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Discrete & Continuous Dynamical Systems

May 2021 , Volume 41 , Issue 5

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Measures and stabilizers of group Cantor actions
Maik Gröger and Olga Lukina
2021, 41(5): 2001-2029 doi: 10.3934/dcds.2020350 +[Abstract](499) +[HTML](236) +[PDF](432.03KB)
Abstract:

We consider a minimal equicontinuous action of a finitely generated group \begin{document}$ G $\end{document} on a Cantor set \begin{document}$ X $\end{document} with invariant probability measure \begin{document}$ \mu $\end{document}, and the stabilizers of points for such an action. We give sufficient conditions under which there exists a subgroup \begin{document}$ H $\end{document} of \begin{document}$ G $\end{document} such that the set of points in \begin{document}$ X $\end{document} whose stabilizers are conjugate to \begin{document}$ H $\end{document} has full measure. The conditions are that the action is locally quasi-analytic and locally non-degenerate. An action is locally quasi-analytic if its elements have unique extensions on subsets of uniform diameter. The condition that the action is locally non-degenerate is introduced in this paper. We apply our results to study the properties of invariant random subgroups induced by minimal equicontinuous actions on Cantor sets and to certain almost one-to-one extensions of equicontinuous actions.

A $ G^{\delta, 1} $ almost conservation law for mCH and the evolution of its radius of spatial analyticity
A. Alexandrou Himonas and Gerson Petronilho
2021, 41(5): 2031-2050 doi: 10.3934/dcds.2020351 +[Abstract](448) +[HTML](190) +[PDF](398.96KB)
Abstract:

The Cauchy problem of the modified Camassa-Holm (mCH) equation with initial data \begin{document}$ u(0) $\end{document} that are analytic on the line and have uniform radius of analyticity \begin{document}$ r(0) $\end{document} is considered. First, by using bilinear estimates for the nonlocal nonlinearity in analytic Bourgain spaces, it is shown that this equation is well-posed in analytic Gevrey spaces \begin{document}$ G^{\delta, s} $\end{document}, with useful solution lifespan \begin{document}$ T_0 $\end{document} and size estimates. This shows that the radius of spatial analyticity \begin{document}$ r(t) $\end{document} persists during the time interval \begin{document}$ [-T_0, T_0] $\end{document}. Then, exploiting the fact that solutions to this equation conserve the \begin{document}$ H^1 $\end{document} norm, and utilizing the available bilinear estimates, an almost conservation low in \begin{document}$ G^{\delta,1} $\end{document} spaces is proved. Finally, using this almost conservation law it is shown that the solution \begin{document}$ u(t) $\end{document} exists for all time \begin{document}$ t $\end{document} and a lower bound for the radius of spatial analyticity is provided.

Cylinder absolute games on solenoids
L. Singhal
2021, 41(5): 2051-2070 doi: 10.3934/dcds.2020352 +[Abstract](571) +[HTML](185) +[PDF](401.73KB)
Abstract:

Let \begin{document}$ A $\end{document} be any affine surjective endomorphism of a solenoid \begin{document}${\Sigma_{{\mathcal{P}}}} $\end{document} over the circle \begin{document}$ S^1 $\end{document} which is not an infinite-order translation of \begin{document}$ {\Sigma_{{\mathcal{P}}}}$\end{document}. We prove the existence of a cylinder absolute winning (CAW) subset \begin{document}$ F \subseteq {\Sigma_{{\mathcal{P}}}} $\end{document} with the property that for any \begin{document}$ x \in F $\end{document}, the orbit closure \begin{document}$ \overline{\{ A^{\ell} x \mid \ell \in {\mathbb{N}} \}} $\end{document} does not contain any periodic orbits. A measure \begin{document}$ \mu $\end{document} on a metric space is said to be Federer if for all small enough balls around any generic point with respect to \begin{document}$ \mu $\end{document}, the measure grows by at most some constant multiple on doubling the radius of the ball. The class of infinite solenoids considered in this paper provides, to the best of our knowledge, some of the early natural examples of non-Federer spaces where absolute games can be played and won. Dimension maximality and incompressibility of CAW sets is also discussed for a number of possibilities in addition to their winning nature for the games known from before.

