共查询到20条相似文献,搜索用时 46 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.
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
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.
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
5.
We study the Cauchy problem for the nonlinear dissipative equations (0.1) uo∂u-αδu + Β|u|2/n
u = 0,x ∃ Rn,t } 0,u(0,x) = u0(x),x ∃ Rn, where α,Β ∃ C, ℜα 0. We are interested in the dissipative case ℜα 0, and ℜδ(α,Β)≥ 0, θ = |∫ u0(x)dx| ⊋ 0, where δ(α, Β) = ##|α|n-1nn/2 / ((n + 1)|α|2 + α2
n/2. Furthermore, we assume that the initial data u0 ∃ Lp are such that (1 + |x|)αu0 ∃ L1, with sufficiently small norm ∃ = (1 + |x|)α u0 1 + u0 p, wherep 1, α ∃ (0,1). Then there exists a unique solution of the Cauchy problem (0.1)u(t, x) ∃ C ((0, ∞); L∞) ∩ C ([0, ∞); L1 ∩ Lp) satisfying the time decay estimates for allt0 u(t)||∞ Cɛt-n/2(1 + η log 〈t〉)-n/2, if hg = θ2/n 2π ℜδ(α, Β) 0; u(t)||∞ Cɛt-n/2(1 + Μ log 〈t〉)-n/4, if η = 0 and Μ = θ4/n 4π)2 (ℑδ(α, Β))2 ℜ((1 + 1/n) υ1-1 υ2) 0; and u(t)||∞ Cɛt-n/2(1 + κ log 〈t〉)-n/6, if η = 0, Μ = 0, κ 0, where υl,l = 1,2 are defined in (1.2), κ is a positive constant defined in (2.31). 相似文献
6.
In the space of functions B
a3+={g(x, t)=−g(−x, t)=g(x+2π, t)=−g(x, t+T3/2)=g(x, −t)}, we establish that if the condition aT
3
(2s−1)=4πk, (4πk, a (2s−1))=1, k ∈ ℤ, s ∈ ℕ, is satisfied, then the linear problem u
u
−a
2
u
xx
=g(x, t), u(0, t)=u(π, t)=0, u(x, t+T
3
)=u(x, t), ℝ2, is always consistent. To prove this statement, we construct an exact solution in the form of an integral operator.
Ternopol’ Pedagogical Institute, Ternopol’. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 2, pp. 302–308,
Feburary, 1997
Ternopol’ Pedagogical Institute, Ternopol’. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 2, pp. 302–308,
Feburary, 1997 相似文献
7.
The object of this paper is to study the existence of a solution of the Cauchy problemu
t=Δum−up, u(x,0)=δ(x) and when a solution exists, to study its behaviour ast→0. 相似文献
8.
We prove that the so-called Smoluchowski-Kramers approximation holds for a class of partial differential equations perturbed
by a non-Gaussian noisy term. Namely, we show that the solution of the one-dimensional semi-linear stochastic damped wave
equations
, u(0) = u0, ut (0) = v0, endowed with Dirichlet boundary conditions, converges as the parameter μ goes to zero to the solution of the semi-linear
stochastic heat equation
, u(0) = u0, endowed with Dirichlet boundary conditions.
Dedicated to Giuseppe Da Prato on the occasion of his 70th birthday 相似文献
9.
John A. Crow 《偏微分方程通讯》2013,38(8-9):1529-1548
The existence and uniqueness of long time classical solutions of the Cauchy problem ut t+μut = div(a(u)▽u), where a(u) = 1+u and μ ≥ 0, are studied for the case of two space dimensions. Let the initial data u(0,.) = φ and ut(0,.) = ψ be supported compactly on R2. Then for every T > 0, such a solution exists on [0,T] whenever (φ,ψ) is small enough in H4 (R2) x H3(R2). A result on the asymptotic relation between the maximal T and the size of the initial data is given. 相似文献
10.
We investigate the linear periodic problem u
tt
−u
xx
=F(x, t), u(x+2π, t)=u(x, t+T)=u(x, t), ∈ ℝ2, and establish conditions for the existence of its classical solution in spaces that are subspaces of the Vejvoda-Shtedry
spaces.
Ternopol’ Pedagogical Institute, Ternopol’. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 2, pp. 302–308,
February, 1997. 相似文献
11.
In this paper we consider the heat flow of harmonic maps between two compact Riemannian Manifolds M and N (without boundary)
with a free boundary condition. That is, the following initial boundary value problem ∂1,u −Δu = Γ(u)(∇u, ∇u) [tT Tu
uN, on M × [0, ∞), u(t, x) ∈ Σ, for x ∈ ∂M, t > 0, ∂u/t6n(t, x) ⊥u Tu(t,x) Σ, for x ∈ ∂M, t > 0, u(o,x) = uo(x), on M, where Σ is a smooth submanifold without boundary in N and n is a unit normal vector field of M along ∂M.
Due to the higher nonlinearity of the boundary condition, the estimate near the boundary poses considerable difficulties,
even for the case N = ℝn, in which the nonlinear equation reduces to ∂tu-Δu = 0.
