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Discrete and Continuous Dynamical Systems
October 2015 , Volume 35 , Issue 10
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2015, 35(10): 4683-4734
doi: 10.3934/dcds.2015.35.4683
+[Abstract](2688)
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Abstract:
We study the set of periods of degree 1 continuous maps from $\sigma$ into itself, where $\sigma$ denotes the space shaped like the letter $\sigma$ (i.e., a segment attached to a circle by one of its endpoints). Since the maps under consideration have degree 1, the rotation theory can be used. We show that, when the interior of the rotation interval contains an integer, then the set of periods (of periodic points of any rotation number) is the set of all integers except maybe $1$ or $2$. We exhibit degree 1 $\sigma$-maps $f$ whose set of periods is a combination of the set of periods of a degree 1 circle map and the set of periods of a $3$-star (that is, a space shaped like the letter $Y$). Moreover, we study the set of periods forced by periodic orbits that do not intersect the circuit of $\sigma$; in particular, when there exists such a periodic orbit whose diameter (in the covering space) is at least $1$, then there exist periodic points of all periods.
We study the set of periods of degree 1 continuous maps from $\sigma$ into itself, where $\sigma$ denotes the space shaped like the letter $\sigma$ (i.e., a segment attached to a circle by one of its endpoints). Since the maps under consideration have degree 1, the rotation theory can be used. We show that, when the interior of the rotation interval contains an integer, then the set of periods (of periodic points of any rotation number) is the set of all integers except maybe $1$ or $2$. We exhibit degree 1 $\sigma$-maps $f$ whose set of periods is a combination of the set of periods of a degree 1 circle map and the set of periods of a $3$-star (that is, a space shaped like the letter $Y$). Moreover, we study the set of periods forced by periodic orbits that do not intersect the circuit of $\sigma$; in particular, when there exists such a periodic orbit whose diameter (in the covering space) is at least $1$, then there exist periodic points of all periods.
2015, 35(10): 4735-4741
doi: 10.3934/dcds.2015.35.4735
+[Abstract](2658)
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Abstract:
We study sectional-Anosov flows on compact $3$-manifolds. First we prove that every periodic orbits represents an infinite order element of the fundamental group outside the strong stable manifolds of the singularities. Next, in the transitive case, we prove that the first Betti number of the manifold is positive, that the number of singularities is given by the Euler characteristic and that every boundary's connected component has nonpositive Euler characteristic. Moreover, there is one component with negative characteristic if and only if the flow has singularities. These results will be used to discuss the existence of transitive sectional-Anosov flows on specific compact 3-manifolds with boundary.
We study sectional-Anosov flows on compact $3$-manifolds. First we prove that every periodic orbits represents an infinite order element of the fundamental group outside the strong stable manifolds of the singularities. Next, in the transitive case, we prove that the first Betti number of the manifold is positive, that the number of singularities is given by the Euler characteristic and that every boundary's connected component has nonpositive Euler characteristic. Moreover, there is one component with negative characteristic if and only if the flow has singularities. These results will be used to discuss the existence of transitive sectional-Anosov flows on specific compact 3-manifolds with boundary.
2015, 35(10): 4743-4764
doi: 10.3934/dcds.2015.35.4743
+[Abstract](3463)
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Abstract:
Spacing shifts were introduced by Lau and Zame in the 1970's to provide accessible examples of maps that are weakly mixing but not mixing. In previous papers by the authors and others, it has been observed that the problem of describing when spacing shifts are topologically transitive appears to be quite difficult in general. In the present paper, we give a characterization of sofic spacing shifts and begin to investigate which sofic spacing shifts are topologically transitive. We show that the canonical graph presentation of such a shift has a rather simple form, for which we introduce the terminology hereditary bunched cycle and discuss the apparently difficult problem of determining which hereditary bunched cycles actually present spacing shifts.
