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Discrete & Continuous Dynamical Systems - A

June 2017 , Volume 37 , Issue 6

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The Chowla and the Sarnak conjectures from ergodic theory point of view
El Houcein El Abdalaoui, Joanna Kułaga-Przymus, Mariusz Lemańczyk and Thierry de la Rue
2017, 37(6): 2899-2944 doi: 10.3934/dcds.2017125 +[Abstract](355) +[HTML](8) +[PDF](805.9KB)
Abstract:

We rephrase the conditions from the Chowla and the Sarnak conjectures in abstract setting, that is, for sequences in \begin{document}$\{-1,0,1\}^{{\mathbb{N}^*}}$\end{document}, and introduce several natural generalizations. We study the relationships between these properties and other notions from topological dynamics and ergodic theory.

Singular cw-expansive flows
Alfonso Artigue
2017, 37(6): 2945-2956 doi: 10.3934/dcds.2017126 +[Abstract](273) +[HTML](1) +[PDF](381.0KB)
Abstract:

We study continuum-wise expansive flows with fixed points on metric spaces and low dimensional manifolds. We give sufficient conditions for a surface flow to be singular cw-expansive and examples showing that cw-expansivity does not imply expansivity. We also construct a singular Axiom A vector field on a three-manifold being singular cw-expansive and with a Lorenz attractor and a Lorenz repeller in its non-wandering set.

Li-Yorke chaos for dendrite maps with zero topological entropy and ω-limit sets
Ghassen Askri
2017, 37(6): 2957-2976 doi: 10.3934/dcds.2017127 +[Abstract](345) +[HTML](10) +[PDF](1098.6KB)
Abstract:

Let X be a dendrite with set of endpoints $E(X)$ closed and let $f:~X \to X$ be a continuous map with zero topological entropy. Let $P(f)$ be the set of periodic points of f and let L be an ω-limit set of f. We prove that if L is infinite then $L\cap P(f)\subset E(X)^{\prime}$, where $E(X)^{\prime}$ is the set of all accumulations points of $E(X)$. Furthermore, if $E(X)$ is countable and L is uncountable then $L\cap P(f)=\emptyset$. We also show that if $E(X)^{\prime}$ is finite and L is uncountable then there is a sequence of subdendrites $(D_k)_{k ≥ 1}$ of X and a sequence of integers $n_k ≥ 2$ satisfying the following properties. For all $k≥1$,
  1. $f^{α_k}(D_k)=D_k$ where $α_k=n_1 n_2 \dots n_k$,
  2. $\cup_{k=0}^{n_j -1}f^{k α_{j-1}}(D_{j}) \subset D_{j-1}$ for all $j≥q 2$,
  3. $L \subset \cup_{i=0}^{α_k -1}f^{i}(D_k)$,
  4. $f(L \cap f^{i}(D_k))=L\cap f^{i+1}(D_k)$ for any $ 0≤q i ≤q α_{k}-1$. In particular, $L \cap f^{i}(D_k) ≠ \emptyset$,
  5. $f^{i}(D_k)\cap f^{j}(D_k)$ has empty interior for any $ 0≤q i≠ j<α_k $.
  As a consequence, if f has a Li-Yorke pair $(x,y)$ with $ω_f(x)$ or $ω_f(y)$ uncountable then f is Li-Yorke chaotic.

An approximation solvability method for nonlocal semilinear differential problems in Banach spaces
Irene Benedetti, Nguyen Van Loi and Valentina Taddei
2017, 37(6): 2977-2998 doi: 10.3934/dcds.2017128 +[Abstract](352) +[HTML](5) +[PDF](461.4KB)
Abstract:

A new approximation solvability method is developed for the study of semilinear differential equations with nonlocal conditions without the compactness of the semigroup and of the nonlinearity. The method is based on the Yosida approximations of the generator of C0-semigroup, the continuation principle, and the weak topology. It is shown how the abstract result can be applied to study the reaction-diffusion models.

Scattering of solutions to the nonlinear Schrödinger equations with regular potentials
Xing Cheng, Ze Li and Lifeng Zhao
2017, 37(6): 2999-3023 doi: 10.3934/dcds.2017129 +[Abstract](286) +[HTML](4) +[PDF](542.8KB)
Abstract:

In this paper, we prove the scattering of radial solutions to high dimensional energy-critical nonlinear Schrödinger equations with regular potentials in the defocusing case.