Entropy conjugacy for Markov multi-maps of the interval
James P. Kelly and Kevin McGoff
2021, 41(5): 2071-2094 doi: 10.3934/dcds.2020353 +[Abstract](380) +[HTML](176) +[PDF](469.86KB)
Abstract:

We consider a class \begin{document}$ \mathcal{F} $\end{document} of Markov multi-maps on the unit interval. Any multi-map gives rise to a space of trajectories, which is a closed, shift-invariant subset of \begin{document}$ [0, 1]^{\mathbb{Z}_+} $\end{document}. For a multi-map in \begin{document}$ \mathcal{F} $\end{document}, we show that the space of trajectories is (Borel) entropy conjugate to an associated shift of finite type. Additionally, we characterize the set of numbers that can be obtained as the topological entropy of a multi-map in \begin{document}$ \mathcal{F} $\end{document}.

Minimal period solutions in asymptotically linear Hamiltonian system with symmetries
Zhiping Fan and Duanzhi Zhang
2021, 41(5): 2095-2124 doi: 10.3934/dcds.2020354 +[Abstract](490) +[HTML](184) +[PDF](429.59KB)
Abstract:

In this paper, applying the Maslov-type index theory for periodic orbits and brake orbits, we study the minimal period problems in asymptotically linear Hamiltonian systems with different symmetries. For the asymptotically linear semipositive even Hamiltonian systems, we prove that for any given \begin{document}$ T>0 $\end{document}, there exists a central symmetric periodic solution with minimal period \begin{document}$ T $\end{document}. Moreover, if the Hamiltonian systems are also reversible, we prove the existence of a central symmetric brake orbit with minimal period being either \begin{document}$ T $\end{document} or \begin{document}$ T/3 $\end{document}. Also we give some other lower bound estimations for brake orbits case.

Gamma convergence and asymptotic behavior for eigenvalues of nonlocal problems
Julián Fernández Bonder, Analía Silva and Juan F. Spedaletti
2021, 41(5): 2125-2140 doi: 10.3934/dcds.2020355 +[Abstract](446) +[HTML](173) +[PDF](356.78KB)
Abstract:

In this paper we analyze the asymptotic behavior of several fractional eigenvalue problems by means of Gamma-convergence methods. This method allows us to treat different eigenvalue problems under a unified framework. We are able to recover some known results for the behavior of the eigenvalues of the \begin{document}$ p- $\end{document}fractional laplacian when the fractional parameter \begin{document}$ s $\end{document} goes to 1, and to extend some known results for the behavior of the same eigenvalue problem when \begin{document}$ p $\end{document} goes to \begin{document}$ \infty $\end{document}. Finally we analyze other eigenvalue problems not previously covered in the literature.

Thermodynamic formalism of $ \text{GL}_2(\mathbb{R}) $-cocycles with canonical holonomies
Clark Butler and Kiho Park
2021, 41(5): 2141-2166 doi: 10.3934/dcds.2020356 +[Abstract](436) +[HTML](189) +[PDF](478.07KB)
Abstract:

We study the norm potentials of Hölder continuous \begin{document}$ \text{GL}_2(\mathbb{R}) $\end{document}-cocycles over hyperbolic systems whose canonical holonomies converge and are Hölder continuous. Such cocycles include locally constant \begin{document}$ \text{GL}_2(\mathbb{R}) $\end{document}-cocycles as well as fiber-bunched \begin{document}$ \text{GL}_2(\mathbb{R}) $\end{document}-cocycles. We show that the norm potentials of irreducible such cocycles have unique equilibrium states. Among the reducible cocycles, we provide a characterization for cocycles whose norm potentials have more than one equilibrium states.

Radially symmetric stationary wave for two-dimensional Burgers equation
Huijiang Zhao and Qingsong Zhao
2021, 41(5): 2167-2185 doi: 10.3934/dcds.2020357 +[Abstract](420) +[HTML](185) +[PDF](410.58KB)
Abstract:

We are concerned with the radially symmetric stationary wave for the exterior problem of two-dimensional Burgers equation. A sufficient and necessary condition to guarantee the existence of such a stationary wave is given and it is also shown that the stationary wave satisfies nice decay estimates and is time-asymptotically nonlinear stable under radially symmetric initial perturbation.

