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1.
The authors localize the blow-up points of positive solutions of the systemu
t
=Δu,v
t
=Δv with conditions
at the boundary of a bounded smooth domain Θ under some restrictions off andg and the initial data (Δu
0, Δν0>c>0).
If Θ is a ball, the hypothesis on the initial data can be removed.
Supported by Universidad de Buenos Aires under grant EX071 and CONICET. 相似文献
2.
M. A. Raupp R. A. Feijóo C. A. de Moura 《Bulletin of the Brazilian Mathematical Society》1978,9(2):39-61
In this paper we present the analysis of an algorithm of Uzawa type to compute solutions of the quasi variational inequality $$\begin{gathered} (QVI)\left( {\frac{{\partial ^2 u}}{{\partial t^2 }},\upsilon - \frac{{\partial u}}{{\partial t}}} \right) + \left( {\frac{{\partial u}}{{\partial x}},\frac{{\partial \upsilon }}{{\partial x}} - \frac{{\partial ^2 u}}{{\partial x\partial t}}} \right) + \left( {\frac{{\partial ^2 u}}{{\partial x\partial t}},\frac{{\partial \upsilon }}{{\partial x}} - \frac{{\partial ^2 u}}{{\partial x\partial t}}} \right) + \hfill \\ + \left[ {u(1,t) + \frac{{\partial u}}{{\partial t}}(1,t)} \right]\left[ {\upsilon (1) - \frac{{\partial u}}{{\partial t}}(1,t)} \right] + J(u;\upsilon ) - J\left( {u;\frac{{\partial u}}{{\partial t}}} \right) \geqslant \hfill \\ \geqslant \left( {f,\upsilon - \frac{{\partial u}}{{\partial t}}} \right) + F(t)\left[ {\upsilon (0) - \frac{{\partial u}}{{\partial t}}(0,t)} \right],t > 0,\forall \upsilon \in H^1 (0,1), \hfill \\ \end{gathered} $$ which is a model for the dynamics of a pile driven into the ground under the action of a pile hammer. In (QVI) (...) is the scalar product inL 2(0, 1) andJ(u;.) is a convex functional onH 1(0, 1), for eachu, describing the soil-pile friction effect. 相似文献
3.
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
4.
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
5.
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
6.
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
7.
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
8.
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
9.
By constructing the corresponding Green's function in a trapezoidal domain, we establish the existence of self-adjoint realizations of
incorporating boundary conditions of the formu(s, 0)=u(s, T)=0. Such operators correspond to the historically important concept of a simultaneous crossing of the axis for vibrating strings. 相似文献
10.
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
11.
A. P. Oskolkov 《Journal of Mathematical Sciences》1985,28(5):751-758
One proves the global unique solvability in class \(W_\infty ^1 (0,T;C^{2,d} (\bar \Omega ) \cap H(\Omega ))\) of the initial-boundary-value problem for the quasilinear system $$\frac{{\partial \vec \upsilon }}{{\partial t}} + \upsilon _k \frac{{\partial \vec \upsilon }}{{\partial x_k }} - \mu _1 \frac{{\partial \Delta \vec \upsilon }}{{\partial t}} - \int\limits_0^t {K(t - \tau )\Delta \vec \upsilon (\tau )d\tau + grad p = \vec f,di\upsilon \bar \upsilon = 0,\upsilon , > 0.}$$ This system described the nonstationary flows of the elastic-viscous Kelvin-Voigt fluids with defining relation $$\left( {1 + \sum\limits_{\ell = 1}^L {\lambda _\ell } \frac{{\partial ^\ell }}{{\partial t^\ell }}} \right)\sigma = 2\left( {v + \sum\limits_{m = 1}^{L + 1} {\user2{\ae }_m } \frac{{\partial ^m }}{{\partial t^m }}} \right)D,L = 0,1,2,...;\lambda _L ,\user2{\ae }_{L + 1} > 0.$$ 相似文献
12.
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
13.
Xiu Hui YANG Fu Cai LI Chun Hong XIE 《数学学报(英文版)》2005,21(4):923-928
Abstract In this paper, we investigate the positive solutions of strongly coupled nonlinear parabolic systems with nonlinear boundary conditions: {ut-a(u, v)△u=g(u, v), vt-b(u, v)△v=h(u, v), δu/δη=d(u, v), δu/δη=f(u, v).Under appropriate hypotheses on the functions a, b, g, h, d and f, we obtain that the solutions may exist globally or blow up in finite time by utilizing upper and lower solution techniques. 相似文献
14.
Yosef Stein 《Journal d'Analyse Mathématique》1990,54(1):237-245
The main result of this work is the following theorem: LetP,QɛC[x, y] satisfy the Jacobian identity
相似文献
15.
A. P. Oskolkov 《Journal of Mathematical Sciences》1978,10(1):95-103
For the system of Navier-Stokes-Voigt equations $$\frac{{\partial \vec v}}{{\partial t}} - v\Delta \vec v - \aleph \frac{{\partial \Delta \vec v}}{{\partial t}} + v_\kappa \frac{{\partial \Delta \vec v}}{{\partial x_\kappa }} + grad \rho = 0, div \vec v = 0$$ and the BBM equation $$\frac{{\partial v}}{{\partial t}} + v\frac{{\partial \Delta v}}{{\partial x}} - \frac{{\partial ^3 v}}{{\partial t\partial x^2 }} = 0$$ characteristic functions \(\mathcal{F}\left( {\vec \theta ;t} \right)\) of the measure μt(ω)=μ(V ?1 t (ω)), describing the evolution in time of the probability measure μ(ω) defined on the set of initial conditions for the first initial boundary-value problem for system (1) or Eq. (2) are constructed and investigated. It is shown that the characteristic functions \(\mathcal{F}\left( {\vec \theta ;t} \right)\) constructed satisfy partial differential equations with an infinite number of independent variables (t; θ1,θ2,...) [the statistical equations of E. Hopf for the system (1) or Eq. (2)]. 相似文献
16.
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
17.
V. A. Kondratiev 《Journal of Mathematical Sciences》2006,135(1):2666-2674
The equations under consideration have the following structure:
18.
L. A. Medeiros J. Limaco S. B. Menezes 《Journal of Computational Analysis and Applications》2002,4(3):211-263
Dedicated to Professor Jacque-Louis Lions on the occasion of his 70th birthday
We consider a mixed problem for the operator
19.
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
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