共查询到20条相似文献,搜索用时 15 毫秒
1.
N. I. Karachalios N. B. Zographopoulos 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2005,56(1):11-30
We study a real Ginzburg-Landau equation, in a bounded domain of
with a variable, generally non-smooth diffusion coefficient having a finite number of zeroes. By using the compactness of the embeddings of the weighted Sobolev spaces involved in the functional formulation of the problem, and the associated energy equation, we show the existence of a global attractor. The extension of the main result in the case of an unbounded domain is also discussed, where in addition, the diffusion coefficient has to be unbounded. Some remarks for the case of a complex Ginzburg-Landau equation are given.Received: May 6, 2002; revised: October 3, 2002 相似文献
2.
Todd Kapitula 《纯数学与应用数学通讯》1994,47(6):831-841
I consider the nonlinear stability of plane wave solutions to a Ginzburg-Landau equation with additional fifth-order terms and cubic terms containing spatial derivatives. I show that, under the constraint that the diffusion coefficient be real, these waves are stable. Furthermore, it is shown that the radial component of the perturbation decays at a faster rate than the phase component of the perturbation as t → ∞. The result is also applicable to the classical Ginzburg-Landau equation. © 1994 John Wiley & Sons, Inc. 相似文献
3.
Youjun Xu Xinguang Zhang Dongyuan Liu 《Nonlinear Analysis: Theory, Methods & Applications》2009,71(11):5885-5894
In this paper, we consider the existence of insensitizing control for a semilinear heat equation involving gradient terms in unbounded domain Ω. In this case, we prove the existence of controls insensitizing the L2-norm of the observation of the solution in an open subset of the domain. The proofs of the main results in this paper involve such inequalities and rely on the study of these linear problems and appropriate fixed point arguments. 相似文献
4.
Nikos I. Karachalios Nikos B. Zographopoulos 《Calculus of Variations and Partial Differential Equations》2006,25(3):361-393
We study the dynamics of a degenerate parabolic equation with a variable, generally non-smooth diffusion coefficient, which
may vanish at some points or be unbounded. We show the existence of a global branch of nonnegative stationary states, covering
both the cases of a bounded and an unbounded domain. The global bifurcation of stationary states, implies-in conjuction with
the definition of a gradient dynamical system in the natural phase space-that at least in the case of a bounded domain, any
solution with nonnegative initial data tends to the trivial or the nonnegative equilibrium. Applications of the global bifurcation
result to general degenerate semilinear as well as to quasilinear elliptic equations, are also discussed.
Mathematics Subject Classification (1991) 35B40, 35B41, 35R05 相似文献
5.
A. V. Filinovskii 《Journal of Mathematical Sciences》2014,197(3):435-446
Let Ω ? ? n , n?≥?2, be an unbounded domain with a smooth (possibly noncompact) star-shaped boundary Γ. For the first mixed problem for a hyperbolic equation with an unbounded coefficient with power growth at infinity, the large-time behavior of the solutions is studied. Estimates for the resolvent of the spectral problem are obtained for various values of the parameters. 相似文献
6.
E. A. Sheina 《Differential Equations》2010,46(3):415-427
In the present paper, we consider a quasilinear elliptic equation in ℝ
N
with a parameter whose values lie in a neighborhood of an eigenvalue of the linear problem. To prove the existence of a nontrivial
solution, we use a modification of the conditional mountain pass method. The difficulties related to the lack of compactness
of the Sobolev operator in the case of an unbounded domain are eliminated with the use of the Lions concentration-compactness
method. 相似文献
7.
The Ginzburg-Landau-type complex equations are simplified mathematical models for various pattern formation systems in mechanics,
physics, and chemistry. In this paper, the derivative complex Ginzburg-Landau (DCGL) equation in an unbounded domain Ω ⊂ ℝ2 is studied. We extend the Gagliardo-Nirenberg inequality to the weighted Sobolev spaces introduced by S. V. Zelik. Applied
this Gagliardo-Nirenberg inequality of the weighted Sobolev spaces and based on the technique for the semi-linear system of
parabolic equations which has been developed by M. A. Efendiev and S. V. Zelik, the global attractor
in the corresponding phase space is constructed, the upper bound of its Kolmogorov’s ɛ-entropy is obtained, and the spatial chaos of the attractor
for DCGL equation in ℝ2 is detailed studied.
相似文献
8.
G. Schneider 《Journal of Nonlinear Science》1998,8(1):17-41
Summary. We consider weakly unstable reaction—diffusion systems defined on domains with one or more unbounded space-directions. In
the systems which we have in mind, at criticality, the most unstable eigenvalue belongs to the wave vector zero and possesses
a nonvanishing imaginary part. This instability leads to an almost spatially homogeneous Hopf-bifurcation in time. A standard
example is the Brusselator in certain parameter ranges. Using multiple scaling analysis we derive a Ginzburg-Landau equation
and show that all small solutions develop in such a way that they can be approximated after a certain time by the solutions
of the Ginzburg-Landau equation. The proof differs essentially from the case when the bifurcating pattern is oscillatory in
space. Our proof is based on normal form methods. As a consequence of the results, the global existence in time of all small
bifurcating solutions and the upper-semicontinuity of the original system attractor towards the associated Ginzburg-Landau
attractor follows.
Original received February 21, 1996; revision accepted April 16, 1997 相似文献
9.
We consider a Ginzburg-Landau type functional on S
2 with a 6
th
order potential and the corresponding selfduality equations. We study the limiting behavior in the two vortex case when
a coupling parameter tends to zero. This two vortex case is a limiting case for the Moser inequality, and we correspondingly
detect a rich and varied asymptotic behavior depending on the position of the vortices. We exploit analogies with the Nirenberg
problem for the prescribed Gauss curvature equation on S
2.