Integral equations on compact CR manifolds
Yazhou Han
2021, 41(5): 2187-2204 doi: 10.3934/dcds.2020358 +[Abstract](419) +[HTML](187) +[PDF](419.93KB)
Abstract:

Assume that \begin{document}$ M $\end{document} is a CR compact manifold without boundary and CR Yamabe invariant \begin{document}$ \mathcal{Y}(M) $\end{document} is positive. Here, we devote to study a class of sharp Hardy-Littlewood-Sobolev inequality as follows

where \begin{document}$ G_\xi^\theta(\eta) $\end{document} is the Green function of CR conformal Laplacian \begin{document}$ \mathcal{L_\theta} = b_n\Delta_b+R $\end{document}, \begin{document}$ \mathcal{Y}_\alpha(M) $\end{document} is sharp constant, \begin{document}$ \Delta_b $\end{document} is Sublaplacian and \begin{document}$ R $\end{document} is Tanaka-Webster scalar curvature. For the diagonal case \begin{document}$ f = g $\end{document}, we prove that \begin{document}$ \mathcal{Y}_\alpha(M)\geq \mathcal{Y}_\alpha(\mathbb{S}^{2n+1}) $\end{document} (the unit complex sphere of \begin{document}$ \mathbb{C}^{n+1} $\end{document}) and \begin{document}$ \mathcal{Y}_\alpha(M) $\end{document} can be attained if \begin{document}$ \mathcal{Y}_\alpha(M)> \mathcal{Y}_\alpha(\mathbb{S}^{2n+1}) $\end{document}. So, we got the existence of the Euler-Lagrange equations

Moreover, we prove that the solution of (1) is \begin{document}$ \Gamma^\alpha(M) $\end{document}. Particular, if \begin{document}$ \alpha = 2 $\end{document}, the previous extremal problem is closely related to the CR Yamabe problem. Hence, we can study the CR Yamabe problem by integral equations.

Quantitative oppenheim conjecture for $ S $-arithmetic quadratic forms of rank $ 3 $ and $ 4 $
Jiyoung Han
2021, 41(5): 2205-2225 doi: 10.3934/dcds.2020359 +[Abstract](382) +[HTML](183) +[PDF](410.35KB)
Abstract:

The celebrated result of Eskin, Margulis and Mozes [8] and Dani and Margulis [7] on quantitative Oppenheim conjecture says that for irrational quadratic forms \begin{document}$ q $\end{document} of rank at least 5, the number of integral vectors \begin{document}$ \mathbf v $\end{document} such that \begin{document}$ q( \mathbf v) $\end{document} is in a given bounded interval is asymptotically equal to the volume of the set of real vectors \begin{document}$ \mathbf v $\end{document} such that \begin{document}$ q( \mathbf v) $\end{document} is in the same interval.

In rank \begin{document}$ 3 $\end{document} or \begin{document}$ 4 $\end{document}, there are exceptional quadratic forms which fail to satisfy the quantitative Oppenheim conjecture. Even in those cases, one can say that two asymptotic limits coincide for almost all quadratic forms([8, Theorem 2.4]). In this paper, we extend this result to the \begin{document}$ S $\end{document}-arithmetic version.

Martingale solution for stochastic active liquid crystal system
Zhaoyang Qiu and Yixuan Wang
2021, 41(5): 2227-2268 doi: 10.3934/dcds.2020360 +[Abstract](403) +[HTML](173) +[PDF](476.21KB)
Abstract:

The global weak martingale solution is built through a four-level approximation scheme to stochastic compressible active liquid crystal system driven by multiplicative noise in a smooth bounded domain in \begin{document}$ \mathbb{R}^{3} $\end{document} with large initial data. The coupled structure makes the analysis challenging, and more delicate arguments are required in stochastic case compared to the deterministic one [11].

Liouville type theorems for fractional and higher-order fractional systems
Daomin Cao and Guolin Qin
2021, 41(5): 2269-2283 doi: 10.3934/dcds.2020361 +[Abstract](469) +[HTML](172) +[PDF](424.93KB)
Abstract:

In this paper, we first establish decay estimates for the fractional and higher-order fractional Hénon-Lane-Emden systems by using a nonlocal average and integral estimates, which deduce a result of non-existence. Next, we apply the method of scaling spheres introduced in [16] to derive a Liouville type theorem. We also construct an interesting example on super \begin{document}$ \frac{\alpha}{2} $\end{document}-harmonic functions (Proposition 1.2).

