinear estimate method initiated by Bourgain. 相似文献
5.
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
6.
An Application of a Mountain Pass Theorem 总被引:3,自引:0,他引:3
We are concerned with the following Dirichlet problem:
−Δu(x) = f(x, u), x∈Ω, u∈H
1
0(Ω), (P)
where f(x, t) ∈C (×ℝ), f(x, t)/t is nondecreasing in t∈ℝ and tends to an L
∞-function q(x) uniformly in x∈Ω as t→ + ∞ (i.e., f(x, t) is asymptotically linear in t at infinity). In this case, an Ambrosetti-Rabinowitz-type condition, that is, for some θ > 2, M > 0,
0 > θF(x, s) ≤f(x, s)s, for all |s|≥M and x∈Ω, (AR)
is no longer true, where F(x, s) = ∫
s
0
f(x, t)dt. As is well known, (AR) is an important technical condition in applying Mountain Pass Theorem. In this paper, without assuming
(AR) we prove, by using a variant version of Mountain Pass Theorem, that problem (P) has a positive solution under suitable
conditions on f(x, t) and q(x). Our methods also work for the case where f(x, t) is superlinear in t at infinity, i.e., q(x) ≡ +∞.
Received June 24, 1998, Accepted January 14, 2000. 相似文献
7.
Given a∈L
1(ℝ) and A the generator of an L
1-integrable family of bounded and linear operators defined on a Banach space X, we prove the existence of almost automorphic solution to the semilinear integral equation u(t)=∫
−∞
t
a(t−s)[Au(s)+f(s,u(s))]ds for each f:ℝ×X→X almost automorphic in t, uniformly in x∈X, and satisfying diverse Lipschitz type conditions. In the scalar case, we prove that a∈L
1(ℝ) positive, nonincreasing and log-convex is already sufficient. 相似文献
8.
In this paper, sufficient conditions are obtained, so that the second order neutral delay differential equation
has a positive and bounded solution, where q, h, f ∈ C ([0, ∞), ℝ) such that q(t) ≥ 0, but ≢ 0, h(t) ≤ t, h(t) → ∞ as t → ∞, r ∈ C
(1) ([0, ∞), (0, ∞)), p ∈ C
(2) [0, ∞), ℝ), G ∈ C(ℝ, ℝ) and τ ∈ ℝ+. In our work r(t) ≡ 1 is admissible and neither we assume G is non-decreasing, xG(x) > 0 for x ≠ 0, nor we take G is Lipschitzian. Hence the results of this paper improve many recent results.
相似文献
9.
10.
Let Ω be a Greenian domain in ℝ d , d≥2, or—more generally—let Ω be a connected -Brelot space satisfying axiom D, and let u be a numerical function on Ω, , which is locally bounded from below. A short proof yields the following result: The function u is the infimum of its superharmonic majorants if and only if each set {x:u(x)>t}, t∈ℝ, differs from an analytic set only by a polar set and , whenever V is a relatively compact open set in Ω and x∈V. 相似文献
11.
P. V. Tsynaiko 《Ukrainian Mathematical Journal》1998,50(9):1478-1482
We study a periodic boundary-value problem for the quasilinear equation u
tt
−u
xx
=F[u, u
t
, u
x
], u(x, 0)=u(x, π)=0, u(x + ω, t) = u(x, t), x ∈ ℝ t ∈ [0, π], and establish conditions that guarantee the validity of a theorem on unique solvability.
Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 9, pp. 1293–1296, September, 1998. 相似文献
12.
T. S. Kopaliani 《Ukrainian Mathematical Journal》2008,60(12):2006-2014
We point out that if the Hardy–Littlewood maximal operator is bounded on the space L
p(t)(ℝ), 1 < a ≤ p(t) ≤ b < ∞, t ∈ ℝ, then the well-known characterization of the spaces L
p
(ℝ), 1 < p < ∞, by the Littlewood–Paley theory extends to the space L
p(t)(ℝ). We show that, for n > 1 , the Littlewood–Paley operator is bounded on L
p(t) (ℝ
n
), 1 < a ≤ p(t) ≤ b < ∞, t ∈ ℝ
n
, if and only if p(t) = const.
Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 12, pp. 1709–1715, December, 2008. 相似文献
13.
