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Discrete & Continuous Dynamical Systems - A

January 2017 , Volume 37 , Issue 1

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Characterization of isoperimetric sets inside almost-convex cones
Eric Baer and Alessio Figalli
2017, 37(1): 1-14 doi: 10.3934/dcds.2017001 +[Abstract](901) +[HTML](28) +[PDF](406.2KB)
Abstract:

In this note we characterize isoperimetric regions inside almost-convex cones. More precisely, as in the case of convex cones, we show that isoperimetric sets are given by intersecting the cone with a ball centered at the origin.

Homogenization of singular quasilinear elliptic problems with natural growth in a domain with many small holes
José Carmona and Pedro J. Martínez-Aparicio
2017, 37(1): 15-31 doi: 10.3934/dcds.2017002 +[Abstract](614) +[HTML](12) +[PDF](436.4KB)
Abstract:

In this paper we consider the homogenization problem for quasilinear elliptic equations with singularities in the gradient, whose model is the following

where Ω is an open bounded set of \begin{document} $\mathbb{R}^N$ \end{document}, \begin{document} $θ ∈ (0,1)$ \end{document} and \begin{document} $f$ \end{document} is positive function that belongs to a certain Lebesgue's space. The homogenization of these equations is posed in a sequence of domains \begin{document} $Ω^\varepsilon$ \end{document} obtained by removing many small holes from a fixed domain Ω. We also give a corrector result.

On the Smoluchowski-Kramers approximation for SPDEs and its interplay with large deviations and long time behavior
Sandra Cerrai, Mark Freidlin and Michael Salins
2017, 37(1): 33-76 doi: 10.3934/dcds.2017003 +[Abstract](527) +[HTML](6) +[PDF](667.3KB)
Abstract:

We discuss here the validity of the small mass limit (the so-called Smoluchowski-Kramers approximation) on a fixed time interval for a class of semi-linear stochastic wave equations, both in the case of the presence of a constant friction term and in the case of the presence of a constant magnetic field. We also consider the small mass limit in an infinite time interval and we see how the approximation is stable in terms of the invariant measure and of the large deviation estimates and the exit problem from a bounded domain of the space of square integrable functions.

Nodal solutions for a generalized quasilinear Schrödinger equation with critical exponents
Kun Cheng and Yinbin Deng
2017, 37(1): 77-103 doi: 10.3934/dcds.2017004 +[Abstract](820) +[HTML](9) +[PDF](572.3KB)
Abstract:

This paper is concerned with constructing nodal radial solutions for generalized quasilinear Schrödinger equations in \begin{document} $\mathbb{R}^N$ \end{document} with critical growth which arise from plasma physics, fluid mechanics, as well as the self-channeling of a high-power ultashort laser in matter. We find the critical exponents for a generalized quasilinear Schrödinger equations and obtain the existence of sign-changing solution with k nodes for any given integer \begin{document} $k ≥ 0$ \end{document}.

Limiting distributions for countable state topological Markov chains with holes
Mark F. Demers, Christopher J. Ianzano, Philip Mayer, Peter Morfe and Elizabeth C. Yoo
2017, 37(1): 105-130 doi: 10.3934/dcds.2017005 +[Abstract](706) +[HTML](13) +[PDF](594.9KB)
Abstract:

We study the dynamics of countable state topological Markov chains with holes, where the hole is a countable union of 1-cylinders. For a large class of positive recurrent potentials and under natural assumptions on the surviving dynamics, we prove the existence of a limiting conditionally invariant distribution, which is the unique limit of regular densities under the renormalized dynamics conditioned on non-escape. We also prove the existence of a Gibbs measure on the survivor set, the set of points that never enter the hole, which is an equilibrium measure for the punctured potential of the open system. We prove that the Gurevic pressure on the survivor set equals the exponential escape rate from the open system. These results extend to the non-compact setting results previously available for finite state topological Markov chains.

On the strong-to-strong interaction case for doubly nonlocal Cahn-Hilliard equations
Ciprian G. Gal
2017, 37(1): 131-167 doi: 10.3934/dcds.2017006 +[Abstract](628) +[HTML](3) +[PDF](644.3KB)
Abstract:

We consider a doubly nonlocal Cahn-Hilliard equation for the nonlocal phase-separation of a two-component material in a bounded domain in the case when mass transport exhibits non-Fickian behavior. Such equations are important for phase-segregation phenomena that exhibit non-standard (anomalous) behaviors. Recently, four different cases were proposed to handle this important equation and the two levels of nonlocality and interaction that are present in the equation. The so-called strong-to-weak interaction case (when one kernel is integrable in some sense while the other is not) was investigated recently for the doubly nonlocal parabolic equation with a regular polynomial potential. In this contribution, we address the so-called strong-to-strong interaction case when both kernels are strongly singular and non-integrable in a suitable sense. We establish well-posedness results along with some regularity and long-time results in terms of finite dimensional global attractors.