Received: December 3, 1997 相似文献
10.
Tadahisa Funaki 《Probability Theory and Related Fields》1991,90(4):519-562
Summary As a microscopic model we consider a system of interacting continuum like spin field overR
d
. Its evolution law is determined by the Ginzburg-Landau type random Hamiltonian and the total spin of the system is preserved by this evolution. We show that the spin field converges, under the hydrodynamic space-time scalling, to a deterministic limit which is a solution of a certain nonlinear diffusion equation. This equation describes the time evolution of the macroscopic field. The hydrodynamic scaling has an effect of the homogenization on the system at the same time. 相似文献
11.
1IntroductionInthispaper,weintendtostudytheexistenceoftheglobalattractorforthefollowinginitialvalueproblemofGinzburgLandauequationcoupledwithBBMequationinanunboundeddomainR=(-oo,oo)wheree(x,f)isacomplexfunction,n(x,t)isarealscalarfunction,pty,6,T,cr1,a2,P1,P2arerealcoustants,andg1(x),g2(x)aregivenrealfunctions.ThisproblemdescribesthenonlinearinteractionsbetweentheLangnluirwaveandtheionacousticwaveinplasmaphysics,E(z,t)denoteselectricfield,n(x,t)betheperturbationofdensity(see[2,12,6l).In[7]… 相似文献
12.
13.
We construct the Carleman matrix for the Cauchy problem for the Helmholtz equation in an unbounded domain ℝ3 with piecewise smooth boundaries.
Translated fromMatematicheskie Zametki, Vol. 68, No. 4, pp. 548–553, October, 2000. 相似文献
14.
We study the variational convergence of a family of twodimensional Ginzburg-Landau functionals arising in the study of superfluidity
or thin-film superconductivity as the Ginzburg-Landau parameter ε tends to 0. In this regime and for large enough applied rotations (for superfluids) or magnetic fields (for superconductors),
the minimizers acquire quantized point singularities (vortices). We focus on situations in which an unbounded number of vortices
accumulate along a prescribed Jordan curve or a simple arc in the domain. This is known to occur in a circular annulus under
uniform rotation, or in a simply connected domain with an appropriately chosen rotational vector field. We prove that if suitably
normalized, the energy functionals Γ-converge to a classical energy from potential theory. Applied to global minimizers, our
results describe the limiting distribution of vortices along the curve in terms of Green equilibrium measures. 相似文献
15.
Danping Yang 《应用数学学报(英文版)》1993,9(3):223-235
Combining difference method and boundary integral equation method, we propose a new numerical method for solving initial-boundary value problem of second order hyperbolic partial differential equations defined on a bounded or unbounded domain inR
3 and obtain the error estimates of the approximate solution in energy norm and local maximum norm.China State Major Key Project for Basic Researches. 相似文献
16.
Finite element solutions for three‐dimensional elliptic boundary value problems on unbounded domains
Hae‐Soo Oh Jae‐Heon Yun Bong Soo Jang 《Numerical Methods for Partial Differential Equations》2006,22(6):1418-1437
The finite element (FE) solutions of a general elliptic equation ?div([aij] ??u) + u = f in an exterior domain Ω, which is the complement of a bounded subset of R 3, is considered. The most common approach to deal with exterior domain problems is truncating an unbounded subdomain Ω∞, so that the remaining part ΩB = Ω\Ω ∞ is bounded, and imposing an artificial boundary condition on the resulted artificial boundary Γa = Ω ∞ ∩ Ω B. In this article, instead of discarding an unbounded subdomain Ω∞ and introducing an artificial boundary condition, the unbounded domain is mapped to a unit ball by an auxiliary mapping. Then, a similar technique to the method of auxiliary mapping, introduced by Babu?ka and Oh for handling the domain singularities, is applied to obtain an accurate FE solution of this problem at low cost. This method thus does have neither artificial boundary nor any restrictions on f. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006 相似文献
17.
We investigate the blow-up of the solution to a complex Ginzburg-Landau like equation in u coupled with a Poisson equation in f\phi defined on the whole space \Bbb Rn, n = 1{\Bbb R}^n, n = 1 or 2. 相似文献
18.
By the interpolation inequality and a priori estimates in the weighted space, the existence of global solutions for generalized
Ginzburg-Landau equation coupled with BBM equation in an unbounded domain is considered, and the existence of the maximal
attractor is obtained.
This research is supported by the National Natural Science Foundation of China (No.19861004). 相似文献
19.
We consider the initial value boundary problem with zero Neumann data for an equation modeled after the porous media equation, with variable coefficients. The spatial domain is unbounded and shaped like a (general) paraboloid, and the solution u is integrable in space and nonnegative. We show that the asymptotic profile for large times of u is one dimensional and given by an explicit function, which can be regarded as the fundamental solution of a one-dimensional differential equation with weights. In the case when the domain is a cone or the whole space (Cauchy problem), we obtain a genuine multidimensional profile given by the well-known Barenblatt solution. 相似文献
20.
Chaosheng Zhu 《Applicable analysis》2013,92(1):59-65
We utilize a new necessary and sufficient condition to verity the asymptotic compactness of an evolution equation defined in an unbounded domain, which involves the Littlewood–Paley projection operators. We then use this condition to prove the existence of an attractor for the damped Benjamin–Bona–Mahony equation in the phase space H 1(R 1) by showing the solutions are point dissipative and asymptotically compact. Moreover the attractor is in fact smoother and it belongs to H 3/2?? for every ?>0. 相似文献