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Discrete & Continuous Dynamical Systems - A

April 2021 , Volume 41 , Issue 4

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Averaging of Hamilton-Jacobi equations along divergence-free vector fields
Hitoshi Ishii and Taiga Kumagai
2021, 41(4): 1519-1542 doi: 10.3934/dcds.2020329 +[Abstract](388) +[HTML](187) +[PDF](430.83KB)

We study the asymptotic behavior of solutions to the Dirichlet problem for Hamilton-Jacobi equations with large drift terms, where the drift terms are given by divergence-free vector fields. This is an attempt to understand the averaging effect for fully nonlinear degenerate elliptic equations. In this work, we restrict ourselves to the case of Hamilton-Jacobi equations. The second author has already established averaging results for Hamilton-Jacobi equations with convex Hamiltonians (\begin{document}$ G $\end{document} below) under the classical formulation of the Dirichlet condition. Here we treat the Dirichlet condition in the viscosity sense and establish an averaging result for Hamilton-Jacobi equations with relatively general Hamiltonian \begin{document}$ G $\end{document}.

Jordan decomposition and the recurrent set of flows of automorphisms
Víctor Ayala, Adriano Da Silva and Philippe Jouan
2021, 41(4): 1543-1559 doi: 10.3934/dcds.2020330 +[Abstract](395) +[HTML](189) +[PDF](405.36KB)

In this paper we show that any linear vector field \begin{document}$ \mathcal{X} $\end{document} on a connected Lie group \begin{document}$ G $\end{document} admits a Jordan decomposition and the recurrent set of the associated flow of automorphisms is given as the intersection of the fixed points of the hyperbolic and nilpotent components of its Jordan decomposition.

A Liouville theorem of parabolic Monge-AmpÈre equations in half-space
Ziwei Zhou, Jiguang Bao and Bo Wang
2021, 41(4): 1561-1578 doi: 10.3934/dcds.2020331 +[Abstract](475) +[HTML](202) +[PDF](414.5KB)

In this paper, we establish the gradient and second derivative estimates for solutions to two kinds of parabolic Monge-Ampère equations in half-space under certain boundary data and growth condition. We also use such estimates to obtain the Liouville theorems for these two kinds of parabolic Monge-Ampère equations and one kind of elliptic Monge-Ampère equation.

Well-posedness for the three dimensional stochastic planetary geostrophic equations of large-scale ocean circulation
Bo You
2021, 41(4): 1579-1604 doi: 10.3934/dcds.2020332 +[Abstract](353) +[HTML](191) +[PDF](464.67KB)

The objective of this paper is to study the well-posedness of solutions for the three dimensional planetary geostrophic equations of large-scale ocean circulation with additive noise. Since strong coupling terms and the noise term create some difficulties in the process of showing the existence of weak solutions, we will first show the existence of weak solutions by the monotonicity methods when the initial data satisfies some "regular" condition. For the case of general initial data, we will establish the existence of weak solutions by taking a sequence of "regular" initial data and proving the convergence in probability as well as some weak convergence of the corresponding solution sequences. Finally, we establish the existence of weak \begin{document}$ \mathcal{D} $\end{document}-pullback mean random attractors in the framework developed in [11,25].

Classification of nonnegative solutions to an equation involving the Laplacian of arbitrary order
Phuong Le
2021, 41(4): 1605-1626 doi: 10.3934/dcds.2020333 +[Abstract](532) +[HTML](197) +[PDF](404.46KB)

We classify all nonnegative nontrivial classical solutions to the equation

where \begin{document}$ 0<\alpha,\beta<n $\end{document}, \begin{document}$ c_1,c_2\ge0 $\end{document}, \begin{document}$ c_1+c_2>0 $\end{document} and \begin{document}$ f,g,h \in C([0, +\infty),[0, +\infty)) $\end{document} are increasing functions such that \begin{document}$ {f(t)}/{t^{\frac{n+\beta}{n-\alpha}}} $\end{document}, \begin{document}$ {g(t)}/{t^{\frac{\alpha+\beta}{n-\alpha}}} $\end{document}, \begin{document}$ {h(t)}/{t^{\frac{n+\alpha}{n-\alpha}}} $\end{document} are nonincreasing in \begin{document}$ (0, +\infty) $\end{document}. We also derive a Liouville type theorem for the equation in the case \begin{document}$ \alpha\ge n $\end{document}. The main tool we use is the method of moving spheres in integral forms.

