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Discrete & Continuous Dynamical Systems - A

February 2018 , Volume 38 , Issue 2

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An estimate on the Hausdorff dimension of stable sets of non-uniformly hyperbolic horseshoes
Carlos Matheus and Jacob Palis
2018, 38(2): 431-448 doi: 10.3934/dcds.2018020 +[Abstract](638) +[HTML](81) +[PDF](546.7KB)
Abstract:

We show that the Hausdorff dimension of stable sets of non-uniformly hyperbolic horseshoes is strictly smaller than two.

Receding horizon control for the stabilization of the wave equation
Behzad Azmi and Karl Kunisch
2018, 38(2): 449-484 doi: 10.3934/dcds.2018021 +[Abstract](581) +[HTML](69) +[PDF](2851.6KB)
Abstract:

Stabilization of the wave equation by the receding horizon framework is investigated. Distributed control, Dirichlet boundary control, and Neumann boundary control are considered. Moreover for each of these control actions, the well-posedness of the control system and the exponential stability of Receding Horizon Control (RHC) with respect to a proper functional analytic setting are investigated. Observability conditions are necessary to show the suboptimality and exponential stability of RHC. Numerical experiments are given to illustrate the theoretical results.

Positive solutions for critically coupled Schrödinger systems with attractive interactions
Hongyu Ye
2018, 38(2): 485-507 doi: 10.3934/dcds.2018022 +[Abstract](626) +[HTML](59) +[PDF](475.3KB)
Abstract:

In this paper, we consider the following coupled Schrödinger system with doubly critical exponents:

where \begin{document}$Ω\subset\mathbb R^4$\end{document} is a smooth bounded domain, \begin{document}$μ_1, μ_2>0$\end{document} and \begin{document}$β>0$\end{document}, \begin{document}$-λ_1(Ω)<λ_1, λ_2<0$\end{document}, here \begin{document}$λ_1(Ω)$\end{document} is the first eigenvalue of \begin{document}$-Δ$\end{document} with the Dirichlet boundary condition. We give the optimal ranges of \begin{document}$β>0$\end{document} for the existence of positive solutions to the problem, which is an open problem proposed by Chen and Zou in [Arch. Rational Mech. Anal. 205 (2012), 515-551]. Finally, as a by-product of our approaches, we extend the existence results to a critically coupled Schrödinger system defined in the whole space:

Linear diffusion with singular absorption potential and/or unbounded convective flow: The weighted space approach
Jesus Ildefonso Díaz, David Gómez-Castro, Jean Michel Rakotoson and Roger Temam
2018, 38(2): 509-546 doi: 10.3934/dcds.2018023 +[Abstract](506) +[HTML](65) +[PDF](623.4KB)
Abstract:

In this paper we prove the existence and uniqueness of very weak solutions to linear diffusion equations involving a singular absorption potential and/or an unbounded convective flow on a bounded open set of $\text{IR}^N$. In most of the paper we consider homogeneous Dirichlet boundary conditions but we prove that when the potential function grows faster than the distance to the boundary to the power -2 then no boundary condition is required to get the uniqueness of very weak solutions. This result is new in the literature and must be distinguished from other previous results in which such uniqueness of solutions without any boundary condition was proved for degenerate diffusion operators (which is not our case). Our approach, based on the treatment on some distance to the boundary weighted spaces, uses a suitable regularity of the solution of the associated dual problem which is here established. We also consider the delicate question of the differentiability of the very weak solution and prove that some suitable additional hypothesis on the data is required since otherwise the gradient of the solution may not be integrable on the domain.

Nonradial least energy solutions of the p-Laplace elliptic equations
Ryuji Kajikiya
2018, 38(2): 547-561 doi: 10.3934/dcds.2018024 +[Abstract](635) +[HTML](56) +[PDF](387.1KB)
Abstract:

We study the p-Laplace elliptic equations in the unit ball under the Dirichlet boundary condition. We call u a least energy solution if it is a minimizer of the Lagrangian functional on the Nehari manifold. A least energy solution becomes a positive solution. Assume that the nonlinear term is radial and it vanishes in \begin{document}$|x| <a$\end{document} and it is positive in \begin{document}$a<|x|<1$\end{document}. We prove that if a is close enough to 1, then no least energy solution is radial. Therefore there exist both a positive radial solution and a positive nonradial solution.

Energy-critical NLS with potentials of quadratic growth
Casey Jao
2018, 38(2): 563-587 doi: 10.3934/dcds.2018025 +[Abstract](486) +[HTML](47) +[PDF](538.9KB)
Abstract:

We consider the global wellposedness problem for the nonlinear Schrödinger equation

where \begin{document}$Σ$\end{document} is the weighted Sobolev space \begin{document}$\dot{H}^1 \cap |x|^{-1} L^2$\end{document}. The case \begin{document}$V(x) = \tfrac{1}{2}|x|^2$\end{document} was recently treated by the author. This note generalizes the results to a class of "approximately quadratic" potentials.