Carleman estimates and Unique Continuation Property for 1-D viscous Camassa-Holm equation
Peng Gao
2017, 37(1): 169-188 doi: 10.3934/dcds.2017007 +[Abstract](678) +[HTML](3) +[PDF](424.6KB)
Abstract:

This paper is devoted to studying the 1-D viscous Camassa-Holm equation on a bounded interval. We first deduce the existence and uniqueness of strong solution to the viscous Camassa-Holm equation by using Galerkin method. Then we establish an identity for a second order parabolic operator, by applying this identity we obtain two global Carleman estimates for the linear viscous Camassa-Holm operator. Based on these estimates, we obtain two types of Unique Continuation Property for the viscous Camassa-Holm equation.

The wind-driven ocean circulation: Applying dynamical systems theory to a climate problem
Michael Ghil
2017, 37(1): 189-228 doi: 10.3934/dcds.2017008 +[Abstract](1336) +[HTML](13) +[PDF](11870.1KB)
Abstract:

The large-scale, near-surface flow of the mid-latitude oceans is dominated by the presence of a larger, anticyclonic and a smaller, cyclonic gyre. The two gyres share the eastward extension of western boundary currents, such as the Gulf Stream or Kuroshio, and are induced by the shear in the winds that cross the respective ocean basins. This physical phenomenology is described mathematically by a hierarchy of systems of nonlinear partial differential equations (PDEs). We study the low-frequency variability of this wind-driven, double-gyre circulation in mid-latitude ocean basins, subject to time-constant, purely periodic and more general forms of time-dependent wind stress. Both analytical and numerical methods of dynamical systems theory are applied to the PDE systems of interest. Recent work has focused on the application of non-autonomous and random forcing to double-gyre models. We discuss the associated pullback and random attractors and the non-uniqueness of the invariant measures that are obtained. The presentation moves from observations of the geophysical phenomena to modeling them and on to a proper mathematical understanding of the models thus obtained. Connections are made with the highly topical issues of climate change and climate sensitivity.

Oscillatory orbits in the restricted elliptic planar three body problem
Marcel Guardia, Tere M. Seara, Pau Martín and Lara Sabbagh
2017, 37(1): 229-256 doi: 10.3934/dcds.2017009 +[Abstract](782) +[HTML](3) +[PDF](548.2KB)
Abstract:

The restricted planar elliptic three body problem models the motion of a massless body under the Newtonian gravitational force of two other bodies, the primaries, which evolve in Keplerian ellipses.

A trajectory is called oscillatory if it leaves every bounded region but returns infinitely often to some fixed bounded region. We prove the existence of such type of trajectories for any values for the masses of the primaries provided the eccentricity of the Keplerian ellipses is small.

Global well-posedness for the derivative nonlinear Schrödinger equation in $H^{\frac 12} (\mathbb{R} )$
Zihua Guo and Yifei Wu
2017, 37(1): 257-264 doi: 10.3934/dcds.2017010 +[Abstract](578) +[HTML](6) +[PDF](349.9KB)
Abstract:

We prove that the derivative nonlinear Schrödinger equation is globally well-posed in \begin{document}$H^{\frac 12} (\mathbb{R} )$\end{document} when the mass of initial data is strictly less than 4π.

Liouville theorems for stable solutions of the weighted Lane-Emden system
Hatem Hajlaoui, Abdellaziz Harrabi and Foued Mtiri
2017, 37(1): 265-279 doi: 10.3934/dcds.2017011 +[Abstract](876) +[HTML](8) +[PDF](413.3KB)
Abstract:

We examine the general weighted Lane-Emden system

where \begin{document}$1 <p≤qθ$\end{document} and \begin{document}$ρ: \mathbb{R}^N \to \mathbb{R}$\end{document} is a radial continuous function satisfying \begin{document}$ρ(x)≥q A(1+|x|^2)^{\frac{α}{2}}$\end{document} in \begin{document}$\mathbb{R}^N$\end{document} for some \begin{document}$α≥q 0$\end{document} and \begin{document}$A>0$\end{document}. We prove some Liouville type results for stable solution and improve the previous works [2, 9, 12]. In particular, we establish a new comparison property (see Proposition 1 below) which is crucial to handle the case \begin{document}$1 < p ≤q \frac{4}{3}$\end{document}. Our results can be applied also to the weighted Lane-Emden equation \begin{document}$-Δ u = ρ(x)u^p$\end{document} in \begin{document}$\mathbb{R}^N$\end{document}.

