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1078-0947
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Discrete & Continuous Dynamical Systems - A
July 2002 , Volume 8 , Issue 3
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2002, 8(3): 519-562
doi: 10.3934/dcds.2002.8.519
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Abstract:
We study the nonlinear eigenvalue problem $f(x) = \lambda x$ for a class of maps $f: K\to K$ which are homogeneous of degree one and order-preserving, where $K\subset X$ is a closed convex cone in a Banach space X. Solutions are obtained, in part, using a theory of the "cone spectral radius" which we develop. Principal technical tools are the generalized measure of noncompactness and related degree-theoretic techniques. We apply our results to a class of problems max
We study the nonlinear eigenvalue problem $f(x) = \lambda x$ for a class of maps $f: K\to K$ which are homogeneous of degree one and order-preserving, where $K\subset X$ is a closed convex cone in a Banach space X. Solutions are obtained, in part, using a theory of the "cone spectral radius" which we develop. Principal technical tools are the generalized measure of noncompactness and related degree-theoretic techniques. We apply our results to a class of problems max
$\max_{t\in J(s)} a(s, t)x(t) = \lambda x(s)$
arising from so-called "max-plus operators," where we seek a nonnegative eigenfunction $ x\in C[0, \mu]$ and eigenvalue $\lambda$. Here $J(s) = [\alpha(s), \beta(s)] \subset [0, \mu]$ for $s\in [0, \mu]$, with $a, \alpha$, and $\beta$ given functions, and the function $a$ nonnegative.
2002, 8(3): 563-584
doi: 10.3934/dcds.2002.8.563
+[Abstract](1586)
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Abstract:
This paper deals with existence and multiplicity of solutions to the nonlinear Schrödinger equation of the type
This paper deals with existence and multiplicity of solutions to the nonlinear Schrödinger equation of the type
$-\Delta u + (\lambda a(x) + a_0(x))u = f(x, u), u\in H^1(\mathbb R^N).$
We improve some previous results in two respects: we do not require $a_0$ to be positive on one hand, and allow $f(x, u)$ to be critical nonlinear on the other hand.
2002, 8(3): 585-597
doi: 10.3934/dcds.2002.8.585
+[Abstract](1155)
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Abstract:
In this paper we study the long time behaviour of dynamical systems acting in topological spaces. We give suitable definitions of attractivity and prove some properties of limit sets, as well as theorems on existence of attractors in the class of regular spaces.
In this paper we study the long time behaviour of dynamical systems acting in topological spaces. We give suitable definitions of attractivity and prove some properties of limit sets, as well as theorems on existence of attractors in the class of regular spaces.
2002, 8(3): 599-604
doi: 10.3934/dcds.2002.8.599
+[Abstract](1317)
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Abstract:
In this paper we show ergodicity of the strong stable foliations for nilpotent extensions of transitive Anosov flows with respect to the lift of the Gibbs measure for any Hölder continuous function.
In this paper we show ergodicity of the strong stable foliations for nilpotent extensions of transitive Anosov flows with respect to the lift of the Gibbs measure for any Hölder continuous function.
2002, 8(3): 605-625
doi: 10.3934/dcds.2002.8.606
+[Abstract](994)
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Abstract:
We consider an initial boundary value problem set in $\{x > 0\}$ for a mixed hyperbolic parabolic system of conservation laws with a small parameter $\varepsilon$, $u_t +F(u)_x =\varepsilon(B(u)u_x)_x$. In the non-characteristic case a boundary layers analysis gives a set of boundary conditions, the set of residual boundary conditions $\mathcal C$, for the inviscid system $u_t+F(u)_x = 0$. We generalize the results of [16] obtained for strictly parabolic perturbations to a realistic setting. We show that the set $\mathcal C$ has the suitable geometric property to construct a solution of the inviscid sytem in a vicinity of a point where the Evans function of the corresponding profile of boundary layer does not vanish at zero. Next we consider multidimensional systems. We show that the Kreiss-Lopatinski determinant for the hyperbolic system linearized about a constant state in $\mathcal C$ is equal to the reduced Evans function for the viscous system linearized about the corresponding profile of boundary layer.
