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1078-0947
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Discrete & Continuous Dynamical Systems - A
March 2006 , Volume 16 , Issue 1
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2006, 16(1): 1-18
doi: 10.3934/dcds.2006.16.1
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Abstract:
A new technique, combining the global energy and entropy balance equations with the local stability theory for dynamical systems, is used for proving that every solution to a non-smooth temperature-driven phase separation model with conserved energy converges pointwise in space to an equilibrium as time tends to infinity. Three main features are observed: the limit temperature is uniform in space, there exists a partition of the physical body into at most three constant limit phases, and the phase separation process has a hysteresis-like character.
A new technique, combining the global energy and entropy balance equations with the local stability theory for dynamical systems, is used for proving that every solution to a non-smooth temperature-driven phase separation model with conserved energy converges pointwise in space to an equilibrium as time tends to infinity. Three main features are observed: the limit temperature is uniform in space, there exists a partition of the physical body into at most three constant limit phases, and the phase separation process has a hysteresis-like character.
2006, 16(1): 19-46
doi: 10.3934/dcds.2006.16.19
+[Abstract](1598)
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Abstract:
We consider the general question of estimating decay of correlations for non-uniformly expanding maps, for classes of observables which are much larger than the usual class of Hölder continuous functions. Our results give new estimates for many non-uniformly expanding systems, including Manneville-Pomeau maps, many one-dimensional systems with critical points, and Viana maps . In many situations, we also obtain a Central Limit Theorem for a much larger class of observables than usual.
Our main tool is an extension of the coupling method introduced by L.-S. Young for estimating rates of mixing on certain non-uniformly expanding tower maps.
We consider the general question of estimating decay of correlations for non-uniformly expanding maps, for classes of observables which are much larger than the usual class of Hölder continuous functions. Our results give new estimates for many non-uniformly expanding systems, including Manneville-Pomeau maps, many one-dimensional systems with critical points, and Viana maps . In many situations, we also obtain a Central Limit Theorem for a much larger class of observables than usual.
Our main tool is an extension of the coupling method introduced by L.-S. Young for estimating rates of mixing on certain non-uniformly expanding tower maps.
2006, 16(1): 47-66
doi: 10.3934/dcds.2006.16.47
+[Abstract](1536)
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Abstract:
This paper is concerned with the asymptotic stability of travelling wave front solutions with algebraic decay for $n$-degree Fisher-type equations. By detailed spectral analysis, each travelling wave front solution with non-critical speed is proved to be locally exponentially stable to perturbations in some exponentially weighted $L^{\infty}$ spaces. Further by Evans function method and detailed semigroup estimates, the travelling wave fronts with non-critical speed are proved to be locally algebraically stable to perturbations in some polynomially weighted $L^{\infty}$ spaces. It's remarked that due to the slow algebraic decay rate of the wave at $+\infty,$ the Evans function constructed in this paper is an extension of the definitions in [1, 3, 7, 11, 21] to some extent, and the Evans function can be extended analytically in the neighborhood of the origin.
This paper is concerned with the asymptotic stability of travelling wave front solutions with algebraic decay for $n$-degree Fisher-type equations. By detailed spectral analysis, each travelling wave front solution with non-critical speed is proved to be locally exponentially stable to perturbations in some exponentially weighted $L^{\infty}$ spaces. Further by Evans function method and detailed semigroup estimates, the travelling wave fronts with non-critical speed are proved to be locally algebraically stable to perturbations in some polynomially weighted $L^{\infty}$ spaces. It's remarked that due to the slow algebraic decay rate of the wave at $+\infty,$ the Evans function constructed in this paper is an extension of the definitions in [1, 3, 7, 11, 21] to some extent, and the Evans function can be extended analytically in the neighborhood of the origin.
2006, 16(1): 67-86
doi: 10.3934/dcds.2006.16.67
+[Abstract](1527)
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Abstract:
Let $\Omega=[0,L_1]\times[0,L_2]\times[0,\epsilon]$ where $L_1,L_2>0$ and $\epsilon\in(0,1)$. We consider the Navier-Stokes equations with periodic boundary conditions and prove that if
Let $\Omega=[0,L_1]\times[0,L_2]\times[0,\epsilon]$ where $L_1,L_2>0$ and $\epsilon\in(0,1)$. We consider the Navier-Stokes equations with periodic boundary conditions and prove that if
$ \|\| \nabla u_0\|\|_{L^2(\Omega)} \le \frac{1}{C(L_1,L_2)\epsilon^{1/6}} $
then there exists a unique global smooth solution with the initial datum $u_0$.