Discrete Schrödinger equation and ill-posedness for the Euler equation
In-Jee Jeong and Benoit Pausader
2017, 37(1): 281-293 doi: 10.3934/dcds.2017012 +[Abstract](758) +[HTML](3) +[PDF](449.5KB)
Abstract:

We consider the 2D Euler equation with periodic boundary conditions in a family of Banach spaces based on the Fourier coefficients, and show that it is ill-posed in the sense that 'norm inflation' occurs. The proof is based on the observation that the evolution of certain perturbations of the 'Kolmogorov flow' given in velocity by

can be well approximated by the linear Schrödinger equation, at least for a short period of time.

Mixed dimensional infinite soliton trains for nonlinear Schrödinger equations
Liren Lin and Tai-Peng Tsai
2017, 37(1): 295-336 doi: 10.3934/dcds.2017013 +[Abstract](569) +[HTML](9) +[PDF](660.3KB)
Abstract:

In this note we construct mixed dimensional infinite soliton trains, which are solutions of nonlinear Schrödinger equations whose asymptotic profiles at time infinity consist of infinitely many solitons of multiple dimensions. For example infinite line-point soliton trains in 2D space, and infinite plane-line-point soliton trains in 3D space. This note extends the works of Le Coz, Li and Tsai [6,7], where single dimensional trains are considered. In our approach, spatial L bounds for lower dimensional trains play an essential role.

Finiteness and existence of attractors and repellers on sectional hyperbolic sets
A. M. López
2017, 37(1): 337-354 doi: 10.3934/dcds.2017014 +[Abstract](621) +[HTML](4) +[PDF](426.3KB)
Abstract:

We study small perturbations of a sectional hyperbolic set of a vector field on a compact manifold. Indeed, we obtain an upper bound for the number of attractors and repellers that can arise from these perturbations. Moreover, no repeller can arise if the unperturbed set has singularities, is connected and consists of nonwandering points.

Perron-Frobenius theory and frequency convergence for reducible substitutions
Martin Lustig and Caglar Uyanik
2017, 37(1): 355-385 doi: 10.3934/dcds.2017015 +[Abstract](547) +[HTML](5) +[PDF](485.7KB)
Abstract:

We prove a general version of the classical Perron-Frobenius convergence property for reducible matrices. We then apply this result to reducible substitutions and use it to produce limit frequencies for factors and hence invariant measures on the associated subshift. The analogous results are well known for primitive substitutions and have found many applications, but for reducible substitutions the tools provided here were so far missing from the theory.

A Hamiltonian approach for nonlinear rotational capillary-gravity water waves in stratified flows
Calin Iulian Martin
2017, 37(1): 387-404 doi: 10.3934/dcds.2017016 +[Abstract](657) +[HTML](1) +[PDF](411.4KB)
Abstract:

Under consideration here are two-dimensional rotational stratified water flows driven by gravity and surface tension, bounded below by a rigid flat bed and above by a free surface. The distribution of vorticity and of density is piecewise constant-with a jump across the interface separating the fluid of bigger density from the lighter fluid adjacent to the free surface. The main result is that the governing equations for the two-layered rotational stratified flows, as described above, admit a Hamiltonian formulation.

Long-time behavior of a fully discrete Lagrangian scheme for a family of fourth order equations
Horst Osberger
2017, 37(1): 405-434 doi: 10.3934/dcds.2017017 +[Abstract](591) +[HTML](17) +[PDF](1117.0KB)
Abstract:

A fully discrete Lagrangian scheme for solving a family of fourth order equations numerically is presented. The discretization is based on the equations' underlying gradient flow structure with respect to the Wasserstein metric, and preserves numerous of their most important structural properties by construction, like conservation of mass and entropy-dissipation.

In this paper, the long-time behavior of our discretization is analysed: We show that discrete solutions decay exponentially to equilibrium at the same rate as smooth solutions of the original problem. Moreover, we give a proof of convergence of discrete entropy minimizers towards Barenblatt-profiles or Gaussians, respectively, using $Γ$-convergence.