We closely follow the previous concentration compactness arguments for the harmonic oscillator. A key technical difference is that in the absence of a concrete formula for the linear propagator, we apply more general tools from microlocal analysis, including a Fourier integral parametrix of Fujiwara.

Bounded and unbounded capillary surfaces derived from the catenoid
Filippo Morabito
2018, 38(2): 589-614 doi: 10.3934/dcds.2018026 +[Abstract](527) +[HTML](45) +[PDF](473.0KB)
Abstract:

We construct two kinds of capillary surfaces by using a perturbation method. Surfaces of first kind are embedded in a solid ball B of \begin{document}$\mathbb{R}^3$\end{document} with assigned mean curvature function and whose boundary curves lie on $\partial B.$ The contact angle along such curves is a non-constant function. Surfaces of second kind are unbounded and embedded in \begin{document}$\mathbb{R}^3 \setminus \tilde B,$\end{document} \begin{document}$\tilde B$\end{document} being a deformation of a solid ball in \begin{document}$\mathbb{R}^3.$\end{document} These surfaces have assigned mean curvature function and one boundary curve on \begin{document}$\partial \tilde B.$\end{document} Also in this case the contact angle along the boundary is a non-constant function.

On the classification of generalized competitive Atkinson-Allen models via the dynamics on the boundary of the carrying simplex
Mats Gyllenberg, Jifa Jiang, Lei Niu and Ping Yan
2018, 38(2): 615-650 doi: 10.3934/dcds.2018027 +[Abstract](668) +[HTML](77) +[PDF](3057.0KB)
Abstract:

We propose the generalized competitive Atkinson-Allen map

which is the classical Atkson-Allen map when \begin{document}$r_i=1$\end{document} and \begin{document}$c_i=c$\end{document} for all \begin{document}$i=1, ..., n$\end{document} and a discretized system of the competitive Lotka-Volterra equations. It is proved that every \begin{document}$n$\end{document}-dimensional map \begin{document}$T$\end{document} of this form admits a carrying simplex Σ which is a globally attracting invariant hypersurface of codimension one. We define an equivalence relation relative to local stability of fixed points on the boundary of Σ on the space of all such three-dimensional maps. In the three-dimensional case we list a total of 33 stable equivalence classes and draw the corresponding phase portraits on each Σ. The dynamics of the generalized competitive Atkinson-Allen map differs from the dynamics of the standard one in that Neimark-Sacker bifurcations occur in two classes for which no such bifurcations were possible for the standard competitive Atkinson-Allen map. We also found Chenciner bifurcations by numerical examples which implies that two invariant closed curves can coexist for this model, whereas those have not yet been found for all other three-dimensional competitive mappings via the carrying simplex. In one class every map admits a heteroclinic cycle; we provide a stability criterion for heteroclinic cycles. Besides, the generalized Atkinson-Allen model is not dynamically consistent with the Lotka-Volterra system.

On an exponential attractor for a class of PDEs with degenerate diffusion and chemotaxis
Messoud Efendiev and Anna Zhigun
2018, 38(2): 651-673 doi: 10.3934/dcds.2018028 +[Abstract](451) +[HTML](47) +[PDF](508.7KB)
Abstract:

In this article we deal with a class of strongly coupled parabolic systems that encompasses two different effects: degenerate diffusion and chemotaxis. Such classes of equations arise in the mesoscale level modeling of biomass spreading mechanisms via chemotaxis. We show the existence of an exponential attractor and, hence, of a finite-dimensional global attractor under certain 'balance conditions' on the order of the degeneracy and the growth of the chemotactic function.

Nonexistence results for elliptic differential inequalities with a potential in bounded domains
Dario D. Monticelli and Fabio Punzo
2018, 38(2): 675-695 doi: 10.3934/dcds.2018029 +[Abstract](412) +[HTML](52) +[PDF](478.0KB)
Abstract:

In this paper we are concerned with a class of elliptic differential inequalities with a potential in bounded domains both of $\mathbb{R}^m$ and of Riemannian manifolds. In particular, we investigate the effect of the behavior of the potential at the boundary of the domain on nonexistence of nonnegative solutions.

Nonlinear Schrödinger Equations on Periodic Metric Graphs
Alexander Pankov
2018, 38(2): 697-714 doi: 10.3934/dcds.2018030 +[Abstract](477) +[HTML](56) +[PDF](421.0KB)
Abstract:

The paper is devoted to the nonlinear Schrödinger equation with periodic linear and nonlinear potentials on periodic metric graphs. Assuming that the spectrum of linear part does not contain zero, we prove the existence of finite energy ground state solution which decays exponentially fast at infinity. The proof is variational and makes use of the generalized Nehari manifold for the energy functional combined with periodic approximations. Actually, a finite energy ground state solution is obtained from periodic solutions in the infinite wave length limit.

