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Discrete & Continuous Dynamical Systems - A

August 2018 , Volume 38 , Issue 8

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Symmetry breaking in a globally coupled map of four sites
Fanni M. Sélley
2018, 38(8): 3707-3734 doi: 10.3934/dcds.2018161 +[Abstract](694) +[HTML](117) +[PDF](885.74KB)
Abstract:

A system of four globally coupled doubling maps is studied in this paper. It is known that such systems have a unique absolutely continuous invariant measure (acim) for weak interaction, but the case of stronger coupling is still unexplored. As in the case of three coupled sites [14], we prove the existence of a critical value of the coupling parameter at which multiple acims appear. Our proof has several new ingredients in comparison to the one presented in [14]. We strongly exploit the symmetries of the dynamics in the course of the argument. This simplifies the computations considerably, and gives us a precise description of the geometry and symmetry properties of the arising asymmetric invariant sets. Some new phenomena are observed which are not present in the case of three sites. In particular, the asymmetric invariant sets arise in areas of the phase space which are transient for weaker coupling and a nontrivial symmetric invariant set emerges, shaped by an underlying centrally symmetric Lorenz map. We state some conjectures on further invariant sets, indicating that unlike the case of three sites, ergodicity breaks down in many steps, and not all of them are accompanied by symmetry breaking.

Degenerate lower dimensional invariant tori in reversible system
Shengqing Hu and Bin Liu
2018, 38(8): 3735-3763 doi: 10.3934/dcds.2018162 +[Abstract](573) +[HTML](108) +[PDF](461.76KB)
Abstract:

In this paper, we are concerned with the existence of lower dimensional invariant tori in nearly integrable reversible systems. By KAM method, we prove that under some reasonable assumptions, there are many so-called degenerate lower dimensional invariant tori, that is one of normal frequencies is zero.

Long-time behavior of a nonlocal Cahn-Hilliard equation with reaction
Annalisa Iuorio and Stefano Melchionna
2018, 38(8): 3765-3788 doi: 10.3934/dcds.2018163 +[Abstract](888) +[HTML](148) +[PDF](526.71KB)
Abstract:

In this paper we study the long-time behavior of a nonlocal Cahn-Hilliard system with singular potential, degenerate mobility, and a reaction term. In particular, we prove the existence of a global attractor with finite fractal dimension, the existence of an exponential attractor, and convergence to equilibria for two physically relevant classes of reaction terms.

Impulsive control of conservative periodic equations and systems: Variational approach
Pavel Drábek and Martina Langerová
2018, 38(8): 3789-3802 doi: 10.3934/dcds.2018164 +[Abstract](568) +[HTML](115) +[PDF](398.01KB)
Abstract:

Using the variational structure of the second order periodic problems we find an optimal impulsive control which forces the conservative system into a periodic motion. In particular, our main results concern the system of charged planar pendulums with external disturbances and neglected friction. Such a system might serve as a model for coupled micromechanical array.

The existence of two non-contractible closed geodesics on every bumpy Finsler compact space form
Hui Liu, Yiming Long and Yuming Xiao
2018, 38(8): 3803-3829 doi: 10.3934/dcds.2018165 +[Abstract](400) +[HTML](85) +[PDF](548.15KB)
Abstract:

Let \begin{document}$M = S^n/ Γ$\end{document} and \begin{document}$h$\end{document} be a nontrivial element of finite order \begin{document}$p$\end{document} in \begin{document}$π_1(M)$\end{document}, where the integer \begin{document}$n≥2$\end{document}, \begin{document}$Γ$\end{document} is a finite group which acts freely and isometrically on the \begin{document}$n$\end{document}-sphere and therefore \begin{document}$M$\end{document} is diffeomorphic to a compact space form. In this paper, we establish first the resonance identity for non-contractible homologically visible minimal closed geodesics of the class \begin{document}$[h]$\end{document} on every Finsler compact space form \begin{document}$(M, F)$\end{document} when there exist only finitely many distinct non-contractible closed geodesics of the class \begin{document}$[h]$\end{document} on \begin{document}$(M, F)$\end{document}. Then as an application of this resonance identity, we prove the existence of at least two distinct non-contractible closed geodesics of the class \begin{document}$[h]$\end{document} on \begin{document}$(M, F)$\end{document} with a bumpy Finsler metric, which improves a result of Taimanov in [39] by removing some additional conditions. Also our results extend the resonance identity and multiplicity results on \begin{document}$\mathbb{R}P^n$\end{document} in [25] to general compact space forms.