2006, 16(1): 87-106
doi: 10.3934/dcds.2006.16.87
+[Abstract](1840)
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Abstract:
We study the scattering problem for the nonlinear wave equation with potential. In the absence of the potential, one has sharp global existence results for the Cauchy problem with small initial data; those require the data to decay at a rate $k\geq k_c$, where $k_c$ is a critical decay rate that depends on the order of the nonlinearity. However, scattering results have appeared only for the supercritical case $k>k_c$. In this paper, we extend the latter results to the critical case and we also allow the presence of a short-range potential.
We study the scattering problem for the nonlinear wave equation with potential. In the absence of the potential, one has sharp global existence results for the Cauchy problem with small initial data; those require the data to decay at a rate $k\geq k_c$, where $k_c$ is a critical decay rate that depends on the order of the nonlinearity. However, scattering results have appeared only for the supercritical case $k>k_c$. In this paper, we extend the latter results to the critical case and we also allow the presence of a short-range potential.
2006, 16(1): 107-119
doi: 10.3934/dcds.2006.16.107
+[Abstract](1696)
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Abstract:
We consider the second order strongly indefinite differential system with superlinearities. By using the approximation method of finite element, we show that bounds on solutions of the restriction functional onto finite dimensional subspace are equivalent to bounds on their relative Morse indices. The obtained results can be used to establish a Morse theory for strongly indefinite functionals with superlinearities.
We consider the second order strongly indefinite differential system with superlinearities. By using the approximation method of finite element, we show that bounds on solutions of the restriction functional onto finite dimensional subspace are equivalent to bounds on their relative Morse indices. The obtained results can be used to establish a Morse theory for strongly indefinite functionals with superlinearities.
2006, 16(1): 121-136
doi: 10.3934/dcds.2006.16.121
+[Abstract](1646)
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Abstract:
The existence of the global attractor of the damped, forced generalized KdV-Benjamin-Ono equation in $L^2( \mathbb{R})$ is proved for forces in $L^2( \mathbb{R})$. Moreover, the global attractor in $L^2( \mathbb{R})$ is actually a compact set in $H^3( \mathbb{R})$.
The existence of the global attractor of the damped, forced generalized KdV-Benjamin-Ono equation in $L^2( \mathbb{R})$ is proved for forces in $L^2( \mathbb{R})$. Moreover, the global attractor in $L^2( \mathbb{R})$ is actually a compact set in $H^3( \mathbb{R})$.
2006, 16(1): 137-156
doi: 10.3934/dcds.2006.16.137
+[Abstract](1311)
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Abstract:
This paper is concerned with the long time behavior for evolution of a curve governed by a curvature flow with constant driving force in the two-dimensional space. This problem has two types of traveling waves: traveling lines and V-shaped fronts, except for stationary circles. Studying the Cauchy problem, we deal with moving curves represented by entire graphs on the $x$-axis. In this paper, we consider the uniform convergence of curves to the V-shaped fronts. Convergence results for a class of spatially non-decaying initial perturbations are established. Our results hold true with no assumptions on the smallness of given perturbations.
This paper is concerned with the long time behavior for evolution of a curve governed by a curvature flow with constant driving force in the two-dimensional space. This problem has two types of traveling waves: traveling lines and V-shaped fronts, except for stationary circles. Studying the Cauchy problem, we deal with moving curves represented by entire graphs on the $x$-axis. In this paper, we consider the uniform convergence of curves to the V-shaped fronts. Convergence results for a class of spatially non-decaying initial perturbations are established. Our results hold true with no assumptions on the smallness of given perturbations.
2006, 16(1): 157-177
doi: 10.3934/dcds.2006.16.157
+[Abstract](1558)
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Abstract:
The cyclicity of period annuli of some classes of reversible and non-Hamiltonian quadratic systems under quadratic perturbations are studied. The argument principle method and the centroid curve method are combined to prove that the related Abelian integral has at most two zeros.
The cyclicity of period annuli of some classes of reversible and non-Hamiltonian quadratic systems under quadratic perturbations are studied. The argument principle method and the centroid curve method are combined to prove that the related Abelian integral has at most two zeros.
2006, 16(1): 179-226
doi: 10.3934/dcds.2006.16.179
+[Abstract](1553)
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Abstract:
In the present paper it is proved that given a maximal invariant attracting homoclinic class for a smooth three dimensional Kupka-Smale diffeomorphism, either the diffeomorphisms is $C^1$ approximated by another one exhibiting a homoclinic tangency or a heterodimensional cycle, or it follows that the homoclinic class is conjugate to a hyperbolic set (in this case we say that the homoclinic class is "topologically hyperbolic").
We also characterize, in any dimension, the dynamics of a topologically hyperbolic homoclinic class and we describe the continuation of this homoclinic class for a perturbation of the initial system.
Moreover, we prove that, under some topological conditions, the homoclinic class is contained in a two dimensional manifold and it is hyperbolic.