Spacing shifts were introduced by Lau and Zame in the 1970's to provide accessible examples of maps that are weakly mixing but not mixing. In previous papers by the authors and others, it has been observed that the problem of describing when spacing shifts are topologically transitive appears to be quite difficult in general. In the present paper, we give a characterization of sofic spacing shifts and begin to investigate which sofic spacing shifts are topologically transitive. We show that the canonical graph presentation of such a shift has a rather simple form, for which we introduce the terminology hereditary bunched cycle and discuss the apparently difficult problem of determining which hereditary bunched cycles actually present spacing shifts.
2015, 35(10): 4765-4789
doi: 10.3934/dcds.2015.35.4765
+[Abstract](2353)
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Abstract:
We present a method designed for computing solutions of infinite dimensional nonlinear operators $f(x)=0$ with a tridiagonal dominant linear part. We recast the operator equation into an equivalent Newton-like equation $x=T(x)=x-Af(x)$, where $A$ is an approximate inverse of the derivative $Df(\overline{x})$ at an approximate solution $\overline{x}$. We present rigorous computer-assisted calculations showing that $T$ is a contraction near $\overline{x}$, thus yielding the existence of a solution. Since $Df(\overline{x})$ does not have an asymptotically diagonal dominant structure, the computation of $A$ is not straightforward. This paper provides ideas for computing $A$, and proposes a new rigorous method for proving existence of solutions of nonlinear operators with tridiagonal dominant linear part.
We present a method designed for computing solutions of infinite dimensional nonlinear operators $f(x)=0$ with a tridiagonal dominant linear part. We recast the operator equation into an equivalent Newton-like equation $x=T(x)=x-Af(x)$, where $A$ is an approximate inverse of the derivative $Df(\overline{x})$ at an approximate solution $\overline{x}$. We present rigorous computer-assisted calculations showing that $T$ is a contraction near $\overline{x}$, thus yielding the existence of a solution. Since $Df(\overline{x})$ does not have an asymptotically diagonal dominant structure, the computation of $A$ is not straightforward. This paper provides ideas for computing $A$, and proposes a new rigorous method for proving existence of solutions of nonlinear operators with tridiagonal dominant linear part.
2015, 35(10): 4791-4804
doi: 10.3934/dcds.2015.35.4791
+[Abstract](3052)
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Abstract:
A global Calderón-Zygmund estimate type estimate in Weighted Lorentz spaces and Lorentz-Morrey spaces is obtained for weak solutions to elliptic obstacle problems of $p$-Laplacian type with discontinuous coefficients over Reifenberg flat domains.
A global Calderón-Zygmund estimate type estimate in Weighted Lorentz spaces and Lorentz-Morrey spaces is obtained for weak solutions to elliptic obstacle problems of $p$-Laplacian type with discontinuous coefficients over Reifenberg flat domains.
2015, 35(10): 4805-4821
doi: 10.3934/dcds.2015.35.4805
+[Abstract](2928)
+[PDF](469.9KB)
Abstract:
We consider a class of linear Schrödinger equations in $\mathbb{R}^d$ with rough Hamiltonian, namely with certain derivatives in the Sjöostrand class $M^{\infty,1}$. We prove that the corresponding propagator is bounded on modulation spaces. The present results improve several contributions recently appeared in the literature and can be regarded as the evolution counterpart of the fundamental result of Sjöstrand about the boundedness of pseudodifferential operators with symbols in that class.
Finally we consider nonlinear perturbations of real-analytic type and we prove local wellposedness of the corresponding initial value problem in certain modulation spaces.
We consider a class of linear Schrödinger equations in $\mathbb{R}^d$ with rough Hamiltonian, namely with certain derivatives in the Sjöostrand class $M^{\infty,1}$. We prove that the corresponding propagator is bounded on modulation spaces. The present results improve several contributions recently appeared in the literature and can be regarded as the evolution counterpart of the fundamental result of Sjöstrand about the boundedness of pseudodifferential operators with symbols in that class.