Existence, nonexistence and multiplicity of positive solutions for the poly-Laplacian and nonlinearities with zeros
Leonelo Iturriaga and Eugenio Massa
2018, 38(8): 3831-3850 doi: 10.3934/dcds.2018166 +[Abstract](464) +[HTML](97) +[PDF](517.61KB)
Abstract:

In this paper we consider the equation \begin{document}$(-Δ)^k\, u = λ f(x, u)+μ g(x, u)$\end{document} with Navier boundary conditions, in a bounded and smooth domain. The main interest is when the nonlinearity is nonnegative but admits a zero and \begin{document}$f, g$\end{document} are, respectively, identically zero above and below the zero. We prove the existence of multiple positive solutions when the parameters lie in a region of the form \begin{document}$λ>\overline λ$\end{document} and \begin{document}$0 < μ< \overlineμ(λ)$\end{document}, then we provide further conditions under which, respectively, the bound \begin{document}$\overlineμ(λ)$\end{document} is either necessary, or can be removed.

Quantization conditions of eigenvalues for semiclassical Zakharov-Shabat systems on the circle
Setsuro Fujiié and Jens Wittsten
2018, 38(8): 3851-3873 doi: 10.3934/dcds.2018167 +[Abstract](442) +[HTML](87) +[PDF](788.35KB)
Abstract:

Bohr-Sommerfeld type quantization conditions of semiclassical eigenvalues for the non-selfadjoint Zakharov-Shabat operator on the unit circle are derived using an exact WKB method. The conditions are given in terms of the action associated with the unit circle or the action associated with turning points following the absence or presence of real turning points.

Global weak solution and boundedness in a three-dimensional competing chemotaxis
Hua Zhong, Chunlai Mu and Ke Lin
2018, 38(8): 3875-3898 doi: 10.3934/dcds.2018168 +[Abstract](531) +[HTML](167) +[PDF](489.94KB)
Abstract:

We consider an initial-boundary value problem for a parabolic-parabolic-elliptic attraction-repulsion chemotaxis model

in a bounded domain \begin{document}$Ω\subset \mathbb{R}^3$\end{document} with positive parameters \begin{document}$χ, ξ, α, β, γ$\end{document} and \begin{document}$δ$\end{document}.

It is firstly proved that if the repulsion dominates in the sense that \begin{document}$ξγ>χα$\end{document}, then for any choice of sufficiently smooth initial data \begin{document}$(u_0, v_0)$\end{document} the corresponding initial-boundary value problem is shown to possess a globally defined weak solution. To the best of our knowledge, this situation provides the first result on global existence of the above system in the three-dimensional setting when \begin{document}$ξγ>χα$\end{document}, and extends the results in Lin et al. (2016) [19] and Jin and Xiang (2017) [14] to more general case.

Secondly, if the initial data is appropriately small or the repulsion is enough strong in the sense that \begin{document}$ξγ$\end{document} is suitable large as related to \begin{document}$χα$\end{document}, then the classical solutions to the above system are uniformly-in-time bounded.

Oscillating solutions for prescribed mean curvature equations: euclidean and lorentz-minkowski cases
Alessio Pomponio
2018, 38(8): 3899-3911 doi: 10.3934/dcds.2018169 +[Abstract](416) +[HTML](92) +[PDF](413.91KB)
Abstract:

This paper deals with the prescribed mean curvature equations

both in the Euclidean case, with the sign "+", and in the Lorentz-Minkowski case, with the sign "-", for N ≥ 1 under the assumption g'(0)>0. We show the existence of oscillating solutions, namely with an unbounded sequence of zeros. Moreover these solutions are periodic, if N = 1, while they are radial symmetric and decay to zero at infinity with their derivatives, if N≥ 2.

Gradient blow-up for a fourth-order quasilinear Boussinesq-type equation
Pablo Álvarez-Caudevilla, Jonathan D. Evans and Victor A. Galaktionov
2018, 38(8): 3913-3938 doi: 10.3934/dcds.2018170 +[Abstract](576) +[HTML](87) +[PDF](1014.31KB)
Abstract:

The Cauchy problem for a fourth-order Boussinesq-type quasilinear wave equation (QWE-4) of the form

and bounded smooth initial data, is considered. Self-similar single-point gradient blow-up solutions are studied. It is shown that such singular solutions exist and satisfy the case of the so-called self-similarity of the second type.