Sliding method for the semi-linear elliptic equations involving the uniformly elliptic nonlocal operators
Meng Qu, Jiayan Wu and Ting Zhang
2021, 41(5): 2285-2300 doi: 10.3934/dcds.2020362 +[Abstract](493) +[HTML](184) +[PDF](354.2KB)
Abstract:

In this paper, we consider the uniformly elliptic nonlocal operators

where \begin{document}$ a(x) $\end{document} is positively uniform bounded satisfying a cylindrical condition. We first establish the narrow region principle in the bounded domain. Then using the sliding method, we obtain the monotonicity of solutions for the semi-linear equation involving \begin{document}$ A_{\alpha} $\end{document} in both the bounded domain and the whole space. In addition, we establish the maximum principle in the unbounded domain and get the non-existence of solutions in the upper half space \begin{document}$ \mathbb R^n_+ $\end{document}.

On the asymptotic properties for stationary solutions to the Navier-Stokes equations
Oleg Imanuvilov
2021, 41(5): 2301-2340 doi: 10.3934/dcds.2020366 +[Abstract](375) +[HTML](173) +[PDF](483.01KB)
Abstract:

In this paper we study solutions of the stationary Navier-Stokes system, and investigate the minimal decay rate for a nontrivial velocity field at infinity in outside of an obstacle. We prove that in an exterior domain if a solution \begin{document}$ v $\end{document} and its derivatives decay like \begin{document}$ O(|x|^{-k}) $\end{document} for sufficiently large \begin{document}$ k $\end{document}, depending on the velocity field, as \begin{document}$ |x|\to \infty $\end{document}, then \begin{document}$ v $\end{document} is zero on that exterior domain. Constructive estimate for \begin{document}$ k $\end{document} is given. In the case where velocity field is only bounded at infinity, we show that the infimum of \begin{document}$ L^2 $\end{document} norm of a velocity field on a unit ball located at distance \begin{document}$ t $\end{document} from an origin is bounded from below as \begin{document}$ Ce^{-\beta t^\frac 43\ln(t)}. $\end{document} The proof of these results are based on the Carleman type estimates, and also the Kelvin transform.

Well-posedness of renormalized solutions for a stochastic $ p $-Laplace equation with $ L^1 $-initial data
Niklas Sapountzoglou and Aleksandra Zimmermann
2021, 41(5): 2341-2376 doi: 10.3934/dcds.2020367 +[Abstract](361) +[HTML](165) +[PDF](483.8KB)
Abstract:

We consider a \begin{document}$ p $\end{document}-Laplace evolution problem with stochastic forcing on a bounded domain \begin{document}$ D\subset\mathbb{R}^d $\end{document} with homogeneous Dirichlet boundary conditions for \begin{document}$ 1<p<\infty $\end{document}. The additive noise term is given by a stochastic integral in the sense of Itô. The technical difficulties arise from the merely integrable random initial data \begin{document}$ u_0 $\end{document} under consideration. Due to the poor regularity of the initial data, estimates in \begin{document}$ W^{1,p}_0(D) $\end{document} are available with respect to truncations of the solution only and therefore well-posedness results have to be formulated in the sense of generalized solutions. We extend the notion of renormalized solution for this type of SPDEs, show well-posedness in this setting and study the Markov properties of solutions.

On the vanishing discount problem from the negative direction
Andrea Davini and Lin Wang
2021, 41(5): 2377-2389 doi: 10.3934/dcds.2020368 +[Abstract](391) +[HTML](165) +[PDF](357.56KB)
Abstract:

It has been proved in [7] that the unique viscosity solution of

\begin{document}$ \begin{equation} \lambda u_\lambda+H(x,d_x u_\lambda) = c(H)\qquad\hbox{in $M$}, \;\;\;\;\;\;\;\;\;(*)\end{equation} $\end{document}

uniformly converges, for \begin{document}$ \lambda\rightarrow 0^+ $\end{document}, to a specific solution \begin{document}$ u_0 $\end{document} of the critical equation

where \begin{document}$ M $\end{document} is a closed and connected Riemannian manifold and \begin{document}$ c(H) $\end{document} is the critical value. In this note, we consider the same problem for \begin{document}$ \lambda\rightarrow 0^- $\end{document}. In this case, viscosity solutions of equation (*) are not unique, in general, so we focus on the asymptotics of the minimal solution \begin{document}$ u_\lambda^- $\end{document} of (*). Under the assumption that constant functions are subsolutions of the critical equation, we prove that the \begin{document}$ u_\lambda^- $\end{document} also converges to \begin{document}$ u_0 $\end{document} as \begin{document}$ \lambda\rightarrow 0^- $\end{document}. Furthermore, we exhibit an example of \begin{document}$ H $\end{document} for which equation (*) admits a unique solution for \begin{document}$ \lambda<0 $\end{document} as well.