Finally we consider nonlinear perturbations of real-analytic type and we prove local wellposedness of the corresponding initial value problem in certain modulation spaces.
2015, 35(10): 4823-4829
doi: 10.3934/dcds.2015.35.4823
+[Abstract](2639)
+[PDF](445.4KB)
Abstract:
We consider special flows over two-dimensional rotations by $(\alpha,\beta)$ on $\mathbb{T}^2$ and under piecewise $C^2$ roof functions $f$ satisfying von Neumann's condition \[\int_{\mathbb{T}^2}f_x(x,y)\,dx\,dy\neq 0\quad\text{ and }\quad \int_{\mathbb{T}^2}f_y(x,y)\,dx \,dy\neq 0.\] For an uncountable set of $(\alpha,\beta)$ with both $\alpha$ and $\beta$ of unbounded partial quotients the mixing property is proved to hold.
We consider special flows over two-dimensional rotations by $(\alpha,\beta)$ on $\mathbb{T}^2$ and under piecewise $C^2$ roof functions $f$ satisfying von Neumann's condition \[\int_{\mathbb{T}^2}f_x(x,y)\,dx\,dy\neq 0\quad\text{ and }\quad \int_{\mathbb{T}^2}f_y(x,y)\,dx \,dy\neq 0.\] For an uncountable set of $(\alpha,\beta)$ with both $\alpha$ and $\beta$ of unbounded partial quotients the mixing property is proved to hold.
2015, 35(10): 4831-4838
doi: 10.3934/dcds.2015.35.4831
+[Abstract](3124)
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Abstract:
We study the ergodic properties of a classical two-particle system with square-well pair potential in an interval.
We study the ergodic properties of a classical two-particle system with square-well pair potential in an interval.
2015, 35(10): 4839-4858
doi: 10.3934/dcds.2015.35.4839
+[Abstract](3283)
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Abstract:
We study the bifurcation curve and exact multiplicity of positive solutions of a two-point boundary value problem arising in a theory of thermal explosion \begin{equation*} \left\{ \begin{array}{l} u^{\prime\prime}(x) + \lambda \exp ( \frac{au}{a+u}) =0, -1 < x < 1, \\ u(-1)=u(1)=0, \end{array} \right. \end{equation*} where $\lambda >0$ is the Frank--Kamenetskii parameter and $a>0$ is the activation energy parameter. By developing some new time-map techniques and applying Sturm's theorem, we prove that, if $a\geq a^{\ast \ast }\approx 4.107$, the bifurcation curve is S-shaped on the $(\lambda ,\Vert u \Vert _{\infty })$-plane. Our result improves one of the main results in Hung and Wang (J. Differential Equations 251 (2011) 223--237).
We study the bifurcation curve and exact multiplicity of positive solutions of a two-point boundary value problem arising in a theory of thermal explosion \begin{equation*} \left\{ \begin{array}{l} u^{\prime\prime}(x) + \lambda \exp ( \frac{au}{a+u}) =0, -1 < x < 1, \\ u(-1)=u(1)=0, \end{array} \right. \end{equation*} where $\lambda >0$ is the Frank--Kamenetskii parameter and $a>0$ is the activation energy parameter. By developing some new time-map techniques and applying Sturm's theorem, we prove that, if $a\geq a^{\ast \ast }\approx 4.107$, the bifurcation curve is S-shaped on the $(\lambda ,\Vert u \Vert _{\infty })$-plane. Our result improves one of the main results in Hung and Wang (J. Differential Equations 251 (2011) 223--237).