The continuum limit of Follow-the-Leader models — a short proof
Helge Holden and Nils Henrik Risebro
2018, 38(2): 715-722 doi: 10.3934/dcds.2018031 +[Abstract](411) +[HTML](57) +[PDF](315.6KB)
Abstract:

We offer a simple and self-contained proof that the Follow-the-Leader model converges to the Lighthill-Whitham-Richards model for traffic flow.

On uniqueness of measure-valued solutions to Liouville's equation of Hamiltonian PDEs
Ammari Zied and Liard Quentin
2018, 38(2): 723-748 doi: 10.3934/dcds.2018032 +[Abstract](687) +[HTML](60) +[PDF](565.5KB)
Abstract:

In this paper, the Cauchy problem of classical Hamiltonian PDEs is recast into a Liouville's equation with measure-valued solutions. Then a uniqueness property for the latter equation is proved under some natural assumptions. Our result extends the method of characteristics to Hamiltonian systems with infinite degrees of freedom and it applies to a large variety of Hamiltonian PDEs (Hartree, Klein-Gordon, Schrödinger, Wave, Yukawa \begin{document}$\dots$\end{document}). The main arguments in the proof are a projective point of view and a probabilistic representation of measure-valued solutions to continuity equations in finite dimension.

Dispersive effects of weakly compressible and fast rotating inviscid fluids
Van-Sang Ngo and Stefano Scrobogna
2018, 38(2): 749-789 doi: 10.3934/dcds.2018033 +[Abstract](468) +[HTML](53) +[PDF](677.2KB)
Abstract:

We consider a system describing the motion of an isentropic, inviscid, weakly compressible, fast rotating fluid in the whole space \begin{document}$\mathbb{R}^3$\end{document}, with initial data belonging to \begin{document}$ H^s \left( \mathbb{R}^3 \right), s>5/2 $\end{document}. We prove that the system admits a unique local strong solution in \begin{document}$ L^\infty \left( [0,T]; H^s\left( \mathbb{R}^3 \right) \right) $\end{document}, where \begin{document}$ T $\end{document} is independent of the Rossby and Mach numbers. Moreover, using Strichartz-type estimates, we prove the longtime existence of the solution, i.e. its lifespan is of the order of \begin{document}$\varepsilon^{-\alpha}, \alpha >0$\end{document}, without any smallness assumption on the initial data (the initial data can even go to infinity in some sense), provided that the rotation is fast enough.

N-barrier maximum principle for degenerate elliptic systems and its application
Chiun-Chuan Chen, Li-Chang Hung and Hsiao-Feng Liu
2018, 38(2): 791-821 doi: 10.3934/dcds.2018034 +[Abstract](584) +[HTML](56) +[PDF](770.7KB)
Abstract:

In this paper, we prove the N-barrier maximum principle, which extends the result in C.-C. Chen and L.-C. Hung (2016) from linear diffusion equations to nonlinear diffusion equations, for a wide class of degenerate elliptic systems of porous medium type. The N-barrier maximum principle provides a priori upper and lower bounds of the solutions to the above-mentioned degenerate nonlinear diffusion equations including the Shigesada-Kawasaki-Teramoto model as a special case. We also apply the N-barrier maximum principle to a coexistence problem in ecology, where we show the nonexistence of traveling waves in a three-species degenerate elliptic system.

A Liouville-type theorem for cooperative parabolic systems
Anh Tuan Duong and Quoc Hung Phan
2018, 38(2): 823-833 doi: 10.3934/dcds.2018035 +[Abstract](557) +[HTML](49) +[PDF](353.1KB)
Abstract:

We prove Liouville-type theorem for semilinear parabolic system of the form \begin{document}$u_t-\Delta u =a_{11}u^{p}+a_{12} u^rv^{s+1}$\end{document}, \begin{document}$v_t-\Delta v =a_{21} u^{r+1}v^{s}+a_{22}v^{p}$\end{document} where \begin{document}$r, s>0$\end{document}, \begin{document}$p=r+s+1$\end{document}. The real matrix \begin{document}$A=(a_{ij})$\end{document} satisfies conditions \begin{document}$ a_{12}, a_{21}\geq 0$\end{document} and \begin{document}$a_{11}, a_{22}>0$\end{document}. This paper is a continuation of Phan-Souplet (Math. Ann., 366,1561-1585,2016) where the authors considered the special case \begin{document}$s=r$\end{document} for the system of \begin{document}$m$\end{document} components. Our tool for the proof of Liouville-type theorem is a refinement of Phan-Souplet, which is based on Gidas-Spruck (Commun. Pure Appl.Math. 34,525–598 1981) and Bidaut-Véron (Équations aux dérivées partielles et applications. Elsevier, Paris, pp 189–198,1998).

