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Generalized solutions to models of compressible viscous fluids Discrete Contin. Dyn. Syst. A (IF 1.338) Pub Date : 2020-10-14 Anna Abbatiello; Eduard Feireisl; Antoní Novotný
We propose a new approach to models of general compressible viscous fluids based on the concept of dissipative solutions. These are weak solutions satisfying the underlying equations modulo a defect measure. A dissipative solution coincides with the strong solution as long as the latter exists (weak–strong uniqueness) and they solve the problem in the classical sense as soon as they are smooth (compatibility)
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\begin{document}$ L^\infty $\end{document}-estimation of generalized Thue-Morse trigonometric polynomials and ergodic maximization Discrete Contin. Dyn. Syst. A (IF 1.338) Pub Date : 2020-10-30 Aihua Fan; Jörg Schmeling; Weixiao Shen
Given an integer $ q\ge 2 $ and a real number $ c\in [0,1) $, consider the generalized Thue-Morse sequence $ (t_n^{(q;c)})_{n\ge 0} $ defined by $ t_n^{(q;c)} = e^{2\pi i c s_q(n)} $, where $ s_q(n) $ is the sum of digits of the $ q $-expansion of $ n $. We prove that the $ L^\infty $-norm of the trigonometric polynomials $ \sigma_{N}^{(q;c)} (x) : = \sum_{n = 0}^{N-1} t_n^{(q;c)} e^{2\pi i n x} $
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On \begin{document}$ \epsilon $\end{document}-escaping trajectories in homogeneous spaces Discrete Contin. Dyn. Syst. A (IF 1.338) Pub Date : 2020-10-30 Federico Rodriguez Hertz; Zhiren Wang
Let $ G/\Gamma $ be a finite volume homogeneous space of a semisimple Lie group $ G $, and $ \{\exp(tD)\} $ be a one-parameter $ \operatorname{Ad} $-diagonalizable subgroup inside a simple Lie subgroup $ G_0 $ of $ G $. Denote by $ Z_{\epsilon,D} $ the set of points $ x\in G/\Gamma $ whose $ \{\exp(tD)\} $-trajectory has an escape for at least an $ \epsilon $-portion of mass along some subsequence
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Entire solutions of the Allen–Cahn–Nagumo equation in a multi-dimensional space Discrete Contin. Dyn. Syst. A (IF 1.338) Pub Date : 2020-10-30 Hirokazu Ninomiya
The Allen–Cahn–Nagumo equation is a reaction-diffusion equation with a bistable nonlinearity. This equation appears to be simple, however, it includes a rich behavior of solutions. The Allen–Cahn–Nagumo equation features a solution that constantly maintains a certain profile and moves with a constant speed, which is referred to as a traveling wave solution. In this paper, the entire solution of the
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Inverse problems for nonlinear hyperbolic equations Discrete Contin. Dyn. Syst. A (IF 1.338) Pub Date : 2020-11-05 Gunther Uhlmann; Jian Zhai
There has been considerable progress in recent years in solving inverse problems for nonlinear hyperbolic equations. One of the striking aspects of these developments is the use of nonlinearity to get new information, which is not possible for the corresponding linear equations. We illustrate this for several examples including Einstein equations and the equations of nonlinear elasticity among others
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Failure of scattering to solitary waves for long-range nonlinear Schrödinger equations Discrete Contin. Dyn. Syst. A (IF 1.338) Pub Date : 2020-09-15 Jason Murphy; Kenji Nakanishi
We consider nonlinear Schrödinger equations with either power-type or Hartree nonlinearity in the presence of an external potential. We show that for long-range nonlinearities, solutions cannot exhibit scattering to solitary waves or more general localized waves. This extends the well-known results concerning non-existence of non-trivial scattering states for long-range nonlinearities.