Together with an essential and, often, key use of numerical methods to describe possible types of gradient blow-up, a "homotopy" approach is applied that traces out the behaviour of such singularity patterns as \begin{document}$n \to 0^+$\end{document}, when the classic linear beam equation occurs

with simple, better-known and understandable evolution properties.

Classification of extremal functions to logarithmic Hardy-Littlewood-Sobolev inequality on the upper half space
Jingbo Dou and Ye Li
2018, 38(8): 3939-3953 doi: 10.3934/dcds.2018171 +[Abstract](542) +[HTML](102) +[PDF](426.03KB)
Abstract:

In this paper we mainly classify the extremal functions of logarithmic Hardy-Littlewood-Sobolev inequality on the upper half space \begin{document}$\mathbb{R}_+^{n}$\end{document}, and also present some remarks on the extremal functions of logarithmic Hardy-Littlewood-Sobolev inequality on the whole space \begin{document}$\mathbb{R}^{n}$\end{document}. Our main techniques are Kelvin transformation and the method of moving spheres in integral forms.

Periodic linear motions with multiple collisions in a forced Kepler type problem
Carlota Rebelo and Alexandre Simões
2018, 38(8): 3955-3975 doi: 10.3934/dcds.2018172 +[Abstract](396) +[HTML](80) +[PDF](475.77KB)
Abstract:

In [7] the author proved the existence of multiple periodic linear motions with collisions for a periodically forced Kepler problem. We extend this result obtaining periodic solutions with multiple collisions for a forced Kepler type problem. In order to do that we apply the Poincaré-Birkhoff theorem.

Continuity of spectral radius over hyperbolic systems
Rui Zou, Yongluo Cao and Gang Liao
2018, 38(8): 3977-3991 doi: 10.3934/dcds.2018173 +[Abstract](601) +[HTML](176) +[PDF](416.32KB)
Abstract:

The continuity of joint and generalized spectral radius is proved for Hölder continuous cocycles over hyperbolic systems. We also prove the periodic approximation of Lyapunov exponents for non-invertible non-uniformly hyperbolic systems, and establish the Berger-Wang formula for general dynamical systems.

Blow-up and superexponential growth in superlinear Volterra equations
John A. D. Appleby and Denis D. Patterson
2018, 38(8): 3993-4017 doi: 10.3934/dcds.2018174 +[Abstract](318) +[HTML](80) +[PDF](517.39KB)
Abstract:

This paper concerns the finite-time blow-up and asymptotic behaviour of solutions to nonlinear Volterra integro-differential equations. Our main contribution is to determine sharp estimates on the growth rates of both explosive and nonexplosive solutions for a class of equations with nonsingular kernels under weak hypotheses on the nonlinearity. In this superlinear setting we must be content with estimates of the form \begin{document}$\lim_{t\toτ}A(x(t), t) = 1$\end{document}, where \begin{document}$τ$\end{document} is the blow-up time if solutions are explosive or \begin{document}$τ = ∞$\end{document} if solutions are global. Our estimates improve on the sharpness of results in the literature and we also recover well-known blow-up criteria via new methods.

On fractional Hardy inequalities in convex sets
Lorenzo Brasco and Eleonora Cinti
2018, 38(8): 4019-4040 doi: 10.3934/dcds.2018175 +[Abstract](536) +[HTML](82) +[PDF](1026.29KB)
Abstract:

We prove a Hardy inequality on convex sets, for fractional Sobolev-Slobodeckiĭ spaces of order \begin{document}$(s, p)$\end{document}. The proof is based on the fact that in a convex set the distance from the boundary is a superharmonic function, in a suitable sense. The result holds for every \begin{document}$1<p<∞$\end{document} and zhongwenzy \begin{document}$0<s<1$\end{document}, with a constant which is stable as \begin{document}$s$\end{document} goes to 1.

Large time behavior of solutions of the heat equation with inverse square potential
Kazuhiro Ishige and Asato Mukai
2018, 38(8): 4041-4069 doi: 10.3934/dcds.2018176 +[Abstract](541) +[HTML](94) +[PDF](558.32KB)
Abstract:

Let \begin{document}$L: = -Δ+V$\end{document} be a nonnegative Schrödinger operator on \begin{document}$L^2({\bf R}^N)$\end{document}, where \begin{document}$N≥ 2$\end{document} and \begin{document}$V$\end{document} is a radially symmetric inverse square potential. In this paper we assume either \begin{document}$L$\end{document} is subcritical or null-critical and we establish a method for obtaining the precise description of the large time behavior of \begin{document}$e^{-tL}\varphi$\end{document}, where \begin{document}$\varphi∈ L^2({\bf R}^N, e^{|x|^2/4}\, dx)$\end{document}.