The relationship between word complexity and computational complexity in subshifts
Ronnie Pavlov and Pascal Vanier
2021, 41(4): 1627-1648 doi: 10.3934/dcds.2020334 +[Abstract](296) +[HTML](157) +[PDF](376.75KB)

We prove several results about the relationship between the word complexity function of a subshift and the set of Turing degrees of points of the subshift, which we call the Turing spectrum. Among other results, we show that a Turing spectrum can be realized via a subshift of linear complexity if and only if it consists of the union of a finite set and a finite number of cones, that a Turing spectrum can be realized via a subshift of exponential complexity (i.e. positive entropy) if and only if it contains a cone, and that every Turing spectrum which either contains degree \begin{document}$ {{\mathbf{0}}}$\end{document} or is a union of cones is realizable by subshifts with a wide range of 'intermediate' complexity growth rates between linear and exponential.

Recurrence for measurable semigroup actions
Michael Blank
2021, 41(4): 1649-1665 doi: 10.3934/dcds.2020335 +[Abstract](350) +[HTML](173) +[PDF](353.96KB)

We study qualitative properties of the set of recurrent points of finitely generated free semigroups of measurable maps. In the case of a single generator the classical Poincare recurrence theorem shows that these properties are closely related to the presence of an invariant measure. Curious, but otherwise it turns out to be possible that almost all points are recurrent, while there is an wandering set of positive (non-invariant) measure. For a general semigroup the assumption about the common invariant measure for all generators looks somewhat unnatural (despite being widely used). Instead we give abstract conditions (of conservativity type) for this problem and propose a weaker version of the recurrent property. Technically, the problem is reduced to the analysis of the recurrence of a specially constructed Markov process. Questions of inheritance of the recurrence property from the semigroup generators to the entire semigroup and vice versa are studied in detail and we demonstrate that this inheritance might be rather unexpected.

Mckean-Vlasov sdes with drifts discontinuous under wasserstein distance
Xing Huang and Feng-Yu Wang
2021, 41(4): 1667-1679 doi: 10.3934/dcds.2020336 +[Abstract](312) +[HTML](160) +[PDF](407.29KB)

Existence and uniqueness are proved for McKean-Vlasov type distribution dependent SDEs with singular drifts satisfying an integrability condition in space variable and the Lipschitz condition in distribution variable with respect to \begin{document}$ {\mathbb W}_0 $\end{document} or \begin{document}$ {\mathbb W}_0+{\mathbb W}_\theta $\end{document} for some \begin{document}$ \theta\ge 1 $\end{document}, where \begin{document}$ {\mathbb W}_0 $\end{document} is the total variation distance and \begin{document}$ {\mathbb W}_\theta $\end{document} is the \begin{document}$ L^\theta $\end{document}-Wasserstein distance. This improves some existing results (see for instance [13]) derived for drifts continuous in the distribution variable with respect to the Wasserstein distance.

