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1.
N. G. Khoma 《Ukrainian Mathematical Journal》1998,50(11):1755-1764
In three spaces, we obtain exact classical solutions of the boundary-value periodic problem u
tt−a
2
u
xx=g(x,t), u(0,t)=u(π,t)=0, u(x,t+T)=u(x,t)=0, x,t∈ĝ
Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 11, pp. 1537–1544, November, 1998. 相似文献
2.
I. V. Dombrovskii 《Ukrainian Mathematical Journal》1999,51(11):1779-1781
We establish conditions for the existence of a smooth solution of a quasilinear hyperbolic equationu
tt
- uxx = ƒ(x, t, u, u, u
x),u (0,t) = u (π,t) = 0,u (x, t+ T) = u (x, t), (x, t) ∈ [0, π] ×R, and prove a theorem on the existence and uniqueness of a solution.
Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 11, pp. 1574–1576, November, 1999. 相似文献
3.
S. G. Khoma 《Ukrainian Mathematical Journal》2000,52(4):655-657
We find conditions for the existence of the classical solution of the boundary-value problem u
tt
-u
xx
= f(x,t), u(0,t)=u(π, t)=0, u(x, 0)=u(x, 2π). 相似文献
4.
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
5.
A. O. Botyuk 《Ukrainian Mathematical Journal》1997,49(7):1120-1124
We study the boundary-value perlodic problem u
tt
−u
xx
=F(x, t), u(0, t)=u(π, t)=0, u(x, t+T)=u(x, t), (x, t) ∈ R
2. By using the Vejvoda-Shtedry operator, we determine a solution of this problem.
Ternopol Pedagogical Institute, Temopol. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 7, pp. 998–1001,
July, 1997. 相似文献
6.
In an attempt to study the scope of a theorem due to Pati, the authors have established that φ(t) logK|t∈B
u
V in (0,π)⟹ΣA
n
(x) is |C, 0,β| forβ>1, at the pointt = x. 相似文献
7.
We study the boundary-value problemu
tt
-u
xx
=g(x, t),u(0,t) =u (π,t) = 0,u(x, t +T) =u(x, t), 0 ≤x ≤ π,t ∈ ℝ. We findexact classical solutions of this problem in three Vejvoda-Shtedry spaces, namely, in the classes of, and-periodic functions (q and s are natural numbers). We obtain the results only for sets of periods, and which characterize the classes of π-, 2π -, and 4π-periodic functions.
Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 2, pp. 281–284, February, 1999. 相似文献
8.
Tomasz Komorowski 《Probability Theory and Related Fields》2001,121(4):525-550
We consider the asymptotic behavior of the solutions ofscaled convection-diffusion equations ∂
t
u
ɛ
(t, x) = κΔ
x
(t, x) + 1/ɛV(t/ɛ2,xɛ) ·∇
x
u
ɛ
(t, x) with the initial condition u
ɛ(0,x) = u
0(x) as the parameter ɛ↓ 0. Under the assumptions that κ > 0 and V(t, x), (t, x) ∈R
d
is a d-dimensional,stationary, zero mean, incompressible, Gaussian random field, Markovian and mixing in t we show that the laws of u
ɛ(t,·), t≥ 0 in an appropriate functional space converge weakly, as ɛ↓ 0, to a δ-type measureconcentrated on a solution of a certain
constant coefficient heat equation.
Received: 23 March 2000 / Revised version: 5 March 2001 / Published online: 9 October 2001 相似文献
9.
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
10.
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
11.
C. Boldrighini R. A. Minlos A. Pellegrinotti 《Probability Theory and Related Fields》1997,109(2):245-273
Summary We consider a model of random walk on ℤν, ν≥2, in a dynamical random environment described by a field ξ={ξ
t
(x): (t,x)∈ℤν+1}. The random walk transition probabilities are taken as P(X
t
+1= y|X
t
= x,ξ
t
=η) =P
0( y−x)+ c(y−x;η(x)). We assume that the variables {ξ
t
(x):(t,x) ∈ℤν+1} are i.i.d., that both P
0(u) and c(u;s) are finite range in u, and that the random term c(u;·) is small and with zero average. We prove that the C.L.T. holds almost-surely, with the same parameters as for P
0, for all ν≥2. For ν≥3 there is a finite random (i.e., dependent on ξ) correction to the average of X
t
, and there is a corresponding random correction of order to the C.L.T.. For ν≥5 there is a finite random correction to the covariance matrix of X
t
and a corresponding correction of order to the C.L.T.. Proofs are based on some new L
p
estimates for a class of functionals of the field.
Received: 4 January 1996/In revised form: 26 May 1997 相似文献
12.
