共查询到20条相似文献,搜索用时 31 毫秒
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.
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
3.
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
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.
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
6.
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
7.
Qi-kang Ran Ai-nong FangDepartment of Applied Mathematics Shanghai University of Finance Economics Shanghai ChinaDepartment of Applied Mathematics Shanghai Jiaotong University Shanghai China 《应用数学学报(英文版)》2002,18(3):461-470
In this paper, using capacity theory and extension theorem of Lipschitz functions we first discuss the uniqueness of weak solution of nonhomogeneous quasilinear elliptic equationsin space W(θ,p)(Ω), which is bigger than W1,p(Ω). Next, using revise reverse Holder inequality we prove that if ωc is uniformly p-think, then there exists a neighborhood U of p, such that for all t ∈U, the weak solutions of equation corresponding t are bounded uniformly. Finally, we get the stability of weak solutions on exponent p. 相似文献
8.
We consider the existence and uniqueness of singular solutions for equations of the formu
1=div(|Du|p−2
Du)-φu), with initial datau(x, 0)=0 forx⇑0. The function ϕ is a nondecreasing real function such that ϕ(0)=0 andp>2.
Under a growth condition on ϕ(u) asu→∞, (H1), we prove that for everyc>0 there exists a singular solution such thatu(x, t)→cδ(x) ast→0. This solution is unique and is called a fundamental solution. Under additional conditions, (H2) and (H3), we show the
existence of very singular solutions, i.e. singular solutions such that ∫|x|≤r
u(x,t)dx→∞ ast→0. Finally, for functions ϕ which behave like a power for largeu we prove that the very singular solution is unique. This is our main result.
In the case ϕ(u)=u
q, 1≤q, there are fundamental solutions forq<p*=p-1+(p/N) and very singular solutions forp-1<q<p*. These ranges are optimal.
Dedicated to Professor Shmuel Agmon 相似文献
9.
Maria E. Schonbek 《Mathematische Annalen》2006,336(3):505-538
This paper considers the existence and large time behavior of solutions to the convection-diffusion equation u
t
−Δu+b(x)·∇(u|u|
q
−1)=f(x, t) in ℝ
n
×[0,∞), where f(x, t) is slowly decaying and q≥1+1/n (or in some particular cases q≥1). The initial condition u
0 is supposed to be in an appropriate L
p
space. Uniform and nonuniform decay of the solutions will be established depending on the data and the forcing term.This work is partially supported by an AMO Grant 相似文献
10.
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
11.
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
12.
We study the large time behaviour of nonnegative solutions of the Cauchy problemu
t=Δu
m −u
p,u(x, 0)=φ(x). Specifically we study the influence of the rate of decay ofφ(x) for large |x|, and the competition between the diffusion and the absorption term. 相似文献
13.
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
14.
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
15.
I. V. Filimonova 《Journal of Mathematical Sciences》2007,143(4):3415-3428
One considers a semilinear parabolic equation u
t
= Lu − a(x)f(u) or an elliptic equation u
tt
+ Lu − a(x)f(u) = 0 in a semi-infinite cylinder Ω × ℝ+ with the nonlinear boundary condition
, where L is a uniformly elliptic divergent operator in a bounded domain Ω ∈ ℝn; a(x) and b(x) are nonnegative measurable functions in Ω. One studies the asymptotic behavior of solutions of such boundary-value problems
for t → ∞.
__________
Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 26, pp. 368–389, 2007. 相似文献
16.
On critical Fujita exponents for heat equations with nonlinear flux conditions on the boundary 总被引:1,自引:0,他引:1
We consider nonnegative solutions of initial-boundary value problems for parabolic equationsu
t=uxx, ut=(um)xxand
(m>1) forx>0,t>0 with nonlinear boundary conditions−u
x=up,−(u
m)x=upand
forx=0,t>0, wherep>0. The initial function is assumed to be bounded, smooth and to have, in the latter two cases, compact support. We prove
that for each problem there exist positive critical valuesp
0,pc(withp
0<pc)such that forp∃(0,p
0],all solutions are global while forp∃(p0,pc] any solutionu≢0 blows up in a finite time and forp>p
csmall data solutions exist globally in time while large data solutions are nonglobal. We havep
c=2,p
c=m+1 andp
c=2m for each problem, whilep
0=1,p
0=1/2(m+1) andp
0=2m/(m+1) respectively.
This work was done during visits of the first author to Iowa State University and the Institute for Mathematics and its Applications
at the University of Minnesota. The second author was supported in part by NSF Grant DMS-9102210. 相似文献
17.
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
18.
In this paper, we study the L
p
(2 ⩽ p ⩽ +∞) convergence rates of the solutions to the Cauchy problem of the so-called p-system with nonlinear damping. Precisely, we show that the corresponding Cauchy problem admits a unique global solution (v(x,t), u(x,t)) and such a solution tends time-asymptotically to the corresponding nonlinear diffusion wave (ῡ(x,t), ū(x,t)) governed by the classical Darcys’s law provided that the corresponding prescribed initial error function (w
0(x), z
0(x)) lies in (H
3 × H
2) (ℝ) and |v
+ − v
−| + ∥w
0∥3 + ∥z
0∥2 is sufficiently small. Furthermore, the L
p
(2 ⩽ p ⩽ +∞) convergence rates of the solutions are also obtained. 相似文献
19.
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
20.
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