Global stability in a multi-dimensional predator-prey system with prey-taxis
Dan Li
2021, 41(4): 1681-1705 doi: 10.3934/dcds.2020337 +[Abstract](373) +[HTML](158) +[PDF](396.46KB)

This paper studies the predator-prey systems with prey-taxis

in a bounded domain \begin{document}$ \Omega\subset\mathbb{R}^{n} $\end{document} \begin{document}$ (n = 2, 3) $\end{document} with Neumann boundary conditions, where the parameters \begin{document}$ \chi $\end{document}, \begin{document}$ \gamma $\end{document}, \begin{document}$ \rho $\end{document}, \begin{document}$ \xi $\end{document} and \begin{document}$ \mu $\end{document} are positive. It is shown that the two-dimensional system possesses a unique global-bounded classical solution. Furthermore, we use some higher-order estimates to obtain the classical solutions with uniform-in-time bounded for suitably small initial data. Finally, we establish that the solution stabilizes towards the prey-only steady state \begin{document}$ (0, 1) $\end{document} if \begin{document}$ \rho>\gamma $\end{document} and towards the co-existence steady state \begin{document}$ (\frac{\mu(\gamma-\rho)}{\xi\rho}, \frac{\rho}{\gamma}) $\end{document} if \begin{document}$ \gamma>\rho $\end{document} under some conditions in the norm of \begin{document}$ L^{\infty}(\Omega) $\end{document} as \begin{document}$ t\rightarrow\infty $\end{document}.

Local well-posedness for the Klein-Gordon-Zakharov system in 3D
Hartmut Pecher
2021, 41(4): 1707-1736 doi: 10.3934/dcds.2020338 +[Abstract](331) +[HTML](159) +[PDF](425.09KB)

We study the Cauchy problem for the Klein-Gordon-Zakharov system in 3D with low regularity data. We lower down the regularity to the critical value with respect to scaling up to the endpoint. The decisive bilinear estimates are proved by means of methods developed by Bejenaru-Herr for the Zakharov system and already applied by Kinoshita to the Klein-Gordon-Zakharov system in 2D.

Dynamical Borel–Cantelli lemmas
Viktoria Xing
2021, 41(4): 1737-1754 doi: 10.3934/dcds.2020339 +[Abstract](337) +[HTML](171) +[PDF](372.56KB)

This paper is a study of Borel–Cantelli lemmas in dynamical systems. D. Kleinbock and G. Margulis [7] have given a very useful sufficient condition for strongly Borel–Cantelli sequences, which is based on the work of W. M. Schmidt [10], [11]. We will obtain a weaker sufficient condition for the strongly Borel–Cantelli sequences. Two versions of the dynamical Borel–Cantelli lemmas will be deduced by extending a theorem by W. M. Schmidt [11], W. J. LeVeque [8], and W. Philipp [9]. Some applications of our theorems will also be discussed. Firstly, a characterization of the strongly Borel–Cantelli sequences in one-dimensional Gibbs–Markov systems will be established. This will improve the theorem of C. Gupta, M. Nicol, and W. Ott in [4]. Secondly, N. Haydn, M. Nicol, T. Persson, and S. Vaienti [5] proved the strong Borel–Cantelli property in sequences of balls in terms of a polynomial decay of correlations for Lipschitz observables. Our theorems will then be applied to relax their inequality assumption.

Invisible tricorns in real slices of rational maps
Russell Lodge and Sabyasachi Mukherjee
2021, 41(4): 1755-1797 doi: 10.3934/dcds.2020340 +[Abstract](399) +[HTML](187) +[PDF](6715.19KB)

One of the conspicuous features of real slices of bicritical rational maps is the existence of Tricorn-type hyperbolic components. Such a hyperbolic component is called invisible if the non-bifurcating sub-arcs on its boundary do not intersect the closure of any other hyperbolic component. Numerical evidence suggests an abundance of invisible Tricorn-type components in real slices of bicritical rational maps. In this paper, we study two different families of real bicritical maps and characterize invisible Tricorn-type components in terms of suitable topological properties in the dynamical planes of the representative maps. We use this criterion to prove the existence of infinitely many invisible Tricorn-type components in the corresponding parameter spaces. Although we write the proofs for two specific families, our methods apply to generic families of real bicritical maps.