Hydrodynamic large scale limit for the Ginzburg-Landau ∇φ interface model was established in [6]. As its next stage this
paper studies the corresponding large deviation problem. The dynamic rate functional is given by
for h=h(t,θ),t∈[0,T],θ∈?
d
, where σ=σ(u) is the surface tension for mean tilt u∈ℝ
d
. Our main tool is H
−1-method expoited by Landim and Yau [9]. The relationship to the rate functional obtained under the static situation by Deuschel
et al. [3] is also discussed.
Received: 22 February 2000 / Revised version: 19 October 2000 / Published online: 5 June 2001 相似文献
14.
Guoxiang Chen Meiying Wang 《分析论及其应用》2007,23(3):266-273
For a continuous, increasing function ω: R → R \{0} of finite exponential type, this paper introduces the set Z(A, ω) of all x in a Banach space X for which the second order abstract differential equation (2) has a mild solution such that [ω(t)]-1u(t,x) is uniformly continues on R , and show that Z(A, ω) is a maximal Banach subspace continuously embedded in X, where A ∈ B(X) is closed. Moreover, A|z(A,ω) generates an O(ω(t))strongly continuous cosine operator function family. 相似文献
15.
N. A. Chalkina 《Moscow University Mathematics Bulletin》2011,66(6):231-234
Sufficient conditions for the existence of an inertial manifold are found for the equation u
tt
+ 2γu
t
− Δu = f(u, u
t
), u = u(x, t), x ∈ Ω ⋐ ℝ
N
, u|
∂Ω = 0, t > 0 under the assumption that the function f satisfies the Lipschitz condition. 相似文献
16.
In this paper we establish some oscillation or nonoscillation criteria for the second order half-linear differential equation
where
(i) r,c ∈ C([t
0, ∞), ℝ := (− ∞, ∞)) and r(t) > 0 on [t
0, ∞) for some t
0 ⩾ 0;
(ii) Φ(u) = |u|p−2
u for some fixed number p > 1.
We also generalize some results of Hille-Wintner, Leighton and Willet. 相似文献
17.
This paper is concerned with solutions of the Hamiltonian system:
[(u)\dot]=JHu(t,u)\dot{u}=\mathcal{J}H_{u}(t,u)
, where
H(t,u)=\frac12u·Lu+W(t,u)H(t,u)=\frac{1}{2}u\cdot Lu+W(t,u)
with L being a 2N×2N symmetric matrix and W∈C
1(ℝ×ℝ2N
,ℝ) being super quadratic at infinity in u∈ℝ2N
. We use variational methods to study this problem. By virtue of some auxiliary system related to the “limit equation” of
the Hamiltonian system, we constructed linking levels of the variational functional Φ such that the minimax value E
l
based on the linking structure of Φ satisfies
0 < El < El00El0E_{l_{0}}
is the least action of the “limit equation”. Thus we can show the (PS)
c
-condition holds true for all
c < El0c相似文献
18.
Kernel regression estimation for continuous spatial processes 总被引:1,自引:0,他引:1
We investigate here a kernel estimate of the spatial regression function r(x) = E(Y
u | X
u = x), x ∈ ℝd, of a stationary multidimensional spatial process { Z
u = (X
u, Y
u), u ∈ ℝ
N
}. The weak and strong consistency of the estimate is shown under sufficient conditions on the mixing coefficients and the
bandwidth, when the process is observed over a rectangular domain of ℝN. Special attention is paid to achieve optimal and suroptimal strong rates of convergence. It is also shown that this suroptimal
rate is preserved by using a suitable spatial sampling scheme.
相似文献
19.
Consider the system with perturbation g
k
∈ ℝ
n
and output z
k
= Cx
k
. Here, A
k
,A
k
(s) ∈ ℝ
n × n
, B
k
(1) ∈ ℝ
n × p
, B
k
(2) ∈ ℝ
n × m
, C ∈ ℝ
p × n
. We construct a special Lyapunov-Krasovskii functional in order to synthesize controls u
k
(1) and u
k
(2) for which the following properties are satisfied:
$
z_{k + 1} = qz_k ,0 < q < 1(outputinvariance)
$
z_{k + 1} = qz_k ,0 < q < 1(outputinvariance)
相似文献
20.
G. I. Laptev 《Journal of Mathematical Sciences》2008,150(5):2384-2394
This paper deals with conditions for the existence of solutions of the equations
|