Reversing and extended symmetries of shift spaces
Michael Baake, John A. G. Roberts and Reem Yassawi
2018, 38(2): 835-866 doi: 10.3934/dcds.2018036 +[Abstract](472) +[HTML](59) +[PDF](471.1KB)
Abstract:

The reversing symmetry group is considered in the setting of symbolic dynamics. While this group is generally too big to be analysed in detail, there are interesting cases with some form of rigidity where one can determine all symmetries and reversing symmetries explicitly. They include Sturmian shifts as well as classic examples such as the Thue–Morse system with various generalisations or the Rudin–Shapiro system. We also look at generalisations of the reversing symmetry group to higher-dimensional shift spaces, then called the group of extended symmetries. We develop their basic theory for faithful \begin{document}$\mathbb{Z}^{d}$\end{document}-actions, and determine the extended symmetry group of the chair tiling shift, which can be described as a model set, and of Ledrappier's shift, which is an example of algebraic origin.

A robustly transitive diffeomorphism of Kan's type
Cheng Cheng, Shaobo Gan and Yi Shi
2018, 38(2): 867-888 doi: 10.3934/dcds.2018037 +[Abstract](657) +[HTML](57) +[PDF](1825.8KB)
Abstract:

We construct a family of partially hyperbolic skew-product diffeomorphisms on \begin{document} $\mathbb{T}^3$ \end{document} that are robustly transitive and admit two physical measures with intermingled basins. In particular, all these diffeomorphisms are not topologically mixing. Moreover, every such example exhibits a dichotomy under perturbation: every perturbation of such example either has a unique physical measure and is robustly topologically mixing, or has two physical measures with intermingled basins.

Parrondo's dynamic paradox for the stability of non-hyperbolic fixed points
Anna Cima, Armengol Gasull and Víctor Mañosa
2018, 38(2): 889-904 doi: 10.3934/dcds.2018038 +[Abstract](406) +[HTML](53) +[PDF](411.7KB)
Abstract:

We show that for periodic non-autonomous discrete dynamical systems, even when a common fixed point for each of the autonomous associated dynamical systems is repeller, this fixed point can became a local attractor for the whole system, giving rise to a Parrondo's dynamic type paradox.

Propagation phenomena for CNNs with asymmetric templates and distributed delays
Zhixian Yu and Xiao-Qiang Zhao
2018, 38(2): 905-939 doi: 10.3934/dcds.2018039 +[Abstract](452) +[HTML](60) +[PDF](1205.7KB)
Abstract:

The aim of this work is to study propagation phenomena for monotone and nonmonotone cellular neural networks with the asymmetric templates and distributed delays. More precisely, for the monotone case, we establish the existence of the leftward (\begin{document} $c_{-}^*$ \end{document}) and rightward (\begin{document} $c_{+}^*$ \end{document}) spreading speeds for CNNs by appealing to the theory developed in [26,27], and \begin{document} $c_{-}^*+c_{+}^*>0$ \end{document}. Especially, if cells possess the symmetric templates and the same delayed interactions, then \begin{document} $c_{-}^*=c_{+}^*>0$ \end{document}. Moreover, if the effect of the self-feedback interaction \begin{document} $α f'(0)$ \end{document} is not less than 1, then both \begin{document} $c_{-}^*>0$ \end{document} and \begin{document} $c_{+}^*>0$ \end{document}. For the non-monotone case, the leftward and rightward spreading speeds are investigated by using the results of the spreading speed for the monotone case and squeezing the given output function between two appropriate nondecreasing functions. It turns out that the leftward and rightward spreading speeds are linearly determinate in these two cases. We further obtain the existence and nonexistence of travelling wave solutions under the weaker conditions than those in [46, 47] and show that the spreading speed coincides with the minimal wave speed.

Renormalization of two-dimensional piecewise linear maps: Abundance of 2-D strange attractors
Antonio Pumariño, José Ángel Rodríguez and Enrique Vigil
2018, 38(2): 941-966 doi: 10.3934/dcds.2018040 +[Abstract](571) +[HTML](54) +[PDF](584.7KB)
Abstract:

For a two parameter family of two-dimensional piecewise linear maps and for every natural number \begin{document} $n$ \end{document}, we prove not only the existence of intervals of parameters for which the respective maps are n times renormalizable but also we show the existence of intervals of parameters where the coexistence of at least \begin{document} $2^n$ \end{document} strange attractors takes place. This family of maps contains the two-dimensional extension of the classical one-dimensional family of tent maps.

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