A p-Laplacian supercritical Neumann problem
Francesca Colasuonno and Benedetta Noris
2017, 37(6): 3025-3057 doi: 10.3934/dcds.2017130 +[Abstract](290) +[HTML](1) +[PDF](610.2KB)
Abstract:

For p > 2, we consider the quasilinear equation \begin{document}$-\Delta_p u+|u|^{p-2}u=g(u)$\end{document} in the unit ball B of \begin{document}$\mathbb R^N$\end{document}, with homogeneous Neumann boundary conditions. The assumptions on g are very mild and allow the nonlinearity to be possibly supercritical in the sense of Sobolev embeddings. We prove the existence of a nonconstant, positive, radially nondecreasing solution via variational methods. In the case \begin{document}$g(u)=|u|^{q-2}u$\end{document}, we detect the asymptotic behavior of these solutions as \begin{document}$q\to \infty$\end{document}.

Onofri inequalities and rigidity results
Jean Dolbeault, Maria J. Esteban and Gaspard Jankowiak
2017, 37(6): 3059-3078 doi: 10.3934/dcds.2017131 +[Abstract](276) +[HTML](1) +[PDF](453.8KB)
Abstract:

This paper is devoted to the Moser-Trudinger-Onofri inequality on smooth compact connected Riemannian manifolds. We establish a rigidity result for the Euler-Lagrange equation and deduce an estimate of the optimal constant in the inequality on two-dimensional closed Riemannian manifolds. Compared to existing results, we provide a non-local criterion which is well adapted to variational methods, introduce a nonlinear flow along which the evolution of a functional related with the inequality is monotone and get an integral remainder term which allows us to discuss optimality issues. As an important application of our method, we also consider the non-compact case of the Moser-Trudinger-Onofri inequality on the two-dimensional Euclidean space, with weights. The standard weight is the one that is computed when projecting the two-dimensional sphere using the stereographic projection, but we also give more general results which are of interest, for instance, for the Keller-Segel model in chemotaxis.

Hyperbolic billiards on polytopes with contracting reflection laws
Pedro Duarte, José Pedro GaivÃo and Mohammad Soufi
2017, 37(6): 3079-3109 doi: 10.3934/dcds.2017132 +[Abstract](300) +[HTML](8) +[PDF](803.4KB)
Abstract:

We study billiards on polytopes in \begin{document}${\mathbb{R}^d} $\end{document} with contracting reflection laws, i.e. non-standard reflection laws that contract the reflection angle towards the normal. We prove that billiards on generic polytopes are uniformly hyperbolic provided there exists a positive integer \begin{document} $k$ \end{document} such that for any \begin{document} $k$ \end{document} consecutive collisions, the corresponding normals of the faces of the polytope where the collisions took place generate \begin{document}${\mathbb{R}^d} $\end{document}. As an application of our main result we prove that billiards on generic polytopes are uniformly hyperbolic if either the contracting reflection law is sufficiently close to the specular or the polytope is obtuse. Finally, we study in detail the billiard on a family of \begin{document} $3$ \end{document}-dimensional simplexes.

On the decoupling of the improved Boussinesq equation into two uncoupled Camassa-Holm equations
H. A. Erbay, S. Erbay and A. Erkip
2017, 37(6): 3111-3122 doi: 10.3934/dcds.2017133 +[Abstract](309) +[HTML](1) +[PDF](400.9KB)
Abstract:

We rigorously establish that, in the long-wave regime characterized by the assumptions of long wavelength and small amplitude, bidirectional solutions of the improved Boussinesq equation tend to associated solutions of two uncoupled Camassa-Holm equations. We give a precise estimate for approximation errors in terms of two small positive parameters measuring the effects of nonlinearity and dispersion. Our results demonstrate that, in the present regime, any solution of the improved Boussinesq equation is split into two waves propagating in opposite directions independently, each of which is governed by the Camassa-Holm equation. We observe that the approximation error for the decoupled problem considered in the present study is greater than the approximation error for the unidirectional problem characterized by a single Camassa-Holm equation. We also consider lower order approximations and we state similar error estimates for both the Benjamin-Bona-Mahony approximation and the Korteweg-de Vries approximation.