Zero sequence entropy and entropy dimension
Yixiao Qiao and Xiaoyao Zhou
2017, 37(1): 435-448 doi: 10.3934/dcds.2017018 +[Abstract](790) +[HTML](6) +[PDF](373.7KB)
Abstract:

Let \begin{document}$(X, T)$\end{document} be a topological dynamical system and \begin{document}$M(X)$\end{document} the set of all Borel probability measures on \begin{document}$X$\end{document} endowed with the weak\begin{document}$^*$\end{document} -topology. In this paper, it is shown that for a given sequence \begin{document}$S$\end{document} , a homeomorphism \begin{document}$T$\end{document} of \begin{document}$X$\end{document} has zero topological sequence entropy if and only if so does the induced homeomorphism \begin{document}$T$\end{document} of \begin{document}$M(X)$\end{document} . This extends the result of Glasner and Weiss [9,Theorem A] for topological entropy and also the result of Kerr and Li [15,Theorem 5.10]for null systems. Moreover, it turns out that the upper entropy dimension of \begin{document}$(X, T)$\end{document} is equal to that of \begin{document}$(M(X), T)$\end{document} . We also obtain the version of ergodic measure-preserving systems related to the sequence entropy and the upper entropy dimension.

Local and global well-posedness results for the Benjamin-Ono-Zakharov-Kuznetsov equation
Francis Ribaud and Stéphane Vento
2017, 37(1): 449-483 doi: 10.3934/dcds.2017019 +[Abstract](610) +[HTML](20) +[PDF](677.9KB)
Abstract:

We show that the initial value problem associated to the dispersive generalized Benjamin-Ono-Zakharov-Kuznetsov equation

is locally well-posed in the spaces \begin{document}$E^s$\end{document}, \begin{document}$s > \frac{2}{\alpha } - \frac{3}{4}$\end{document}, endowed with the norm \begin{document}$\|f{{\|}_{{{E}^{s}}}}=\|{{\left\langle {{\left| \xi \right|}^{\alpha }}+{{\mu }^{2}} \right\rangle }^{s}}\hat{f}{{\|}_{{{L}^{2}}({{\mathbb{R}}^{2}})}}.$\end{document} As a consequence, we get the global well-posedness in the energy space\begin{document}$E^{1/2}$\end{document} as soon as \begin{document}$α>\frac 85$\end{document}. The proof is based on the approach of the short time Bourgain spaces developed by Ionescu, Kenig and Tataru [10] combined with new Strichartz estimates and a modified energy.

Monotone dynamical systems: Reflections on new advances & applications
Hal L. Smith
2017, 37(1): 485-504 doi: 10.3934/dcds.2017020 +[Abstract](871) +[HTML](18) +[PDF](463.0KB)
Abstract:

The article contains the author's reflections on recent developments in a very select portion of the now vast subject of monotone dynamical systems. Continuous timesystems generated by cooperative systems of ordinary differential equations, delay differential equations, parabolic partial differential equations, and controlsystems are the main focus and results are included which the author feels have had a major impact in the applications. These include the theory of competition betweentwo species or two teams and the theory of monotone control systems.

Stationary and time-periodic patterns of two-predator and one-prey systems with prey-taxis
Ke Wang, Qi Wang and Feng Yu
2017, 37(1): 505-543 doi: 10.3934/dcds.2017021 +[Abstract](1270) +[HTML](17) +[PDF](999.2KB)
Abstract:

This paper concerns pattern formation in a class of reaction-advection-diffusion systems modeling the population dynamics of two predators and one prey. We consider the biological situation that both predators forage along the population density gradient of the preys which can defend themselves as a group. We prove the global existence and uniform boundedness of positive classical solutions for the fully parabolic system over a bounded domain with space dimension \begin{document}$ N=1,2 $\end{document} and for the parabolic-parabolic-elliptic system over higher space dimensions. Linearized stability analysis shows that prey-taxis stabilizes the positive constant equilibrium if there is no group defense while it destabilizes the equilibrium otherwise. Then we obtain stationary and time-periodic nontrivial solutions of the system that bifurcate from the positive constant equilibrium. Moreover, the stability of these solutions is also analyzed in detail which provides a wave mode selection mechanism of nontrivial patterns for this strongly coupled system. Finally, we perform numerical simulations to illustrate and support our theoretical results.