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Sharp regularity for degenerate obstacle type problems: A geometric approach Discrete Contin. Dyn. Syst. A (IF 1.338) Pub Date : 2020-09-03 João Vitor da Silva; Hernán Vivas
We prove sharp regularity estimates for solutions of obstacle type problems driven by a class of degenerate fully nonlinear operators. More specifically, we consider viscosity solutions of $ \begin{equation*} \left\{ \begin{array}{rcll} |D u|^\gamma F(x, D^2u)& = & f(x)\chi_{\{u>\phi\}} & \ \rm{ in } \ B_1 \\ u(x) & \geq & \phi(x) & \ \rm{ in } \ B_1 \\ u(x) & = & g(x) & \ \rm{on } \ \partial B_1,
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Global large solutions and optimal time-decay estimates to the Korteweg system Discrete Contin. Dyn. Syst. A (IF 1.338) Pub Date : 2020-09-03 Xiaoping Zhai; Yongsheng Li
We prove the global solutions to the Korteweg system without smallness condition imposed on the vertical component of the incompressible part of the velocity. The weighted Chemin-Lerner-norm technique which is well-known for the incompressible Navier-Stokes equations is introduced to derive the a priori estimates. As a byproduct, we obtain the optimal time decay rates of the solutions by using the
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A sharp scattering threshold level for mass-subcritical nonlinear Schrödinger system Discrete Contin. Dyn. Syst. A (IF 1.338) Pub Date : 2020-09-03 Masaru Hamano; Satoshi Masaki
In this paper, we consider the quadratic nonlinear Schrödinger system in three space dimensions. Our aim is to obtain sharp scattering criteria. Because of the mass-subcritical nature, it is difficult to do so in terms of conserved quantities. The corresponding single equation is studied by the second author and a sharp scattering criterion is established by introducing a distance from a trivial scattering
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Mean-square random invariant manifolds for stochastic differential equations Discrete Contin. Dyn. Syst. A (IF 1.338) Pub Date : 2020-09-03 Bixiang Wang
We develop a theory of mean-square random invariant manifolds for mean-square random dynamical systems generated by stochastic differential equations. This theory is applicable to stochastic partial differential equations driven by nonlinear noise. The existence of mean-square random invariant unstable manifolds is proved by the Lyapunov-Perron method based on a backward stochastic differential equation
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Global graph of metric entropy on expanding Blaschke products Discrete Contin. Dyn. Syst. A (IF 1.338) Pub Date : 2020-09-03 Yunping Jiang
We study the global picture of the metric entropy on the space of expanding Blaschke products. We first construct a smooth path in the space tending to a parabolic Blaschke product. We prove that the metric entropy on this path tends to 0 as the path tends to this parabolic Blaschke product. It turns out that the limiting parabolic Blaschke product on the unit circle is conjugate to the famous Boole
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Existence of nodal solutions for the sublinear Moore-Nehari differential equation Discrete Contin. Dyn. Syst. A (IF 1.338) Pub Date : 2020-09-03 Ryuji Kajikiya
We study the existence of symmetric and asymmetric nodal solutions for the sublinear Moore-Nehari differential equation, $ u''+h(x, \lambda)|u|^{p-1}u = 0 $ in $ (-1, 1) $ with $ u(-1) = u(1) = 0 $, where $ 00 $. For integers $ m, n \geq 0 $, we call a solution $ u $ an $ (m, n) $-solution if it has exactly $ m $ zeros in $ (-1, 0) $ and exactly $ n $ zeros in $ (0, 1) $. We show the existence of an
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Asymptotic stability in a chemotaxis-competition system with indirect signal production Discrete Contin. Dyn. Syst. A (IF 1.338) Pub Date : 2020-08-26 Pan Zheng
This paper deals with a fully parabolic inter-species chemotaxis-competition system with indirect signal production $ \begin{eqnarray*} \label{1a} \left\{ \begin{split}{} &u_{t} = \text{div}(d_{u}\nabla u+\chi u\nabla w)+\mu_{1}u(1-u-a_{1}v), &(x,t)\in \Omega\times (0,\infty), \\ &v_{t} = d_{v}\Delta v+\mu_{2}v(1-v-a_{2}u), &(x,t)\in \Omega\times (0,\infty), \\ & w_{t} = d_{w}\Delta w-\lambda w+\alpha
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On asymptotic stability of ground states of some systems of nonlinear Schrödinger equations Discrete Contin. Dyn. Syst. A (IF 1.338) Pub Date : 2020-08-26 Andrew Comech; Scipio Cuccagna
We extend to a specific class of systems of nonlinear Schrödinger equations (NLS) the theory of asymptotic stability of ground states already proved for the scalar NLS. Here the key point is the choice of an adequate system of modulation coordinates and the novelty, compared to the scalar NLS, is the fact that the group of symmetries of the system is non-commutative.