2015, 35(10): 4859-4887
doi: 10.3934/dcds.2015.35.4859
+[Abstract](2928)
+[PDF](550.0KB)
Abstract:
Let $\Omega$ be a smooth bounded domain in $\mathbb{R}^n$, $n\ge 3$, $0 < m \le \frac{n-2}{n}$, $a_1,a_2,\dots, a_{i_0}\in\Omega$, $\delta_0 = \min_{1 \le i \le i_0} \mbox{dist} (a_i,∂\Omega)$ and let $\Omega_{\delta}=\Omega\setminus\cup_{i=1}^{i_0}B_{\delta}(a_i)$ and $\hat{\Omega}=\Omega\setminus\{a_1\,\dots,a_{i_0}\}$. For any $0<\delta<\delta_0$ we will prove the existence and uniqueness of positive solution of the Neumann problem for the equation $u_t=\Delta u^m$ in $\Omega_{\delta}\times (0,T)$ for some $T>0$. We will prove the existence of singular solutions of this equation in $\hat{\Omega}\times (0,T)$ for some $T>0$ that blow-up at the points $a_1,\dots, a_{i_0}$.
Let $\Omega$ be a smooth bounded domain in $\mathbb{R}^n$, $n\ge 3$, $0 < m \le \frac{n-2}{n}$, $a_1,a_2,\dots, a_{i_0}\in\Omega$, $\delta_0 = \min_{1 \le i \le i_0} \mbox{dist} (a_i,∂\Omega)$ and let $\Omega_{\delta}=\Omega\setminus\cup_{i=1}^{i_0}B_{\delta}(a_i)$ and $\hat{\Omega}=\Omega\setminus\{a_1\,\dots,a_{i_0}\}$. For any $0<\delta<\delta_0$ we will prove the existence and uniqueness of positive solution of the Neumann problem for the equation $u_t=\Delta u^m$ in $\Omega_{\delta}\times (0,T)$ for some $T>0$. We will prove the existence of singular solutions of this equation in $\hat{\Omega}\times (0,T)$ for some $T>0$ that blow-up at the points $a_1,\dots, a_{i_0}$.
2015, 35(10): 4889-4903
doi: 10.3934/dcds.2015.35.4889
+[Abstract](3823)
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The Cauchy problem of Klein-Gordon equations is considered for power and exponential type nonlinear terms with singular weights. Time local and global solutions are shown to exist in the energy class. The Caffarelli-Kohn-Nirenberg inequality and the Trudinger-Moser type inequality with singular weights are applied to the problem.
The Cauchy problem of Klein-Gordon equations is considered for power and exponential type nonlinear terms with singular weights. Time local and global solutions are shown to exist in the energy class. The Caffarelli-Kohn-Nirenberg inequality and the Trudinger-Moser type inequality with singular weights are applied to the problem.
2015, 35(10): 4905-4929
doi: 10.3934/dcds.2015.35.4905
+[Abstract](3629)
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We show that the fractional Laplacian can be viewed as a Dirichlet-to-Neumann map for a degenerate hyperbolic problem, namely, the wave equation with an additional diffusion term that blows up at time zero. A solution to this wave extension problem is obtained from the Schrödinger group by means of an oscillatory subordination formula, which also allows us to find kernel representations for such solutions. Asymptotics of related oscillatory integrals are analysed in order to determine the correct domains for initial data in the general extension problem involving non-negative self-adjoint operators. An alternative approach using Bessel functions is also described.
We show that the fractional Laplacian can be viewed as a Dirichlet-to-Neumann map for a degenerate hyperbolic problem, namely, the wave equation with an additional diffusion term that blows up at time zero. A solution to this wave extension problem is obtained from the Schrödinger group by means of an oscillatory subordination formula, which also allows us to find kernel representations for such solutions. Asymptotics of related oscillatory integrals are analysed in order to determine the correct domains for initial data in the general extension problem involving non-negative self-adjoint operators. An alternative approach using Bessel functions is also described.