Random attractor for stochastic non-autonomous damped wave equation with critical exponent
Zhaojuan Wang and Shengfan Zhou
2017, 37(1): 545-573 doi: 10.3934/dcds.2017022 +[Abstract](718) +[HTML](3) +[PDF](514.3KB)
Abstract:

In this paper, we prove the existence of random attractor and obtainan upper bound of fractal dimension of random attractor forstochastic non-autonomous damped wave equation with criticalexponent and additive white noise. We first prove the existence of arandom attractor by carefully splitting the positivity of the linearoperator in the corresponding random evolution equation of the firstorder in time and by carefully decomposing the solutions of systemthrough two different modes, and we show the boundedness of randomattractor in a higher regular space by a recurrence method. Then weestablish a criterion to bound the fractal dimension of a randominvariant set for a cocycle and applied these conditions to get anupper bound of fractal dimension of the random attractor ofconsidered system.

The attractors for 2nd-order stochastic delay lattice systems
Chengjian Zhang and Lu Zhao
2017, 37(1): 575-590 doi: 10.3934/dcds.2017023 +[Abstract](612) +[HTML](1) +[PDF](462.2KB)
Abstract:

This paper deals with the long-time dynamical behavior of a classof 2nd-order stochastic delay lattice systems. It is shown under thedissipative and sublinear growth conditions that such a systempossesses a compact global random attractor within the set oftempered random bounded sets. A numerical example is given toillustrate the obtained theoretical result.

Direct approach to detect the heteroclinic bifurcation of the planar nonlinear system
Ling-Hao Zhang and Wei Wang
2017, 37(1): 591-604 doi: 10.3934/dcds.2017024 +[Abstract](630) +[HTML](2) +[PDF](551.9KB)
Abstract:

In this paper, we present a novel way of directly detecting the heteroclinic bifurcation of nonlinear systems without iteration or Melnikov type integration. The method regards the phase and fundamental frequency in a hyperbolic function solution and bifurcation parameter as the unknown components. A global collocation point, obtained from the energy balance method, together with two special points on the orbit are used to determine these unknown components. The feasibility analysis is presented to have a clear insight into the method. As an example, in a third-order nonlinear system, an expression for the orbit and the critical value of bifurcation are directly obtained, maintaining the precision but reducing the complication of bifurcation analysis. A second-order collocation point improves the accuracy of computation. For a broader application, the effectiveness of this new approach is verified for systems with a large perturbation parameter and the homoclinic bifurcation problem evolving from the even order nonlinearity.

Bound state solutions of Schrödinger-Poisson system with critical exponent
Xu Zhang, Shiwang Ma and Qilin Xie
2017, 37(1): 605-625 doi: 10.3934/dcds.2017025 +[Abstract](869) +[HTML](3) +[PDF](464.5KB)
Abstract:

In this paper, we consider the following Schrödinger-Poisson problem

where \begin{document}$2^*=6 $\end{document} is the critical exponent in \begin{document} $\mathbb R^3$ \end{document}, \begin{document} $ K∈ L^{\frac{1}{2}}(\mathbb{R}^3)$ \end{document} and \begin{document} $V∈ L^{\frac{3}{2}}(\mathbb{R}^3)$ \end{document} are given nonnegative functions. When \begin{document} $|V|_{\frac{3}{2}}+|K|_{\frac{1}{2}}$ \end{document} is suitable small, we prove that problem (P) has at least one bound state solution via a linking theorem.

Boundedness of solutions to a quasilinear higher-dimensional chemotaxis-haptotaxis model with nonlinear diffusion
Jiashan Zheng
2017, 37(1): 627-643 doi: 10.3934/dcds.2017026 +[Abstract](715) +[HTML](5) +[PDF](500.6KB)
Abstract:

This paper deals with the Neumann problem for the coupled quasilinear chemotaxis-haptotaxis model of cancer invasion given by

where the parameter $m≥q1$ and $\mathbb{R}^N(N≥q2)$ is a bounded domain with smooth boundary. If $m>\frac{2N}{N+2}$, then for any sufficiently smooth initial data there exists a classical solution which is global in time and bounded. The results of this paper partly extend previous results of several authors.

Blow-up of solutions to the periodic generalized modified Camassa-Holm equation with varying linear dispersion
Min Zhu and Ying Wang
2017, 37(1): 645-661 doi: 10.3934/dcds.2017027 +[Abstract](718) +[HTML](1) +[PDF](469.7KB)
Abstract:

Considered herein is the blow-up mechanism to the periodic generalized modified Camassa-Holm equation with varying linear dispersion. The first one is designed for the case when linear dispersion is absent and derive a finite-time blow-up result. The key feature is the ratio between solution and its gradient. The second one handles the general situation when the weak linear dispersion is at present. Fortunately, there exist some conserved quantities that bound the \begin{document}$\|u_x\|_{L^4} $\end{document} for the periodic generalized modified Camassa-Holm equation, then the breakdown mechanisms are set up for the general case.

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