We proved the local existence and the uniqueness of the regular solution by a localized reflection method and the Leray-Schauder
fixed point theorem. We then established the energy monotonicity formula and small energy regularity theorem for the regular
solutions. These facts are used in this paper to construct various examples to show that the regular solutions may develop
singularities in a finite time. A general blow-up theorem is also proven. Moreover, various a priori estimates are discussed
to obtain a lower bound of the blow-up time. We also proved a global existence theorem of regular solutions under some geometrical
conditions on N and Σ which are weaker than KN <-0 and Σ is totally geodesic in N. 相似文献
12.
Yong-hui Wu 《Mathematical Methods in the Applied Sciences》1997,20(11):933-943
In this paper, we consider the Cauchy problem: (ECP) ut−Δu+p(x)u=u(x,t)∫u2(y,t)/∣x−y∣dy; x∈ℝ3, t>0, u(x, 0)=u0(x)⩾0 x∈ℝ3, (0.2) The stationary problem for (ECP) is the famous Choquard–Pekar problem, and it has a unique positive solution ū(x) as long as p(x) is radial, continuous in ℝ3, p(x)⩾ā>0, and lim∣x∣→∞p(x)=p¯>0. In this paper, we prove that if the initial data 0⩽u0(x)⩽(≢)ū(x), then the corresponding solution u(x, t) exists globally and it tends to the zero steady-state solution as t→∞, if u0(x)⩾(≢)ū(x), then the solution u(x,t) blows up in finite time. © 1997 B. G. Teubner Stuttgart–John Wiley & Sons Ltd. 相似文献
13.
Evangelos A. Latos Dimitrios E. Tzanetis 《NoDEA : Nonlinear Differential Equations and Applications》2010,17(2):137-151
We investigate the behaviour of solution u = u(x, t; λ) at λ = λ* for the non-local porous medium equation ${u_t = (u^n)_{xx} + {\lambda}f(u)/({\int_{-1}^1} f(u){\rm d}x)^2}We investigate the behaviour of solution u = u(x, t; λ) at λ = λ* for the non-local porous medium equation ut = (un)xx + lf(u)/(ò-11 f(u)dx)2{u_t = (u^n)_{xx} + {\lambda}f(u)/({\int_{-1}^1} f(u){\rm d}x)^2} with Dirichlet boundary conditions and positive initial data. The function f satisfies: f(s),−f ′ (s) > 0 for s ≥ 0 and s
n-1
f(s) is integrable at infinity. Due to the conditions on f, there exists a critical value of parameter λ, say λ*, such that for λ > λ* the solution u = u(x, t; λ) blows up globally in finite time, while for λ ≥ λ* the corresponding steady-state problem does not have any solution.
For 0 < λ < λ* there exists a unique steady-state solution w = w(x; λ) while u = u(x, t; λ) is global in time and converges to w as t → ∞. Here we show the global grow-up of critical solution u* = u(x, t; λ*) (u* (x, t) → ∞, as t → ∞ for all x ? (-1,1){x\in(-1,1)}. 相似文献
14.
Sopra un principio di trasformazione integrale dei problemi differenziali ed alcune sue applicazioni
Guido Ascoli 《Annali di Matematica Pura ed Applicata》1955,40(1):167-182
Sunto Sviluppando ampiamente un'idea diWhittaker, si studiano trasformazioni del tipo
che mutano le soluzioni di una equazione differenziale lineareL
tx+λM
tx=0, appartenenti ad una classe lineare (G), opportunamente precisata, in soluzioni di una seconda equazione differenzialeP
uy+λQ
uy=0, appartenenti ad un'altra classe lineare (Γ). Il primo problema si suppone autoaggiunto. Il nucleo K è una soluzione dell'equazione
(L
t
Q
u −M
t
P
u)K=0, di classe (G) come funzione di t, di classe (Γ) come funzione di u. Il procedimento permette di stabilire proprietà
integrali in base a sole verifiche qualitative; ne vengono fatte applicazioni alle funzioni diMathieue diBessele ai polinomi diLegendree diHermite.
A Mauro Picone nel suo 70mo compleanno. 相似文献
15.
N. G. Khoma 《Ukrainian Mathematical Journal》1995,47(12):1964-1967
We study a periodic boundary-value problem for the quasilinear equationu
tt–uxx=F[u, ut], u(0, t)=u(, t)=0,u(x, t+2)=u(x, t). We establish conditions that guarantee the validity of the uniqueness theorem.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 12, pp. 1717–1719, December, 1995. 相似文献
16.
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
17.
Uğur Yüksel 《Advances in Applied Clifford Algebras》2010,20(1):201-209
This paper deals with the initial value problem of the type
\frac?u(t,x) ?t = Lu(t,x), u(0,x) = u0(x)\frac{\partial u(t,x)} {\partial t} = {\mathcal{L}}u(t,x), \quad u(0,x) = u_{0}(x) 相似文献
18.
Khoang Van Lai 《Ukrainian Mathematical Journal》1990,42(8):1006-1015
We construct an approximate solution for an initial boundary-value problem of the formu
t
(x, t) + a (x, t) ux
(x, t)=b (x, t, u), u (x, 0)=u0 (x),u (0,t)=u1 (t) by the method of characteristics. It is proved that the approximate solution converges to the exact one with rate of convergence of second order.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 8, pp. 1128–1138, August, 1990. 相似文献
19.
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
20.
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
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