We consider an initial boundary value problem set in $\{x > 0\}$ for a mixed hyperbolic parabolic system of conservation laws with a small parameter $\varepsilon$, $u_t +F(u)_x =\varepsilon(B(u)u_x)_x$. In the non-characteristic case a boundary layers analysis gives a set of boundary conditions, the set of residual boundary conditions $\mathcal C$, for the inviscid system $u_t+F(u)_x = 0$. We generalize the results of [16] obtained for strictly parabolic perturbations to a realistic setting. We show that the set $\mathcal C$ has the suitable geometric property to construct a solution of the inviscid sytem in a vicinity of a point where the Evans function of the corresponding profile of boundary layer does not vanish at zero. Next we consider multidimensional systems. We show that the Kreiss-Lopatinski determinant for the hyperbolic system linearized about a constant state in $\mathcal C$ is equal to the reduced Evans function for the viscous system linearized about the corresponding profile of boundary layer.
2002, 8(3): 627-632
doi: 10.3934/dcds.2002.8.627
+[Abstract](1488)
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Abstract:
In this paper, we prove a generalized shadowing lemma. Let $f \in$ Diff$(M)$. Assume that $\Lambda$ is a closed invariant set of $f$ and there is a continuous invariant splitting $T\Lambda M = E\oplus F$ on $\Lambda$. For any $\lambda \in (0, 1)$ there exist $L > 0, d_0> 0$ such that for any $d \in (0, d_0]$ and any $\lambda$-quasi-hyperbolic d-pseudoorbit $\{x_i, n_i\}_{i=-\infty}^\infty$, there exists a point $x$ which Ld-shadows $\{x_i, n_i\}_{i=-\infty}^\infty$. Moreover, if $\{x_i, n_i\}_{i=-\infty}^\infty$ is periodic, i.e., there exists an $m > 0$ such that $x_{i+m}= x_i$ and $n_{i+m} = n_i$ for all $i$, then the point $x$ can be chosen to be periodic.
In this paper, we prove a generalized shadowing lemma. Let $f \in$ Diff$(M)$. Assume that $\Lambda$ is a closed invariant set of $f$ and there is a continuous invariant splitting $T\Lambda M = E\oplus F$ on $\Lambda$. For any $\lambda \in (0, 1)$ there exist $L > 0, d_0> 0$ such that for any $d \in (0, d_0]$ and any $\lambda$-quasi-hyperbolic d-pseudoorbit $\{x_i, n_i\}_{i=-\infty}^\infty$, there exists a point $x$ which Ld-shadows $\{x_i, n_i\}_{i=-\infty}^\infty$. Moreover, if $\{x_i, n_i\}_{i=-\infty}^\infty$ is periodic, i.e., there exists an $m > 0$ such that $x_{i+m}= x_i$ and $n_{i+m} = n_i$ for all $i$, then the point $x$ can be chosen to be periodic.
2002, 8(3): 633-646
doi: 10.3934/dcds.2002.8.633
+[Abstract](1226)
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Abstract:
We analyze a renormalization group transformation $\mathcal R$ for partially analytic Hamiltonians, with emphasis on what seems to be needed for the construction of non-integrable fixed points. Under certain assumptions, which are supported by numerical data in the golden mean case, we prove that such a fixed point has a critical invariant torus. The proof is constructive and can be used for numerical computations. We also relate $\mathcal R$ to a renormalization group transformation for commuting maps.
We analyze a renormalization group transformation $\mathcal R$ for partially analytic Hamiltonians, with emphasis on what seems to be needed for the construction of non-integrable fixed points. Under certain assumptions, which are supported by numerical data in the golden mean case, we prove that such a fixed point has a critical invariant torus. The proof is constructive and can be used for numerical computations. We also relate $\mathcal R$ to a renormalization group transformation for commuting maps.
2002, 8(3): 647-670
doi: 10.3934/dcds.2002.8.647
+[Abstract](1106)
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Abstract:
This paper is concerned with two types of nonlinear parabolic equations, which arise from the nonlinear filtration problems for non-Newtonian fluids. These equations include as special cases the porous medium equations $u_t =$ div$(u^l\nabla u)$ and the evolution equation governed by p-Laplacian $u_t =$ div $(|\nabla u|^{p-2}\nabla u)$. Because of the degeneracy or singularity caused by the terms $u^l$ and $|\nabla u|^{p-2}$, one can not expect the existence of global (in time) classical solutions for these equations except for special cases. Therefore most of works have been devoted to the study of weak solutions. The main purpose of this paper is to investigate the existence of much more regular (not necessarily global) solutions. The existence of local solutions in $W^{1,\infty}(\Omega)$ is assured under the assumption that initial data are non-negative functions in $W_0^{1,\infty}(\Omega)$, and that the mean curvature of the boundary $\partial \Omega$ of the domain $\Omega$ is non-positive. We here introduce a new method "$L^\infty$-energy method", which provides a main tool for our arguments and would be useful for other situations.