Dynamic rays of bounded-type transcendental self-maps of the punctured plane
Núria Fagella and David Martí-Pete
2017, 37(6): 3123-3160 doi: 10.3934/dcds.2017134 +[Abstract](474) +[HTML](5) +[PDF](10144.7KB)
Abstract:

We study the escaping set of functions in the class \begin{document} $\mathcal{B}^*$ \end{document}, that is, transcendental self-maps of \begin{document} $\mathbb{C}^*$ \end{document} for which the set of singular values is contained in a compact annulus of \begin{document} $\mathbb{C}^*$ \end{document} that separates zero from infinity. For functions in the class \begin{document} $\mathcal{B}^*$ \end{document}, escaping points lie in their Julia set. If \begin{document} $f$ \end{document} is a composition of finite order transcendental self-maps of \begin{document} $\mathbb{C}^*$ \end{document} (and hence, in the class \begin{document} $\mathcal{B}^*$ \end{document}), then we show that every escaping point of \begin{document} $f$ \end{document} can be connected to one of the essential singularities by a curve of points that escape uniformly. Moreover, for every sequence \begin{document} $e∈\{0,∞\}^{\mathbb{N}_0}$ \end{document}, we show that the escaping set of \begin{document} $f$ \end{document} contains a Cantor bouquet of curves that accumulate to the set \begin{document} $\{0,∞\}$ \end{document} according to \begin{document} $e$ \end{document} under iteration by \begin{document} $f$ \end{document}.

Minimality of p-adic rational maps with good reduction
Aihua Fan, Shilei Fan, Lingmin Liao and Yuefei Wang
2017, 37(6): 3161-3182 doi: 10.3934/dcds.2017135 +[Abstract](372) +[HTML](5) +[PDF](468.8KB)
Abstract:

A rational map with good reduction in the field \begin{document} $\mathbb{Q}_p$ \end{document} of \begin{document} $p$ \end{document}-adic numbers defines a \begin{document} $1$ \end{document}-Lipschitz dynamical system on the projective line \begin{document} $\mathbb{P}^1(\mathbb{Q}_p)$ \end{document} over \begin{document} $\mathbb{Q}_p$ \end{document}. The dynamical structure of such a system is completely described by a minimal decomposition. That is to say, \begin{document} $\mathbb{P}^1(\mathbb{Q}_p)$ \end{document} is decomposed into three parts: finitely many periodic orbits; finite or countably many minimal subsystems each consisting of a finite union of balls; and the attracting basins of periodic orbits and minimal subsystems. For any prime \begin{document} $p$ \end{document}, a criterion of minimality for rational maps with good reduction is obtained. When \begin{document} $p=2$ \end{document}, a condition in terms of the coefficients of the rational map is proved to be necessary for the map being minimal and having good reduction, and sufficient for the map being minimal and \begin{document} $1$ \end{document}-Lipschitz. It is also proved that a rational map having good reduction of degrees \begin{document} $2$ \end{document}, \begin{document} $3$ \end{document} and \begin{document} $4$ \end{document} can never be minimal on the whole space \begin{document} $\mathbb{P}^1(\mathbb{Q}_2)$ \end{document}.

Existence of the solution for the viscous bipolar quantum hydrodynamic model
Boling Guo and Guangwu Wang
2017, 37(6): 3183-3210 doi: 10.3934/dcds.2017136 +[Abstract](247) +[HTML](8) +[PDF](561.3KB)
Abstract:

In this paper, we investigate the existence of classical solution of the viscous bipolar quantum hydrodynamic(QHD) models for ir-rotational fluid in a periodic domain. By applying the iteration method, we prove that the viscous bipolar QHD model has a local classical solution. Then we prove this solution is global with small initial data, based on a series of a priori estimates. Finally, we obtained the inviscid limit of this viscous quantum hydrodynamic model.

Sharp estimation for the solutions of delay differential and Halanay type inequalities
István Győri and László Horváth
2017, 37(6): 3211-3242 doi: 10.3934/dcds.2017137 +[Abstract](370) +[HTML](2) +[PDF](501.6KB)
Abstract:

The present paper develops a framework for a Halanay type nonautonomous delay differential inequality with maxima, and establishes necessary and/or sufficient conditions for the global attractivity of the zero solution. The emphasis is put on the rate of convergence based on the theory of the generalized characteristic equation. The applicability and the sharpness of the results are illustrated by examples. This work aspires to serve as a remarkable step towards a unified theory of the nonautonomous Halanay inequality.