For the equation K(t)u
xx
+ u
tt
− b
2
K(t)u = 0 in the rectangular domain D = “(x, t)‖ 0 < x < 1, −α < t < β”, where K(t) = (sgnt)|t|
m
, m > 0, and b > 0, α > 0, and β > 0 are given real numbers, we use the spectral method to obtain necessary and sufficient conditions for the unique solvability
of the boundary value problem u(0, t) = u(1, t), u
x
(0, t) = u
x
(1, t), −α ≤ t ≤ β, u(x, β) = φ(x), u(x,−α) = ψ(x), 0 ≤ x ≤ 1. 相似文献
13.
H. Brézis 《Israel Journal of Mathematics》1971,9(4):513-534
Let φ be a convex l.s.c. function fromH (Hilbert) into ] - ∞, ∞ ] andD(φ)={u ∈H; φ(u)<+∞}. It is proved that for everyu
0 ∈D(φ) the equation − (du/dt)(t ∈ ∂φ(u(t)),u(0)=u
0 has a solution satisfying ÷(du(t)/dt)÷ ≦(c
1/t)+c
2. The behavior ofu(t) in the neighborhood oft=0 andt=+∞ as well as the inhomogeneous equation (du(t)/dt)+∂φ(u(t)) ∈f(t) are then studied. Solutions of some nonlinear boundary value problems are given as applications.
相似文献
14.
Pierre Collet Servet Martínez Jaime San Martín 《Probability Theory and Related Fields》2000,116(3):303-316
We study the asymptotic behaviour of the transition density of a Brownian motion in ?, killed at ∂?, where ?
c
is a compact non polar set. Our main result concern dimension d = 2, where we show that the transition density p
?
t
(x, y) behaves, for large t, as
u(x)u(y)(t(log t)2)−1 for x, y∈?, where u is the unique positive harmonic function vanishing on (∂?)
r
, such that u(x) ∼ log ∣x∣.
Received: 29 January 1999 / Revised version: 11 May 1999 相似文献
15.
We study the problem of existence of periodic and almost periodic solutions of the scalar equation x′ (t) = − δx(t) + pmax
u∈[t − h, t]
x(u) + f(t) where δ, p ∈ R, with a periodic (almost periodic) perturbation f(t). For these solutions, we establish conditions of global exponential stability and prove uniqueness theorems.
Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 6, pp. 747–754, June, 1998. 相似文献
16.
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
17.
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
18.
Shu-Yu Hsu 《Mathematische Annalen》2006,334(1):153-197
Let a1,a2, . . . ,am ∈ ℝ2, 2≤f ∈ C([0,∞)), gi ∈ C([0,∞)) be such that 0≤gi(t)≤2 on [0,∞) ∀i=1, . . . ,m. For any p>1, we prove the existence and uniqueness of solutions of the equation ut=Δ(logu), u>0, in satisfying and logu(x,t)/log|x|→−f(t) as |x|→∞, logu(x,t)/log|x−ai|→−gi(t) as |x−ai|→0, uniformly on every compact subset of (0,T) for any i=1, . . . ,m under a mild assumption on u0 where We also obtain similar existence and uniqueness of solutions of the above equation in bounded smooth convex domains of ℝ2 with prescribed singularities at a finite number of points in the domain. 相似文献
19.
We consider the periodic boundary-value problem u
tt
− u
xx
= g(x, t), u(0, t) = u(π, t) = 0, u(x, t + ω) = u(x, t). By representing a solution of this problem in the form u(x, t) = u
0(x, t) + ũ(x, t), where u
0(x, t) is a solution of the corresponding homogeneous problem and ũ(x, t) is the exact solution of the inhomogeneous equation such that ũ(x, t + ω) u x = ũ(x, t), we obtain conditions for the solvability of the inhomogeneous periodic boundary-value problem for certain values of the
period ω. We show that the relation obtained for a solution includes known results established earlier.
__________
Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 7, pp. 912–921, July, 2005. 相似文献
20.
Tetsutaro Shibata 《Journal d'Analyse Mathématique》2001,83(1):109-120
We consider the perturbed elliptic Sine-Gordon equation on an interval-u″t+γsinu(t)=μf(u(t)),t ∈I := (-T, T),u(t) > 0,t ∈I,u(±T)=0 where λ, μ>0 are parameters andT>0 is a constant. By applying variational methods subject to the constraint depending on λ, we obtain eigenpairs (μ,u)=(μ(λ),u
λ) which solve this eigenvalue problem for a given λ>0. Then we study the asymptotic behavior ofu
λ and μ(λ) as λ→∞. Especially, we study the location of interior transition layers ofu
λ as λ→∞.
This research has been supported by the Japan Society for the Promotion of Science. 相似文献