2015, 35(10): 4931-4954
doi: 10.3934/dcds.2015.35.4931
+[Abstract](3091)
+[PDF](452.3KB)
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This paper deals with a diffusive stage structured model with state-dependent delay which is assumed to be an increasing function of the population density. Compared with the constant delay, the state--dependent delay makes the dynamic behavior more complex. For the state--dependent delay system, the dynamic behavior is dependent of the diffusion coefficients, while the equilibrium state of constant delay system is not destabilized by diffusion. Through calculating the minimum wave speed, we find that the wave is slowed down by the state-dependent delay. Then, the existence of traveling waves is obtained by constructing a pair of upper--lower solutions and using Schauder's fixed point theorem. Finally, the traveling wavefront solutions for large wave speed are also discussed, and the fronts appear to be all monotone, regardless of the state dependent delay. This is an interesting property, since many findings are frequently reported that delay causes a loss of monotonicity, with the front developing a prominent hump in some other delay models.
This paper deals with a diffusive stage structured model with state-dependent delay which is assumed to be an increasing function of the population density. Compared with the constant delay, the state--dependent delay makes the dynamic behavior more complex. For the state--dependent delay system, the dynamic behavior is dependent of the diffusion coefficients, while the equilibrium state of constant delay system is not destabilized by diffusion. Through calculating the minimum wave speed, we find that the wave is slowed down by the state-dependent delay. Then, the existence of traveling waves is obtained by constructing a pair of upper--lower solutions and using Schauder's fixed point theorem. Finally, the traveling wavefront solutions for large wave speed are also discussed, and the fronts appear to be all monotone, regardless of the state dependent delay. This is an interesting property, since many findings are frequently reported that delay causes a loss of monotonicity, with the front developing a prominent hump in some other delay models.
2015, 35(10): 4955-4986
doi: 10.3934/dcds.2015.35.4955
+[Abstract](3705)
+[PDF](3112.5KB)
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Stability analysis is performed for a linear differential equation with two delays. Geometric arguments show that when the two delays are rationally dependent, then the region of stability increases. When the ratio has the form $1/n$, this study finds the asymptotic shape and size of the stability region. For example, a delay ratio of $1/3$ asymptotically produces a stability region about 44.3% larger than any nearby delay ratios, showing extreme sensitivity in the delays. The study provides a systematic and geometric approach to finding the eigenvalues on the boundary of stability for this delay differential equation. A nonlinear model with two delays illustrates how our methods can be applied.
Stability analysis is performed for a linear differential equation with two delays. Geometric arguments show that when the two delays are rationally dependent, then the region of stability increases. When the ratio has the form $1/n$, this study finds the asymptotic shape and size of the stability region. For example, a delay ratio of $1/3$ asymptotically produces a stability region about 44.3% larger than any nearby delay ratios, showing extreme sensitivity in the delays. The study provides a systematic and geometric approach to finding the eigenvalues on the boundary of stability for this delay differential equation. A nonlinear model with two delays illustrates how our methods can be applied.
2015, 35(10): 4987-5001
doi: 10.3934/dcds.2015.35.4987
+[Abstract](2813)
+[PDF](435.8KB)
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The aim of this work is to show the existence of free boundary minimal surfaces of Saddle Tower type which are embedded in a vertical solid cylinder in $\mathbb{R}^3$ and invariant with respect to a vertical translation. The number of boundary curves equals $2l$, $l \ge 2$. These surfaces come in families depending on one parameter and they converge to $2l$ vertical stripes having a common vertical intersection line. Such surfaces are obtained by perturbing the symmetrically modified Saddle Tower minimal surfaces.
The aim of this work is to show the existence of free boundary minimal surfaces of Saddle Tower type which are embedded in a vertical solid cylinder in $\mathbb{R}^3$ and invariant with respect to a vertical translation. The number of boundary curves equals $2l$, $l \ge 2$. These surfaces come in families depending on one parameter and they converge to $2l$ vertical stripes having a common vertical intersection line. Such surfaces are obtained by perturbing the symmetrically modified Saddle Tower minimal surfaces.