Two-phase incompressible flows with variable density: An energetic variational approach
Jie Jiang, Yinghua Li and Chun Liu
2017, 37(6): 3243-3284 doi: 10.3934/dcds.2017138 +[Abstract](499) +[HTML](4) +[PDF](630.1KB)
Abstract:

In this paper, we study a diffuse-interface model for two-phase incompressible flows with different densities. First, we present a derivation of the model using an energetic variational approach. Our model allows large density ratio between the two phases and moreover, it is thermodynamically consistent and admits a dissipative energy law. Under suitable assumptions on the average density function, we establish the global existence of a weak solution in the 3D case as well as the global well-posedness of strong solutions in the 2D case to an initial-boundary problem for the resulting Allen-Cahn-Navier-Stokes system. Furthermore, we investigate the longtime behavior of the 2D strong solutions. In particular, we obtain existence of a maximal compact attractor and prove that the solution will converge to an equilibrium as time goes to infinity.

Local well-posedness of the Camassa-Holm equation on the real line
Jae Min Lee and Stephen C. Preston
2017, 37(6): 3285-3299 doi: 10.3934/dcds.2017139 +[Abstract](303) +[HTML](1) +[PDF](434.6KB)
Abstract:

In this paper we prove the local well-posedness of the Camassa-Holm equation on the real line in the space of continuously differentiable diffeomorphisms with an appropriate decaying condition. This work was motivated by G. Misiolek who proved the same result for the Camassa-Holm equation on the periodic domain. We use the Lagrangian approach and rewrite the equation as an ODE on the Banach space. Then by using the standard ODE technique, we prove existence and uniqueness. Finally, we show the continuous dependence of the solution on the initial data by using the topological group property of the diffeomorphism group.

The Cauchy problem and blow-up phenomena for a new integrable two-component peakon system with cubic nonlinearities
Xiuting Li and Lei Zhang
2017, 37(6): 3301-3325 doi: 10.3934/dcds.2017140 +[Abstract](251) +[HTML](1) +[PDF](542.8KB)
Abstract:

This paper studies the local well-posedness and blow-up phenomena for a new integrable two-component peakon system in the Besov space. Firstly, by utilizing the Littlewood-Paley theory, the logarithmic interpolation inequality and the Osgood's Lemma, we investigate the existence and uniqueness of the solution to the system in the critical Besov space $B_{2, 1}^{\frac{1}{2}}(\mathbb{R})× B_{2, 1}^{\frac{1}{2}}(\mathbb{R})$, and show that the data-to-solution mapping is Hölder continuous. Secondly, we derive a blow-up criteria for the Cauchy problem in the critical Besov space. Finally, with suitable conditions on the initial data, a new blow-up criteria for the system is obtained by virtue of the global conservative property of the potential densities $m$ and $n$ along the characteristics and the blow-up criteria at hand.

On the positive vector solutions for nonlinear fractional Laplacian systems with linear coupling
Dengfeng Lü and Shuangjie Peng
2017, 37(6): 3327-3352 doi: 10.3934/dcds.2017141 +[Abstract](421) +[HTML](4) +[PDF](559.6KB)
Abstract:

In this paper, a class of systems of two coupled nonlinear fractional Laplacian equations are investigated. Under very weak assumptions on the nonlinear terms $f$ and $g$, we establish some results about the existence of positive vector solutions and vector ground state solutions for the fractional Laplacian systems by using variational methods. In addition, we also study the asymptotic behavior of these solutions as the coupling parameter $β$ tends to zero.

Limit cycles for quadratic and cubic planar differential equations under polynomial perturbations of small degree
Ricardo M. Martins and Otávio M. L. Gomide
2017, 37(6): 3353-3386 doi: 10.3934/dcds.2017142 +[Abstract](357) +[HTML](11) +[PDF](575.3KB)
Abstract:

In this paper we consider planar systems of differential equations of the form

where \begin{document}$δ, \varepsilon$\end{document} are small parameters, $(p, q)$ are quadratic or cubic homogeneous polynomials such that the unperturbed system ($\varepsilon=0$) has an isochronous center at the origin and $P_n, Q_n$ are arbitrary perturbations. Estimates for the maximum number of limit cycles are provided and these estimatives are sharp for $n≤q 6$ (when $p, q$ are quadratic). When $p, q$ are cubic polynomials and $P_n, Q_n$ are linear, the problem is addressed from a numerical viewpoint and we also study the existence of limit cycles.

