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1.
A. Yu. Kolesov 《Mathematical Notes》1998,63(5):614-623
We consider the boundary value problem
. Hereu ∈ ℝ2,D = diag{d
1,d
2},d
1,d
2 > 0, and the functionF is jointly smooth in (u, μ) and satisfies the following condition: for 0 <μ ≪ 1 the boundary value problem has a homogeneous (independent ofx) cycle bifurcating from a loop of the separatrix of a saddle. We establish conditions for stability and instability of this
cycle and give a geometric interpretation of these conditions.
Translated fromMatematicheskie Zametki, Vol. 63, No. 5, pp. 697–708, May, 1998.
This research was supported by the Russian Foundation for Basic Research under grant No. 96-01-00207. 相似文献
2.
G. P. Lopushanskaya 《Ukrainian Mathematical Journal》1999,51(1):51-65
We prove certain properties of solutions of the equation
in a domain ω ⊂R
3, which are similar to the properties of harmonic functions. By using the potential method, we investigate basic boundary-value
problems for this equation.
Lvov University, Lvov. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 1, pp. 48–59, January, 1999. 相似文献
3.
The paper is devoted to the study of the behavior of the following mixed problem for large values of time:
where Ω is an unbounded region of ℝ
n
with, generally speaking, noncompact boundary
; the surface Γ is star-shaped (relative to the origin), ν is the unit outer normal to ∂Ω; and the initial functionsf andg are assumed to be sufficiently smooth and finite. Under certain restrictions on the part of the boundary Γ2 constrained by the impedance condition, we establish that one can match the impedanceg≥0 (characterizing the absorption of energy by the surface Γ2) to the geometric properties of this surface so that the energy on an arbitrary compact set will decay at a rate characteristic
for the first mixed problem.
Translated fromMatematicheskie Zametki, Vol. 66, No. 3, pp. 393–400, September, 1999. 相似文献
4.
A. E. Shishkov 《Journal of Mathematical Sciences》1999,97(3):4066-4084
The paper deals with localization properties of solutions to the Cauchy problem with the initial data u0(x) ∈ L2(ℝn) for a wide class of equations in the divergence form. This class contains, e.g., the following equation:
, Restrictions are obtained, sharp in a sense, on the behavior of the function
ensuring the instantaneous compactification of the support of an arbitrary energy solution to the problem as well as the
compactification of the support after a finite waiting-time.
Translated from Trudy Seminara imeni l. G. Petrovskogo, No. 20, pp. 121–154, 1997. 相似文献
5.
G. S. Srivastava 《分析论及其应用》1996,12(4):96-104
The regular solutions of generalized axisymmetric potential equation
, a>−1/2 are called generalized axisymmetric potentials. In this paper, the characterizations of lower order and lower type
of entire GASP in terms of their approximation error {En} have been obtained. 相似文献
6.
Local and Global Existence of Solutions to Initial Value Problems of Nonlinear Kaup-Kupershmidt Equations 总被引:6,自引:0,他引:6
Shuang Ping TAO Shang Bin CUI 《数学学报(英文版)》2005,21(4):881-892
This paper is devoted to studying the initial value problems of the nonlinear Kaup Kupershmidt equations δu/δt + α1 uδ^2u/δx^2 + βδ^3u/δx^3 + γδ^5u/δx^5 = 0, (x,t)∈ E R^2, and δu/δt + α2 δu/δx δ^2u/δx^2 + βδ^3u/δx^3 + γδ^5u/δx^5 = 0, (x, t) ∈R^2. Several important Strichartz type estimates for the fundamental solution of the corresponding linear problem are established. Then we apply such estimates to prove the local and global existence of solutions for the initial value problems of the nonlinear Kaup- Kupershmidt equations. The results show that a local solution exists if the initial function u0(x) ∈ H^s(R), and s ≥ 5/4 for the first equation and s≥301/108 for the second equation. 相似文献
7.