2015, 35(10): 5003-5036
doi: 10.3934/dcds.2015.35.5003
+[Abstract](2951)
+[PDF](540.9KB)
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In this paper we deal with Robin and Neumann parametric elliptic equations driven by a nonhomogeneous differential operator and with a reaction that exhibits competing nonlinearities (concave-convex nonlinearities). For the Robin problem and without employing the Ambrosetti-Rabinowitz condition, we prove a bifurcation theorem for the positive solutions for small values of the parameter $\lambda>0$. For the Neumann problem with a different geometry and using the Ambrosetti-Rabinowitz condition we prove bifurcation for large values of $\lambda>0$.
In this paper we deal with Robin and Neumann parametric elliptic equations driven by a nonhomogeneous differential operator and with a reaction that exhibits competing nonlinearities (concave-convex nonlinearities). For the Robin problem and without employing the Ambrosetti-Rabinowitz condition, we prove a bifurcation theorem for the positive solutions for small values of the parameter $\lambda>0$. For the Neumann problem with a different geometry and using the Ambrosetti-Rabinowitz condition we prove bifurcation for large values of $\lambda>0$.
2015, 35(10): 5037-5054
doi: 10.3934/dcds.2015.35.5037
+[Abstract](2636)
+[PDF](447.5KB)
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We study partially hyperbolic diffeomorphisms satisfying a trapping property which makes them look as if they were Anosov at large scale. We show that, as expected, they share several properties with Anosov diffeomorphisms. We construct an expansive quotient of the dynamics and study some dynamical consequences related to this quotient.
We study partially hyperbolic diffeomorphisms satisfying a trapping property which makes them look as if they were Anosov at large scale. We show that, as expected, they share several properties with Anosov diffeomorphisms. We construct an expansive quotient of the dynamics and study some dynamical consequences related to this quotient.
2015, 35(10): 5055-5082
doi: 10.3934/dcds.2015.35.5055
+[Abstract](2921)
+[PDF](1338.4KB)
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This article is concerned with the study of Mather's $\beta$-function associated to Birkhoff billiards. This function corresponds to the minimal average action of orbits with a prescribed rotation number and, from a different perspective, it can be related to the maximal perimeter of periodic orbits with a given rotation number, the so-called Marked length spectrum. After having recalled its main properties and its relevance to the study of the billiard dynamics, we stress its connections to some intriguing open questions: Birkhoff conjecture and the isospectral rigidity of convex billiards. Both these problems, in fact, can be conveniently translated into questions on this function. This motivates our investigation aiming at understanding its main features and properties. In particular, we provide an explicit representation of the coefficients of its (formal) Taylor expansion at zero, only in terms of the curvature of the boundary. In the case of integrable billiards, this result provides a representation formula for the $\beta$-function near $0$. Moreover, we apply and check these results in the case of circular and elliptic billiards.
This article is concerned with the study of Mather's $\beta$-function associated to Birkhoff billiards. This function corresponds to the minimal average action of orbits with a prescribed rotation number and, from a different perspective, it can be related to the maximal perimeter of periodic orbits with a given rotation number, the so-called Marked length spectrum. After having recalled its main properties and its relevance to the study of the billiard dynamics, we stress its connections to some intriguing open questions: Birkhoff conjecture and the isospectral rigidity of convex billiards. Both these problems, in fact, can be conveniently translated into questions on this function. This motivates our investigation aiming at understanding its main features and properties. In particular, we provide an explicit representation of the coefficients of its (formal) Taylor expansion at zero, only in terms of the curvature of the boundary. In the case of integrable billiards, this result provides a representation formula for the $\beta$-function near $0$. Moreover, we apply and check these results in the case of circular and elliptic billiards.