Generalized inhomogeneous Strichartz estimates
Robert Schippa
2017, 37(6): 3387-3410 doi: 10.3934/dcds.2017143 +[Abstract](252) +[HTML](2) +[PDF](486.0KB)
Abstract:

We prove new inhomogeneous generalized Strichartz estimates, which do not follow from the homogeneous generalized estimates by virtue of the Christ-Kiselev lemma. Instead, we make use of the bilinear interpolation argument worked out by Keel and Tao and refined by Foschi presented in a unified framework. Finally, we give a sample application.

Topological conjugacy of linear systems on Lie groups
Adriano Da Silva, Alexandre J. Santana and Simão N. Stelmastchuk
2017, 37(6): 3411-3421 doi: 10.3934/dcds.2017144 +[Abstract](261) +[HTML](3) +[PDF](380.1KB)
Abstract:

In this paper we study a classification of linear systems on Lie groups with respect to the conjugacy of the corresponding flows. We also describe stability according to Lyapunov exponents.

Global existence and low Mach number limit to a 3D compressible micropolar fluids model in a bounded domain
Jingrui Su
2017, 37(6): 3423-3434 doi: 10.3934/dcds.2017145 +[Abstract](295) +[HTML](1) +[PDF](353.9KB)
Abstract:

This paper is devoted to investigating the global existence of strong solutions to the three-dimensional compressible micropolar fluids model in a bounded domain with small initial data. Furthermore, we present the low Mach number limit to the corresponding problem.

Asymptotic stability of stationary solutions for magnetohydrodynamic equations
Zhong Tan and Leilei Tong
2017, 37(6): 3435-3465 doi: 10.3934/dcds.2017146 +[Abstract](309) +[HTML](2) +[PDF](533.6KB)
Abstract:

In this paper, we are concerned with the compressible magnetohydrodynamic equations with Coulomb force in three-dimensional space. We show the asymptotic stability of solutions to the Cauchy problem near the non-constant equilibrium state provided that the initial perturbation is sufficiently small. Moreover, the convergence rates are obtained by combining the linear Lp-Lq decay estimates and the higher-order energy estimates.

Global asymptotic stability of pushed traveling fronts for monostable delayed reaction-diffusion equations
Shi-Liang Wu, Tong-Chang Niu and Cheng-Hsiung Hsu
2017, 37(6): 3467-3486 doi: 10.3934/dcds.2017147 +[Abstract](388) +[HTML](1) +[PDF](490.5KB)
Abstract:

In this work we consider the global asymptotic stability of pushed traveling fronts for one-dimensional monostable reaction-diffusion equations with monotone delayed reactions. Pushed traveling front is a special type of critical wave front which converges to zero more rapidly than the near non-critical wave fronts. Recently, Trofimchuk et al. [16] proved the existence and uniqueness of pushed traveling fronts of the considered equation when the reaction term lost the sub-tangency condition. In this article, using the comparison method via a pair of super-and sub-solution and squeezing technique, we prove that the pushed traveling fronts are globally exponentially stable. This also gives an affirmative answer to an open problem presented by Solar and Trofimchuk [14].

Global exponential κ-dissipative semigroups and exponential attraction
Jin Zhang, Peter E. Kloeden, Meihua Yang and Chengkui Zhong
2017, 37(6): 3487-3502 doi: 10.3934/dcds.2017148 +[Abstract](269) +[HTML](5) +[PDF](453.5KB)
Abstract:

Globally exponential \begin{document} $κ-$ \end{document}dissipativity, a new concept of dissipativity for semigroups, is introduced. It provides a more general criterion for the exponential attraction of some evolutionary systems. Assuming that a semigroup \begin{document} $\{S(t)\}_{t≥q 0}$ \end{document} has a bounded absorbing set, then \begin{document} $\{S(t)\}_{t≥q 0}$ \end{document} is globally exponentially \begin{document} $κ-$ \end{document}dissipative if and only if there exists a compact set \begin{document} $\mathcal{A}^*$ \end{document} that is positive invariant and attracts any bounded subset exponentially. The set \begin{document} $\mathcal{A}^*$ \end{document} need not be finite dimensional. This result is illustrated with an application to a damped semilinear wave equation on a bounded domain.

The Cauchy problem for a generalized Novikov equation
Rudong Zheng and Zhaoyang Yin
2017, 37(6): 3503-3519 doi: 10.3934/dcds.2017149 +[Abstract](407) +[HTML](1) +[PDF](443.9KB)
Abstract:

We establish the local well-posedness for a generalized Novikov equation in nonhomogeneous Besov spaces. Besides, we obtain a blow-up criteria and provide a sufficient condition for strong solutions to blow up in finite time.

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