The regularity of solutions to some variational problems, including the p-Laplace equation for 3≤p < 4
Arrigo Cellina
2018, 38(8): 4071-4085 doi: 10.3934/dcds.2018177 +[Abstract](423) +[HTML](76) +[PDF](445.29KB)
Abstract:

We consider the higher differentiability of solutions to the problem of minimising

where \begin{document}$\Omega\subset \mathbb R^N$\end{document}, \begin{document}$L(ξ) = l(|ξ|) = \frac{1}{p}|ξ|^p$\end{document} and \begin{document}$ u^0∈ W^{1, p}(Ω)$\end{document} and hence, in particular, the higher differentiability of weak solution to the equation

We show that, for \begin{document}$3≤ p < 4$\end{document}, under suitable assumptions on \begin{document}$g$\end{document}, there exists a solution \begin{document}$ u^*$\end{document} to the Euler-Lagrange equation associated to the minimisation problem, such that

for \begin{document}$0 < s < 4-p$\end{document}. In particular, for \begin{document}$p = 3$\end{document}, we show that the solution \begin{document}$u^*$\end{document} is such that \begin{document}$\nabla u^*∈ W^{s, 2}_{loc}(\Omega)$\end{document} for every \begin{document}$s < 1$\end{document}. This result is independent of \begin{document}$N$\end{document}. We present an example for \begin{document}$N = 1$\end{document} and \begin{document}$p = 3$\end{document} whose solution \begin{document}$u$\end{document} is such that \begin{document}$\nabla u^*$\end{document} is not in \begin{document}$W^{1, 2}_{loc}(\Omega)$\end{document}, thus showing that our result is sharp.

Automatic sequences as good weights for ergodic theorems
Tanja Eisner and Jakub Konieczny
2018, 38(8): 4087-4115 doi: 10.3934/dcds.2018178 +[Abstract](481) +[HTML](115) +[PDF](557.75KB)
Abstract:

We study correlation estimates of automatic sequences (that is, sequences computable by finite automata) with polynomial phases. As a consequence, we provide a new class of good weights for classical and polynomial ergodic theorems. We show that automatic sequences are good weights in \begin{document}$ L^2$\end{document} for polynomial averages and totally ergodic systems. For totally balanced automatic sequences (i.e., sequences converging to zero in mean along arithmetic progressions) the pointwise weighted ergodic theorem in \begin{document}$ L^1$\end{document} holds. Moreover, invertible automatic sequences are good weights for the pointwise polynomial ergodic theorem in \begin{document}$ L^r$\end{document}, \begin{document}$ r>1$\end{document}.

Lower spectral radius and spectral mapping theorem for suprema preserving mappings
Vladimir Müller and Aljoša Peperko
2018, 38(8): 4117-4132 doi: 10.3934/dcds.2018179 +[Abstract](440) +[HTML](82) +[PDF](414.66KB)
Abstract:

We study Lipschitz, positively homogeneous and finite suprema preserving mappings defined on a max-cone of positive elements in a normed vector lattice. We prove that the lower spectral radius of such a mapping is always a minimum value of its approximate point spectrum. We apply this result to show that the spectral mapping theorem holds for the approximate point spectrum of such a mapping. By applying this spectral mapping theorem we obtain new inequalites for the Bonsall cone spectral radius of max-type kernel operators.

Global regularity for the 2D micropolar equations with fractional dissipation
Bo-Qing Dong, Jiahong Wu, Xiaojing Xu and Zhuan Ye
2018, 38(8): 4133-4162 doi: 10.3934/dcds.2018180 +[Abstract](522) +[HTML](150) +[PDF](583.9KB)
Abstract:

Micropolar equations, modeling micropolar fluid flows, consist of coupled equations obeyed by the evolution of the velocity \begin{document}$ u$\end{document} and that of the microrotation \begin{document}$ w$\end{document}. This paper focuses on the two-dimensional micropolar equations with the fractional dissipation \begin{document}$ (-Δ)^{α} u$\end{document} and \begin{document}$ (-Δ)^{β}w$\end{document}, where \begin{document}$ 0<α, β<1$\end{document}. The goal here is the global regularity of the fractional micropolar equations with minimal fractional dissipation. Recent efforts have resolved the two borderline cases \begin{document}$ α = 1$, $β = 0$\end{document} and \begin{document}$ α = 0$\end{document}, \begin{document}$ β = 1$\end{document}. However, the situation for the general critical case \begin{document}$ α+β = 1$\end{document} with \begin{document}$ 0<α<1$\end{document} is far more complex and the global regularity appears to be out of reach. When the dissipation is split among the equations, the dissipation is no longer as efficient as in the borderline cases and different ranges of \begin{document}$ α$\end{document} and \begin{document}$ β$\end{document} require different estimates and tools. We aim at the subcritical case \begin{document}$\alpha+\beta>1$\end{document} and divide \begin{document}$\alpha\in (0,1)$\end{document} into five sub-intervals to seek the best estimates so that we can impose the minimal requirements on \begin{document}$\alpha$\end{document} and \begin{document}$\beta$\end{document}. The proof of the global regularity relies on the introduction of combined quantities, sharp lower bounds for the fractional dissipation and delicate upper bounds for the nonlinearity and associated commutators.

On the Cauchy problem for a higher-order μ-Camassa-Holm equation
Feng Wang, Fengquan Li and Zhijun Qiao
2018, 38(8): 4163-4187 doi: 10.3934/dcds.2018181 +[Abstract](566) +[HTML](89) +[PDF](519.54KB)
Abstract:

In this paper, we study the Cauchy problem of a higher-order μ-Camassa-Holm equation. We first establish the Green's function of \begin{document}$(μ-\partial_{x}^{2}+\partial_{x}^{4})^{-1}$\end{document} and local well-posedness for the equation in Sobolev spaces \begin{document}$H^{s}(\mathbb{S})$\end{document}, \begin{document}$s>\frac{7}{2}$\end{document}. Then we provide the global existence results for strong solutions and weak solutions. Moreover, we show that the solution map is non-uniformly continuous in \begin{document}$H^{s}(\mathbb{S})$\end{document}, \begin{document}$s≥ 4$\end{document}. Finally, we prove that the equation admits single peakon solutions which have continuous second derivatives and jump discontinuities in the third derivatives.

Bifurcation of limit cycles for a family of perturbed Kukles differential systems
Salomón Rebollo-Perdomo and Claudio Vidal
2018, 38(8): 4189-4202 doi: 10.3934/dcds.2018182 +[Abstract](573) +[HTML](107) +[PDF](537.28KB)
Abstract:

We consider an integrable non-Hamiltonian system, which belongs to the quadratic Kukles differential systems. It has a center surrounded by a bounded period annulus. We study polynomial perturbations of such a Kukles system inside the Kukles family. We apply averaging theory to study the limit cycles that bifurcate from the period annulus and from the center of the unperturbed system. First, we show that the periodic orbits of the period annulus can be parametrized explicitly through the Lambert function. Later, we prove that at most one limit cycle bifurcates from the period annulus, under quadratic perturbations. Moreover, we give conditions for the non-existence, existence, and stability of the bifurcated limit cycles. Finally, by using averaging theory of seventh order, we prove that there are cubic systems, close to the unperturbed system, with 1 and 2 small limit cycles.

Convergence rates in homogenization of higher-order parabolic systems
Weisheng Niu and Yao Xu
2018, 38(8): 4203-4229 doi: 10.3934/dcds.2018183 +[Abstract](529) +[HTML](95) +[PDF](528.97KB)
Abstract:

This paper is concerned with the optimal convergence rate in homogenization of higher order parabolic systems with bounded measurable, rapidly oscillating periodic coefficients. The sharp \begin{document}$O(\varepsilon )$\end{document} convergence rate in the space \begin{document}$L^2(0, T; H^{m-1}(\Omega ))$\end{document} is obtained for both the initial-Dirichlet problem and the initial-Neumann problem. The duality argument inspired by [25] is used here.

Second order regularity for degenerate nonlinear elliptic equations
Annamaria Canino, Elisa De Giorgio and Berardino Sciunzi
2018, 38(8): 4231-4242 doi: 10.3934/dcds.2018184 +[Abstract](499) +[HTML](93) +[PDF](434.79KB)
Abstract:

We investigate the second order regularity of solutions to degenerate nonlinear elliptic equations.

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