In the present paper it is proved that given a maximal invariant attracting homoclinic class for a smooth three dimensional Kupka-Smale diffeomorphism, either the diffeomorphisms is $C^1$ approximated by another one exhibiting a homoclinic tangency or a heterodimensional cycle, or it follows that the homoclinic class is conjugate to a hyperbolic set (in this case we say that the homoclinic class is "topologically hyperbolic").
We also characterize, in any dimension, the dynamics of a topologically hyperbolic homoclinic class and we describe the continuation of this homoclinic class for a perturbation of the initial system.
Moreover, we prove that, under some topological conditions, the homoclinic class is contained in a two dimensional manifold and it is hyperbolic.
2006, 16(1): 227-234
doi: 10.3934/dcds.2006.16.227
+[Abstract](2015)
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Abstract:
In this note, it is shown that some Hamiltonian partial differential equations such as semi-linear Schrödinger equations, semi-linear wave equations and semi-linear beam equations are partially integrable, i.e., they possess integrable invariant manifolds foliated by invariant tori which carry periodic or quasi-periodic solutions. The linear stability of the obtained invariant manifolds is also concluded. The proofs are based on a special invariant property of the considered equations and a symplectic change of variables first observed in [26].
In this note, it is shown that some Hamiltonian partial differential equations such as semi-linear Schrödinger equations, semi-linear wave equations and semi-linear beam equations are partially integrable, i.e., they possess integrable invariant manifolds foliated by invariant tori which carry periodic or quasi-periodic solutions. The linear stability of the obtained invariant manifolds is also concluded. The proofs are based on a special invariant property of the considered equations and a symplectic change of variables first observed in [26].
Traveling pulses for the Klein-Gordon equation on a lattice or continuum with long-range interaction
2006, 16(1): 235-252
doi: 10.3934/dcds.2006.16.235
+[Abstract](2030)
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Abstract:
We study traveling pulses on a lattice and in a continuum where all pairs of particles interact, contributing to the potential energy. The interaction may be positive or negative, depending on the particular pair but overall is positive in a certain sense. For such an interaction kernel $J$ with unit integral (or sum), the operator 1/ε2[J∗u-u], with ∗ continuous or discrete convolution, shares some common features with the spatial second derivative operator, especially when ε is small. Therefore, the equation $u_{t t}$ - 1/ε2[J∗u-u] + f(u)=0 may be compared with the nonlinear Klein Gordon equation $u_{t t}$ - $u_{x x}$$ + f(u)=0$. If $f$ is such that the Klein-Gordon equation has supersonic traveling pulses, we show that the same is true for the nonlocal version, both the continuum and lattice cases.
We study traveling pulses on a lattice and in a continuum where all pairs of particles interact, contributing to the potential energy. The interaction may be positive or negative, depending on the particular pair but overall is positive in a certain sense. For such an interaction kernel $J$ with unit integral (or sum), the operator 1/ε2[J∗u-u], with ∗ continuous or discrete convolution, shares some common features with the spatial second derivative operator, especially when ε is small. Therefore, the equation $u_{t t}$ - 1/ε2[J∗u-u] + f(u)=0 may be compared with the nonlinear Klein Gordon equation $u_{t t}$ - $u_{x x}$$ + f(u)=0$. If $f$ is such that the Klein-Gordon equation has supersonic traveling pulses, we show that the same is true for the nonlocal version, both the continuum and lattice cases.
2006, 16(1): 253-277
doi: 10.3934/dcds.2006.16.253
+[Abstract](1919)
+[PDF](334.3KB)
Abstract:
In this paper, we study the Cauchy problem for the Boltzmann equation with an external force and the Vlasov-Poisson-Boltzmann system in infinite vacuum. The global existence of solutions is first proved for the Boltzmann equation with an external force which is integrable with respect to time in some sense under the smallness assumption on initial data in weighted norms. For the Vlasov-Poisson-Boltzmann system, the smallness assumption on initial data leads to the decay of the potential field which in turn gives the global existence of solutions by the result on the case with external forces and an iteration argument. The results obtained here generalize those previous works on these topics and they hold for a class of general cross sections including the hard-sphere model.
In this paper, we study the Cauchy problem for the Boltzmann equation with an external force and the Vlasov-Poisson-Boltzmann system in infinite vacuum. The global existence of solutions is first proved for the Boltzmann equation with an external force which is integrable with respect to time in some sense under the smallness assumption on initial data in weighted norms. For the Vlasov-Poisson-Boltzmann system, the smallness assumption on initial data leads to the decay of the potential field which in turn gives the global existence of solutions by the result on the case with external forces and an iteration argument. The results obtained here generalize those previous works on these topics and they hold for a class of general cross sections including the hard-sphere model.
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