Anomalous time-scaling of extreme events in infinite systems and Birkhoff sums of infinite observables
Stefano Galatolo, Mark Holland, Tomas Persson and Yiwei Zhang
2021, 41(4): 1799-1841 doi: 10.3934/dcds.2020341 +[Abstract](372) +[HTML](185) +[PDF](542.39KB)

We establish quantitative results for the statistical behaviour of infinite systems. We consider two kinds of infinite system:

ⅰ) a conservative dynamical system \begin{document}$ (f,X,\mu) $\end{document} preserving a \begin{document}$ \sigma $\end{document}-finite measure \begin{document}$ \mu $\end{document} such that \begin{document}$ \mu(X) = \infty $\end{document};

ⅱ) the case where \begin{document}$ \mu $\end{document} is a probability measure but we consider the statistical behaviour of an observable \begin{document}$ \phi\colon X\to[0,\infty) $\end{document} which is non-integrable: \begin{document}$ \int \phi \, d\mu = \infty $\end{document}.

In the first part of this work we study the behaviour of Birkhoff sums of systems of the kind ii). For certain weakly chaotic systems, we show that these sums can be strongly oscillating. However, if the system has superpolynomial decay of correlations or has a Markov structure, then we show this oscillation cannot happen. In this case we prove a general relation between the behaviour of \begin{document}$ \phi $\end{document}, the local dimension of \begin{document}$ \mu $\end{document}, and the scaling rate of the growth of Birkhoff sums of \begin{document}$ \phi $\end{document} as time tends to infinity. We then establish several important consequences which apply to infinite systems of the kind i). This includes showing anomalous scalings in extreme event limit laws, or entrance time statistics. We apply our findings to non-uniformly hyperbolic systems preserving an infinite measure, establishing anomalous scalings for the power law behaviour of entrance times (also known as logarithm laws), dynamical Borel–Cantelli lemmas, almost sure growth rates of extremes, and dynamical run length functions.

Spatial dynamics and optimization method for a network propagation model in a shifting environment
Linhe Zhu and Wenshan Liu
2021, 41(4): 1843-1874 doi: 10.3934/dcds.2020342 +[Abstract](318) +[HTML](171) +[PDF](2105.79KB)

In this paper, a reaction-diffusion \begin{document}$ ISCT $\end{document} rumor propagation model with general incidence rate is proposed in a spatially heterogeneous environment. We first summarize the well-posedness of global solutions. Then the basic reproduction number \begin{document}$ \mathcal{R}_0 $\end{document} is introduced for the model which contains the spatial homogeneity as a special case. The threshold-type dynamics are also established in terms of \begin{document}$ \mathcal{R}_0 $\end{document}, including the global asymptotic stability of the rumor-free steady state and the uniform persistence of all positive solutions. Furthermore, by applying a controller to this model, we investigate the optimal control problem. Employing the operator semigroup theory, we prove the existence, uniqueness and some estimates of the positive strong solution to the controlled system. Subsequently, the existence of the optimal control strategy is established with the aid of minimal sequence techniques and the first order necessary optimality conditions for the optimal control is deduced. Finally, some numerical simulations are performed to validate the main analysis. The results of our study can theoretically promote the regulation of rumor propagation on the Internet.

Reversible perturbations of conservative Hénon-like maps
Marina Gonchenko, Sergey Gonchenko and Klim Safonov
2021, 41(4): 1875-1895 doi: 10.3934/dcds.2020343 +[Abstract](375) +[HTML](180) +[PDF](600.99KB)

For area-preserving Hénon-like maps and their compositions, we consider smooth perturbations that keep the reversibility of the initial maps but destroy their conservativity. For constructing such perturbations, we use two methods, a new method based on reversible properties of maps written in the so-called cross-form, and the classical Quispel-Roberts method based on a variation of involutions of the initial map. We study symmetry breaking bifurcations of symmetric periodic orbits in reversible families containing quadratic conservative orientable and nonorientable Hénon maps as well as a product of two Hénon maps whose Jacobians are mutually inverse.

