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Discrete & Continuous Dynamical Systems - A

1998 , Volume 4 , Issue 2

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On the global existence of a cross-diffusion system
Yuan Lou, Wei-Ming Ni and Yaping Wu
1998, 4(2): 193-203 doi: 10.3934/dcds.1998.4.193 +[Abstract](402) +[PDF](171.8KB)
Abstract:
We consider a strongly-coupled nonlinear parabolic system which arises from population dynamics. The global existence of classical solutions is established when the space dimension is two and one of the cross-diffusion pressures is zero.
Determining nodes for the Ginzburg-Landau equations of superconductivity
Hans G. Kaper, Bixiang Wang and Shouhong Wang
1998, 4(2): 205-224 doi: 10.3934/dcds.1998.4.205 +[Abstract](293) +[PDF](271.7KB)
Abstract:
It is shown that a solution of the time-independent Ginzburg-Landau equations of superconductivity is determined completely and exactly by its values at a finite but sufficiently dense set of determining nodes in the domain. If the applied magnetic field is time dependent and asymptotically stationary, the large-time asymptotic behavior of a solution of the time-dependent Ginzburg-Landau equations of superconductivity is determined similarly by its values at a finite set of determining nodes, whose positions may vary with time.
Exponential attractors for nonautonomous first-order evolution equations
Pierre Fabrie and Alain Miranville
1998, 4(2): 225-240 doi: 10.3934/dcds.1998.4.225 +[Abstract](261) +[PDF](187.3KB)
Abstract:
Our aim in this article is to study the existence of exponential attractors for nonautonomous dissipative evolution equations. We follow the approach of Chepyzhov and Vishik, which consists in studying a semigroup on an extended space.
Sensitivity analysis for state constrained optimal control problems
Kazimierz Malanowski and Helmut Maurer
1998, 4(2): 241-272 doi: 10.3934/dcds.1998.4.241 +[Abstract](412) +[PDF](620.4KB)
Abstract:
A family of parameter dependent optimal control poblems for nonlinear ODEs is considered. The problems are subject to pointwise control and state inequality type constraints. It is assumed that, at the reference value of the parameter the reference optimal solution exists and is regular. Regularity conditions are formulated under which the original problems are locally equivalent to some problems subject to equality type constraints only. The classical implicit function theorem is applied to these new problems to investigate Fréchet differentiability of the solutions with respect to the parameter. A numerical example is provided.
Existence results for general systems of differential equations on one-dimensional networks and prewavelets approximation
Denis Mercier and Serge Nicaise
1998, 4(2): 273-300 doi: 10.3934/dcds.1998.4.273 +[Abstract](209) +[PDF](290.2KB)
Abstract:
In this paper, we first prove existence results for general systems of differential equations of parabolic and hyperbolic type in a Hilbert space setting using the notion of Agmon-Douglis-Nirenberg elliptic systems on a half-line and finding a necessary and sufficient condition on the boundary and/or transmission conditions which insures the dissipativity of the (spatial) operators. Our second goal is to take advantage of the one-dimensional structure of networks in order to build appropriate prewavelet bases in view to the numerical approximation of the above problems. Indeed we show that the use of such bases for their approximation (by the Galerkin method for elliptic operators and a fully discrete scheme for parabolic ones) leads to linear systems which can be preconditioned by a diagonal matrix and then can be reduced to systems with a condition number uniformly bounded (with respect to the mesh parameter).
Minimal sets of periods for torus maps
Boju Jiang and Jaume Llibre
1998, 4(2): 301-320 doi: 10.3934/dcds.1998.4.301 +[Abstract](299) +[PDF](256.7KB)
Abstract:
Let $T^r$ be the $r$-dimensional torus, and let $f:T^r\to T^r$ be a map. If $\Per(f)$ denotes the set of periods of $f$, the minimal set of periods of $f$, denoted by $\MPer(f)$, is defined as $\bigcap_{g\cong f}\Per(g)$ where $g:T^r\to T^r$ is homotopic to $f$. First, we characterize the set $\MPer(f)$ in terms of the Nielsen numbers of the iterates of $f$. Second, we distinguish three types of the set $\MPer(f)$ and show that for each type and any given dimension $r$, the variation of $\MPer(f)$ is uniformly bounded in a suitable sense. Finally, we classify all the sets $\MPer(f)$ for self-maps of the $3$-dimensional torus.
Scattering in domains with many small obstacles
Rodica Toader
1998, 4(2): 321-338 doi: 10.3934/dcds.1998.4.321 +[Abstract](269) +[PDF](231.1KB)
Abstract:
We study the behaviour of the wave operators for the relaxed wave equations corresponding to a $\gamma$-convergent sequence of measures. The model case is that of a sequence of domains with many small obstacles.
Longtime behavior of a homogenized model in viscoelastodynamics
M. Grasselli and Vittorino Pata
1998, 4(2): 339-358 doi: 10.3934/dcds.1998.4.339 +[Abstract](214) +[PDF](243.3KB)
Abstract:
A material with heterogeneous structure at microscopic level is considered. The microscopic mechanical behavior is described by a stress-strain law of Kelvin-Voigt type. It has been shown that a homogenization process leads to a macroscopic stress-strain relation containing a time convolution term which accounts for memory effects. Consequently, the displacement field $\mathbf{u}$ obeys to a Volterra integrodifferential motion equation. The longtime behavior of $\mathbf{u}$ is here investigated proving the existence of a uniform attractor when the body forces vary in a suitable metric space.
Periodic systems for the higher-dimensional Laplace transformation
N. Kamran and K. Tenenblat
1998, 4(2): 359-378 doi: 10.3934/dcds.1998.4.359 +[Abstract](242) +[PDF](213.1KB)
Abstract:
We consider overdetermined systems of linear partial differential equations of the form

$ y_{,k\l}+a_{k\l}^ky_{,k}+a_{k\l}^\ly_{,\l}+c_{k\l}y=0\ , \quad 1\le k\ne\l\le n\ , $

where the coefficients are smooth functions satisfying certain integrability conditions. Generalizing the classical theory of second order linear hyperbolic partial differential equation in the plane, we consider higher-dimensional Laplace invariants of a system of the above class. These invariants are characterized as functions which must satisfy a set of differential equations. We establish a normal form for any system of the above class in terms of these invariants. Moreover, we solve the periodicity problem for the higher-dimensional Laplace transformation applied to such systems, generalizing a classical theorem of Darboux which shows that for $n=2$ a 1-periodic equation is equivalent to the Klein-Gordon equation.

Fixed point theorems in the Arnol'd model about instability of the action-variables in phase-space
Paolo Perfetti
1998, 4(2): 379-391 doi: 10.3934/dcds.1998.4.379 +[Abstract](256) +[PDF](269.3KB)
Abstract:
We consider the hamiltonian $H=1/2(I_1^2+I_2^2)+\varepsilon(\cos\varphi_1-1) (1+\mu(\sin\varphi_2+\cos t))$ $I\in\mathbb{R}^2$ ("Arnol'd model about diffusion"); by means of fixed point theorems, the existence of the stable and unstable manifolds (whiskers) of invariant, "a priori unstable tori", for any vector-frequency $(\omega,1)\in\mathbb{R}^2$ is proven. Our aim is to provide detailed proofs which are missing in Arnol'd's paper, namely prove the content of the Assertion B pag.583 of [A]. Our proofs are based on technical tools suggested by Arnol'd i.e. the contraction mapping method together with the "conical metric" (see the footnote ** of pag. 583 of [A]).

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