L. P. Kuptsov 《Mathematical Notes》1974,15(3):280-286
For an equation of the form $$\begin{gathered} \frac{{\partial u}}{{\partial t}} - \sum\nolimits_{ij = 1}^n {{\text{ }}\alpha ^{ij} } \frac{{\partial ^2 u}}{{\partial x^i \partial x^j }} + \sum\nolimits_{ij = 1}^n {\beta _j^i x^i } \frac{{\partial u}}{{\partial x^i }} = 0, \hfill \\ {\text{ }}x \in R^n ,{\text{ }}t \in R^1 , \hfill \\ \end{gathered}$$ where α=(αij) is a constant nonnegative matrix andΒ=(Β i i ) is a constant matrix, subject to certain conditions, we construct a fundamental solution, similar in its structure to the fundamental solution of the heat conduction equation; we prove a mean value theorem and show that u(x0, t0) can be represented in the form of the mean value of u(x, t) with a nonnegative density over a level surface of the fundamental solution of the adjoint equation passing through the point (x0, t0); finally, we prove a parabolic maximum principle. 相似文献
8.
Local and Global Existence of Solutions to
Initial Value Problems of Modified Nonlinear
Kawahara Equations 总被引:3,自引:0,他引:3
Shuang Ping TAO Shang Bin CUI 《数学学报(英文版)》2005,21(5):1035-1044
This paper is devoted to studying the initial value problem of the modified nonlinear Kawahara equation the first partial dervative of u to t ,the second the third +α the second partial dervative of u to x ,the second the third +β the third partial dervative of u to x ,the second the thire +γ the fifth partial dervative of u to x = 0,(x,t)∈R^2.We first establish several Strichartz type estimates for the fundamental solution of the corresponding linear problem. Then we apply such estimates to prove local and global existence of solutions for the initial value problem of the modified nonlinear Karahara equation. The results show that a local solution exists if the initial function uo(x) ∈ H^s(R) with s ≥ 1/4, and a global solution exists if s ≥ 2. 相似文献
9.
Résumé Dans un célèbre papier ([3]), B. GIDAS et J.SPRUCK ont utilisé-sous des hypothèses adéquates- la technique du “blow up” pour
montrer que les solutionsu ∈C
0
∩C
1 (Ω) du problème
admettent une estimation a priori dansC
0
.
Dans ce travail, on montre que, si les solutionsu sont juste supposéesC
0
, une telle estimation a priori n’existe plus.
In a famous paper ([3]), B. GIDAS and J. SPRUCK used a “blow-up” argument to show that, under appropriate assumptions, all
the solutionsu ∈C
0
∩C
1 (Ω) of the problem
admit an a priori estimate inC
0
.
In this work, we show that, if one supposes the solutions are only inC
0
, such an a priori estimate does not hold. 相似文献
10.
Sirkka -Liisa Eriksson-Bique Kirsti Oja-Kontio 《Advances in Applied Clifford Algebras》2001,11(2):181-189
We considerC
2-solutionsf=u+iv+jw of the system
calledH-solutions introduced by H. Leutwiler. Iff is anH-solution in ω, thenf | Ω∩ℂ is holomorphic. SinceH-solutions are real analytic, a non-zeroH-solution cannot vanish in an open subdomain of ℝ3. Our object is, by the way of examples, to show that there are many kinds of null-sets ofH-solutions in ℝ3. This is in sharp contrast to a holomorphic functionf in ℂ, where the setf
−1 ({0}) consists of discrete points only unlessf≡0.
This research is supported by the Academy of Finland 相似文献
11.
Christer Borell 《Probability Theory and Related Fields》1991,87(3):403-409
Suppose a, b, and are reals witha<b and consider the following diffusion equation
相似文献
12.
In [2], [6], [7], methods are discussed for solving initial value problems
13.