Strichartz estimates and local regularity for the elastic wave equation with singular potentials
Seongyeon Kim, Yehyun Kwon and Ihyeok Seo
2021, 41(4): 1897-1911 doi: 10.3934/dcds.2020344 +[Abstract](380) +[HTML](159) +[PDF](417.32KB)

We obtain weighted \begin{document}$ L^2 $\end{document} estimates for the elastic wave equation perturbed by singular potentials including the inverse-square potential. We then deduce the Strichartz estimates under the sole ellipticity condition for the Lamé operator \begin{document}$ -\Delta^\ast $\end{document}. This improves upon the previous result in [1] which relies on a stronger condition to guarantee the self-adjointness of \begin{document}$ -\Delta^\ast $\end{document}. Furthermore, by establishing local energy estimates for the elastic wave equation we also prove that the solution has local regularity.

Diffeomorphisms with a generalized Lipschitz shadowing property
Manseob Lee, Jumi Oh and Xiao Wen
2021, 41(4): 1913-1927 doi: 10.3934/dcds.2020346 +[Abstract](286) +[HTML](156) +[PDF](357.12KB)

Shadowing property and structural stability are important dynamics with close relationship. Pilyugin and Tikhomirov proved that Lipschitz shadowing property implies the structural stability[5]. Todorov gave a similar result that Lipschitz two-sided limit shadowing property also implies structural stability for diffeomorpshisms[10]. In this paper, we define a generalized Lipschitz shadowing property which unifies these two kinds of Lipschitz shadowing properties, and prove that if a diffeomorphism \begin{document}$ f $\end{document} of a compact smooth manifold \begin{document}$ M $\end{document} has this generalized Lipschitz shadowing property then it is structurally stable. The only if part is also considered.

Splitting theorems on complete Riemannian manifolds with nonnegative Ricci curvature
Alberto Farina and Jesús Ocáriz
2021, 41(4): 1929-1937 doi: 10.3934/dcds.2020347 +[Abstract](317) +[HTML](164) +[PDF](297.67KB)

In this paper we provide some local and global splitting results on complete Riemannian manifolds with nonnegative Ricci curvature. We achieve the splitting through the analysis of some pointwise inequalities of Modica type which hold true for every bounded solution to a semilinear Poisson equation. More precisely, we prove that the existence of a nonconstant bounded solution \begin{document}$ u $\end{document} for which one of the previous inequalities becomes an equality at some point leads to the splitting results as well as to a classification of such a solution \begin{document}$ u $\end{document}.

Steady asymmetric vortex pairs for Euler equations
Zineb Hassainia and Taoufik Hmidi
2021, 41(4): 1939-1969 doi: 10.3934/dcds.2020348 +[Abstract](359) +[HTML](176) +[PDF](450.05KB)

In this paper, we study the existence of co-rotating and counter-rotating unequal-sized pairs of simply connected patches for Euler equations. In particular, we prove the existence of curves of steadily co-rotating and counter-rotating asymmetric vortex pairs passing through a point vortex pairs with unequal circulations. We also provide a careful study of the asymptotic behavior for the angular velocity and the translating speed close to the point vortex pairs.

Asymptotic stability of viscous contact wave for the inflow problem of the one-dimensional radiative Euler equations
Lili Fan, Lizhi Ruan and Wei Xiang
2021, 41(4): 1971-1999 doi: 10.3934/dcds.2020349 +[Abstract](316) +[HTML](152) +[PDF](484.59KB)

This paper is devoted to the study of the inflow problem governed by the radiative Euler equations in the one-dimensional half space. We establish the unique global-in-time existence and the asymptotic stability of the viscous contact discontinuity solution. It is different from the case involved with the rarefaction wave for the inflow problem in our previous work [6], since the rarefaction wave is a nonlinear expansive wave, while the contact discontinuity wave is a linearly degenerate diffusive wave. So we need to take good advantage of properties of the viscous contact discontinuity wave instead. Moreover, series of tricky argument on the boundary is done carefully based on the construction and the properties of the viscous contact discontinuity wave for the radiative Euler equations. Our result shows that radiation contributes to the stabilization effect for the supersonic inflow problem.

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