This paper is concerned with two types of nonlinear parabolic equations, which arise from the nonlinear filtration problems for non-Newtonian fluids. These equations include as special cases the porous medium equations $u_t =$ div$(u^l\nabla u)$ and the evolution equation governed by p-Laplacian $u_t =$ div $(|\nabla u|^{p-2}\nabla u)$. Because of the degeneracy or singularity caused by the terms $u^l$ and $|\nabla u|^{p-2}$, one can not expect the existence of global (in time) classical solutions for these equations except for special cases. Therefore most of works have been devoted to the study of weak solutions. The main purpose of this paper is to investigate the existence of much more regular (not necessarily global) solutions. The existence of local solutions in $W^{1,\infty}(\Omega)$ is assured under the assumption that initial data are non-negative functions in $W_0^{1,\infty}(\Omega)$, and that the mean curvature of the boundary $\partial \Omega$ of the domain $\Omega$ is non-positive. We here introduce a new method "$L^\infty$-energy method", which provides a main tool for our arguments and would be useful for other situations.
2002, 8(3): 671-674
doi: 10.3934/dcds.2002.8.671
+[Abstract](1283)
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Abstract:
We prove that there is a residual subset $C$, in the space of all $\mathcal C^1$ vector fields of a closed $n$-manifold $M$, such that for every $X \in \mathcal R$ the set of points in $M$ with Lyapunov stable $\omega$-limit set is residual in $M$. This improves a result in Arnaud [1] and gives a partial solution to a conjecture in Hurley [8].
We prove that there is a residual subset $C$, in the space of all $\mathcal C^1$ vector fields of a closed $n$-manifold $M$, such that for every $X \in \mathcal R$ the set of points in $M$ with Lyapunov stable $\omega$-limit set is residual in $M$. This improves a result in Arnaud [1] and gives a partial solution to a conjecture in Hurley [8].
2002, 8(3): 675-695
doi: 10.3934/dcds.2002.8.675
+[Abstract](1342)
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Abstract:
The linear Euler-Bernoulli viscoelastic equation
The linear Euler-Bernoulli viscoelastic equation
$u_{t t} +\Delta^2 u-\int_0^t g(t-\tau) \Delta^2 u(\tau)d\tau = 0\quad$ in $\Omega \times (0,\infty)$
subject to nonlinear boundary conditions is considered. We prove existence and uniform decay rates of the energy by assuming a nonlinear and nonlocal feedback acting on the boundary and provided that the kernel of the memory decays exponentially.
2002, 8(3): 697-724
doi: 10.3934/dcds.2002.8.697
+[Abstract](1106)
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Abstract:
The travel time brachistochrone curves in a general relativistic framework are timelike curves, satisfying a suitable conservation law with respect to a an observer field, that are stationary points of the travel time functional. We develop a global variational theory for brachistochrones joining an event p and the worldline of an observer $\gamma$ in a stationary spacetime $\mathcal M$.
The travel time brachistochrone curves in a general relativistic framework are timelike curves, satisfying a suitable conservation law with respect to a an observer field, that are stationary points of the travel time functional. We develop a global variational theory for brachistochrones joining an event p and the worldline of an observer $\gamma$ in a stationary spacetime $\mathcal M$.
2002, 8(3): 725-735
doi: 10.3934/dcds.2002.8.725
+[Abstract](1164)
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Abstract:
We consider suspension semiflows over abelian extensions of one-sided mixing subshifts of finite type. Although these are not uniquely ergodic, we identify (in the "ergodic" case) all tail-invariant, locally finite measures which are quasiinvariant for the semiflow.
We consider suspension semiflows over abelian extensions of one-sided mixing subshifts of finite type. Although these are not uniquely ergodic, we identify (in the "ergodic" case) all tail-invariant, locally finite measures which are quasiinvariant for the semiflow.