2015, 35(10): 5083-5105
doi: 10.3934/dcds.2015.35.5083
+[Abstract](3047)
+[PDF](530.6KB)
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In this paper, the compressible magnetohydrodynamic equations without heat conductivity are considered in $\mathbb{R}^3$. The global solution is obtained by combining the local existence and a priori estimates under the smallness assumption on the initial perturbation in $H^l (l>3)$. But we don't need the bound of $L^1$ norm. This is different from the work [5]. Our proof is based on pure estimates to get the time decay estimates on the pressure, velocity and magnet field. In particular, we use a fast decay of velocity gradient to get the uniform bound of the non-dissipative entropy, which is sufficient to close the priori estimates. In addition, we study the optimal convergence rates of the global solution.
In this paper, the compressible magnetohydrodynamic equations without heat conductivity are considered in $\mathbb{R}^3$. The global solution is obtained by combining the local existence and a priori estimates under the smallness assumption on the initial perturbation in $H^l (l>3)$. But we don't need the bound of $L^1$ norm. This is different from the work [5]. Our proof is based on pure estimates to get the time decay estimates on the pressure, velocity and magnet field. In particular, we use a fast decay of velocity gradient to get the uniform bound of the non-dissipative entropy, which is sufficient to close the priori estimates. In addition, we study the optimal convergence rates of the global solution.
2015, 35(10): 5107-5131
doi: 10.3934/dcds.2015.35.5107
+[Abstract](3073)
+[PDF](477.9KB)
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A delayed lattice dynamical system with non-local diffusion and interaction is considered in this paper. The exact asymptotics of the wave profile at both wave tails is derived, and all the wave profiles are shown to be strictly increasing. Moreover, we prove that the wave profile with a given admissible speed is unique up to translation. These results generalize earlier monotonicity, asymptotics and uniqueness results in the literature.
A delayed lattice dynamical system with non-local diffusion and interaction is considered in this paper. The exact asymptotics of the wave profile at both wave tails is derived, and all the wave profiles are shown to be strictly increasing. Moreover, we prove that the wave profile with a given admissible speed is unique up to translation. These results generalize earlier monotonicity, asymptotics and uniqueness results in the literature.
2015, 35(10): 5133-5152
doi: 10.3934/dcds.2015.35.5133
+[Abstract](2657)
+[PDF](455.1KB)
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We present here a construction of horseshoes for any $\mathcal{C}^{1+\alpha}$ mapping $f$ preserving an ergodic hyperbolic measure $\mu$ with $h_{\mu}(f)>0$ and then deduce that the exponential growth rate of the number of periodic points for any $\mathcal{C}^{1+\alpha}$ mapping $f$ is greater than or equal to $h_{\mu}(f)$. We also prove that the exponential growth rate of the number of hyperbolic periodic points is equal to the hyperbolic entropy. The hyperbolic entropy means the entropy resulting from hyperbolic measures.
We present here a construction of horseshoes for any $\mathcal{C}^{1+\alpha}$ mapping $f$ preserving an ergodic hyperbolic measure $\mu$ with $h_{\mu}(f)>0$ and then deduce that the exponential growth rate of the number of periodic points for any $\mathcal{C}^{1+\alpha}$ mapping $f$ is greater than or equal to $h_{\mu}(f)$. We also prove that the exponential growth rate of the number of hyperbolic periodic points is equal to the hyperbolic entropy. The hyperbolic entropy means the entropy resulting from hyperbolic measures.
2015, 35(10): 5153-5169
doi: 10.3934/dcds.2015.35.5153
+[Abstract](3132)
+[PDF](490.5KB)
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This paper is concerned with a four-component Camassa-Holm type system proposed in [37], where its bi-Hamiltonian structure and infinitely many conserved quantities were constructed. In the paper, we first establish the local well-posedness for the system. Then we present several global existence and blow-up results for two integrable two-component subsystems.
This paper is concerned with a four-component Camassa-Holm type system proposed in [37], where its bi-Hamiltonian structure and infinitely many conserved quantities were constructed. In the paper, we first establish the local well-posedness for the system. Then we present several global existence and blow-up results for two integrable two-component subsystems.
2020
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