V. V. Grushin 《Mathematical Notes》1971,10(2):499-501
A direct construction is given of a functionf(x1, x2) ∈ C°, such that the equation $$\frac{{\partial u}}{{\partial x_1 }} + ix_1^{2k - 1} \frac{{\partial u}}{{\partial x_2 }} = f$$ has no solution in any neighborhood of the origin; the functionf and all its derivatives vanish for x1=0. 相似文献
14.
Another method for computing the densities of integrals of motion for the Korteweg-de Vries equation
B. M. Levitan 《Mathematical Notes》1977,22(1):562-565
In the first section of this article a new method for computing the densities of integrals of motion for the KdV equation is given. In the second section the variation with respect to q of the functional ∫ 0 π w (x,t,x,;q)dx (t is fixed) is computed, where W(x, t, s; q) is the Riemann function of the problem $$\begin{gathered} \frac{{\partial ^z u}}{{\partial x^2 }} - q(x)u = \frac{{\partial ^2 u}}{{\partial t^2 }} ( - \infty< x< \infty ), \hfill \\ u|_{t = 0} = f(x), \left. {\frac{{\partial u}}{{\partial t}}} \right|_{t = 0} = 0. \hfill \\ \end{gathered} $$ 相似文献
15.
Explicit inversion formulas of Balakrishnan–Rubin type and a characterization of Bessel potentials associated with the Laplace–Bessel differential operator
are obtained. As an auxiliary tool the B-metaharmonic semigroup is introduced and some of its properties are investigated. 相似文献
16.
The results of dispersion analysis of the equation
and relevant computer-assisted experiments are presented. The existence of solutions with sharpenings (collapses) and solutions
of oscillatory type is discovered. Bibliography: 6 titles.
Translated fromObchyslyuval’na ta Prykladna Matematyka, No. 76, 1992, pp. 13–18. 相似文献
17.
Jan Brzeziński 《Rendiconti del Circolo Matematico di Palermo》1979,28(2):325-336
At first Cauchy-problem for the equation: \(L[u(X,t)] \equiv \sum\limits_{i = 1}^n {\frac{{\partial ^2 u}}{{\partial x_1^2 }} + \frac{{2v}}{{\left| X \right|^2 }}} \sum\limits_{i = 1}^n {x_i \frac{{\partial u}}{{\partial x_i }} - \frac{{\partial u}}{{\partial t}} = 0} \) wheren≥1,v—an arbitrary constant,t>0,X=(x 1, …, xn)∈E n/{0}, |X|= =(x 1 2 +…+x n 2 )1/2, with 0 being a centre of coordinate system, is studied. Basing on the above, the solution of Cauchy-Nicolescu problem is given which consist in finding a solution of the equationL p [u (X, t)]=0, withp∈N subject the initial conditions \(\mathop {\lim }\limits_{t \to \infty } L^k [u(X,t)] = \varphi _k (X)\) ,k=0, 1,…,p?1 and ?k(X) are given functions. 相似文献
18.
Bruno Pini 《Annali di Matematica Pura ed Applicata》1959,48(1):305-332
Sunto Si studia il problema della determinazione di una soluzione dell'equazione
ak(x)∂ku/∂xk=f(x, y) entro la semistriscia a≤x≤b, y≥0, che assuma assegnati valori per y=0 e per x=a, x1, x2, b (a<x1<x2<b). Analogamente si studia il problema della determinazione di una soluzione dell' equazione
ak(x)∂ku/∂xk+b(x)∂u/∂y=f(x,y), entro la medesima semistriscia, cha assuma assegnati valori per y=0 e per x=a, x1, x2, b e la cui ∂/∂y assuma assegnati valori per y=0.
A Giovanni Sansone nel suo 70mo compleanno. 相似文献
19.
It is proved that the Dirichlet problem is correct in the characteristic rectangle D
ab
= [0, a] × [0, b] for the linear hyperbolic equation
20.
A. A. Kon'kov 《Journal of Mathematical Sciences》2006,134(3):2073-2237
The paper considers solutions of the coercive inequalities
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