The general multi-dimensional Riemann problem for hyperbolic systems with real constant coefficients
2002, 8(3): 737-744
doi: 10.3934/dcds.2002.8.737
+[Abstract](1054)
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Abstract:
In this paper, we give the explicit solution to the general multi-dimensional Riemann problem for the canonical form of $2\times 2$ hyperbolic systems with real constant coefficients.
In this paper, we give the explicit solution to the general multi-dimensional Riemann problem for the canonical form of $2\times 2$ hyperbolic systems with real constant coefficients.
2002, 8(3): 745-756
doi: 10.3934/dcds.2002.8.747
+[Abstract](1248)
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Abstract:
In two previous works we improved some earlier results of Imanuvilov, Li and Zhang, and of Zuazua on the boundary exact controllability of one-dimensional semilinear wave equations by weakening the growth assumptions on the nonlinearity. Our growth assumption is in a sense optimal. Here we adapt our method for the case of one-sided control actions. This also enables us to obtain rather general internal controllability results.
In two previous works we improved some earlier results of Imanuvilov, Li and Zhang, and of Zuazua on the boundary exact controllability of one-dimensional semilinear wave equations by weakening the growth assumptions on the nonlinearity. Our growth assumption is in a sense optimal. Here we adapt our method for the case of one-sided control actions. This also enables us to obtain rather general internal controllability results.
2002, 8(3): 757-780
doi: 10.3934/dcds.2002.8.757
+[Abstract](1284)
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Abstract:
In this paper we introduce an obstacle thermistor system. The existence of weak solutions to the steady-state systems and capacity solutions to the time dependent systems are obtained by a penalized method under reasonable assumptions for the initial and boundary data. At the same time, we prove that there exists a uniform absorbing set for nonnegative initial data in $L_2(\Omega)$. Finally for smooth initial data a global attractor to the system is obtained by a series of Campanato space arguments.
In this paper we introduce an obstacle thermistor system. The existence of weak solutions to the steady-state systems and capacity solutions to the time dependent systems are obtained by a penalized method under reasonable assumptions for the initial and boundary data. At the same time, we prove that there exists a uniform absorbing set for nonnegative initial data in $L_2(\Omega)$. Finally for smooth initial data a global attractor to the system is obtained by a series of Campanato space arguments.
2002, 8(3): 781-794
doi: 10.3934/dcds.2002.8.781
+[Abstract](1227)
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Abstract:
In this paper we study a degenerate evolution system $\mathbf H_t +\nabla \times [|\nabla \times \mathbf H|^{p-2}\nabla \times \mathbf H]=\mathbf F$ in a bounded domain as well as its limit as $p\to \infty$ subject to appropriate initial and boundary conditions. This system governs the evolution of the magnetic field $\mathbf H$ in a conductive medium under the influence of a system force $\mathbf F$. The system is an approximation of Bean's critical-state model for type-II superconductors. The existence, uniqueness and regularity of solutions to the system are established. Moreover, it is shown that the limit of $\mathbf H(x, t)$ as $p\to \infty$ is a solution to the Bean model.
In this paper we study a degenerate evolution system $\mathbf H_t +\nabla \times [|\nabla \times \mathbf H|^{p-2}\nabla \times \mathbf H]=\mathbf F$ in a bounded domain as well as its limit as $p\to \infty$ subject to appropriate initial and boundary conditions. This system governs the evolution of the magnetic field $\mathbf H$ in a conductive medium under the influence of a system force $\mathbf F$. The system is an approximation of Bean's critical-state model for type-II superconductors. The existence, uniqueness and regularity of solutions to the system are established. Moreover, it is shown that the limit of $\mathbf H(x, t)$ as $p\to \infty$ is a solution to the Bean model.
2002, 8(3): 795-811
doi: 10.3934/dcds.2002.8.795
+[Abstract](1247)
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Abstract:
We consider the problem of Arnold Diffusion for nearly integrable partially isochronous Hamiltonian systems with three time scales. By means of a careful shadowing analysis, based on a variational technique, we prove that, along special directions, Arnold diffusion takes place with fast (polynomial) speed, even though the "splitting determinant" is exponentially small.
We consider the problem of Arnold Diffusion for nearly integrable partially isochronous Hamiltonian systems with three time scales. By means of a careful shadowing analysis, based on a variational technique, we prove that, along special directions, Arnold diffusion takes place with fast (polynomial) speed, even though the "splitting determinant" is exponentially small.
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