Genetics of iterative roots for PM functions
Liu Liu and Weinian Zhang
2021, 41(5): 2391-2409 doi: 10.3934/dcds.2020369 +[Abstract](362) +[HTML](178) +[PDF](335.56KB)
Abstract:

It is known that the time-one mapping of a flow defines a discrete dynamical system with the same dynamical behaviors as the flow, but conversely one wants to know whether a flow embedded by a homeomorphism preserves the dynamical behaviors of the homeomorphism. In this paper we consider iterative roots, a weak version of embedded flows, for the preservation. We refer an iterative root to be genetic if it is topologically conjugate to its parent function. We prove that none of PM functions with height being \begin{document}$ >1 $\end{document} has a genetic root and none of iterative roots of height being \begin{document}$ >1 $\end{document} is genetic even if the height of its parent function is equal to 1. This shows that most functions do not have a genetic iterative root. Further, we obtain a necessary and sufficient conditions under which a PM function \begin{document}$ f $\end{document} has a genetic iterative root in the case that \begin{document}$ f $\end{document} and the iterative root are both of height 1.

Strong blow-up instability for standing wave solutions to the system of the quadratic nonlinear Klein-Gordon equations
Hayato Miyazaki
2021, 41(5): 2411-2445 doi: 10.3934/dcds.2020370 +[Abstract](482) +[HTML](161) +[PDF](457.21KB)
Abstract:

This paper is concerned with strong blow-up instability (Definition 1.3) for standing wave solutions to the system of the quadratic nonlinear Klein-Gordon equations. In the single case, namely the nonlinear Klein-Gordon equation with power type nonlinearity, stability and instability for standing wave solutions have been extensively studied. On the other hand, in the case of our system, there are no results concerning the stability and instability as far as we know.

In this paper, we prove strong blow-up instability for the standing wave to our system. The proof is based on the techniques in Ohta and Todorova [27]. It turns out that we need the mass resonance condition in two or three space dimensions whose cases are the mass-subcritical case.

Stabilizability in optimization problems with unbounded data
Anna Chiara Lai and Monica Motta
2021, 41(5): 2447-2474 doi: 10.3934/dcds.2020371 +[Abstract](379) +[HTML](166) +[PDF](455.32KB)
Abstract:

In this paper we extend the notions of sample and Euler stabilizability to a set of a control system to a wide class of systems with unbounded controls, which includes nonlinear control-polynomial systems. In particular, we allow discontinuous stabilizing feedbacks, which are unbounded approaching the target. As a consequence, sampling trajectories may present a chattering behaviour and Euler solutions have in general an impulsive character. We also associate to the control system a cost and provide sufficient conditions, based on the existence of a special Lyapunov function, which allow for the existence of a stabilizing feedback that keeps the cost of all sampling and Euler solutions starting from the same point below the same value, in a uniform way.

Orbital and asymptotic stability of a train of peakons for the Novikov equation
José Manuel Palacios
2021, 41(5): 2475-2518 doi: 10.3934/dcds.2020372 +[Abstract](374) +[HTML](176) +[PDF](595.58KB)
Abstract:

The Novikov equation is an integrable Camassa-Holm type equation with cubic nonlinearity. One of the most important features of this equation is the existence of peakon and multi-peakon solutions, i.e. peaked traveling waves behaving as solitons. This paper aims to prove both, the orbital and asymptotic stability of peakon trains solutions, i.e. multi-peakon solutions such that their initial configuration is increasingly ordered. Furthermore, we give an improvement of the orbital stability of a single peakon so that we can drop the non-negativity hypothesis on the momentum density. The same result also holds for the orbital stability for peakon trains, i.e. in this latter case we can also avoid assuming non-negativity of the initial momentum density. Finally, as a corollary of these results together with some asymptotic formulas for the position and momenta vectors for multi-peakon solutions, we obtain the orbital and asymptotic stability for initially not well-ordered multipeakons.

2019  Impact Factor: 1.338

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