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On entropy of \begin{document}$ \Phi $\end{document}-irregular and \begin{document}$ \Phi $\end{document}-level sets in maps with the shadowing property Discrete Contin. Dyn. Syst. A (IF 1.338) Pub Date : 2020-08-26 Magdalena Foryś-Krawiec; Jiří Kupka; Piotr Oprocha; Xueting Tian
We study the properties of $ \Phi $-irregular sets (sets of points for which the Birkhoff average diverges) in dynamical systems with the shadowing property. We estimate the topological entropy of $ \Phi $-irregular set in terms of entropy on chain recurrent classes and prove that $ \Phi $-irregular sets of full entropy are typical. We also consider $ \Phi $-level sets (sets of points whose Birkhoff
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Blow-up and bounded solutions for a semilinear parabolic problem in a saturable medium Discrete Contin. Dyn. Syst. A (IF 1.338) Pub Date : 2020-08-26 Juliana Fernandes; Liliane Maia
The present paper is on the existence and behaviour of solutions for a class of semilinear parabolic equations, defined on a bounded smooth domain and assuming a nonlinearity asymptotically linear at infinity. The behavior of the solutions when the initial data varies in the phase space is analyzed. Global solutions are obtained, which may be bounded or blow-up in infinite time (grow-up). The main
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Entropy production in random billiards Discrete Contin. Dyn. Syst. A (IF 1.338) Pub Date : 2020-08-26 Timothy Chumley; Renato Feres
We consider a class of random mechanical systems called random billiards to study the problem of quantifying the irreversibility of nonequilibrium macroscopic systems. In a random billiard model, a point particle evolves by free motion through the interior of a spatial domain, and reflects according to a reflection operator, specified in the model by a Markov transition kernel, upon collision with
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Gromov-Hausdorff stability for group actions Discrete Contin. Dyn. Syst. A (IF 1.338) Pub Date : 2020-08-26 Meihua Dong; Keonhee Lee; Carlos Morales
We will extend the topological Gromov-Hausdorff stability [2] from homeomorphisms to finitely generated actions. We prove that if an action is expansive and has the shadowing property, then it is topologically GH-stable. From this we derive examples of topologically GH-stable actions of the discrete Heisenberg group on tori. Finally, we prove that the topological GH-stability is an invariant under
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Local limit theorems for suspended semiflows Discrete Contin. Dyn. Syst. A (IF 1.338) Pub Date : 2020-08-06 Jon Aaronson; Dalia Terhesiu
We prove local limit theorems for a cocycle over a semiflow by establishing topological, mixing properties of the associated skew product semiflow. We also establish conditional rational weak mixing of certain skew product semiflows and various mixing properties including order 2 rational weak mixing of hyperbolic geodesic flows on the tangent spaces of cyclic covers.
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The Mandelbrot set is the shadow of a Julia set Discrete Contin. Dyn. Syst. A (IF 1.338) Pub Date : 2020-07-11 François Berteloot; Tien-Cuong Dinh
Working within the polynomial quadratic family, we introduce a new point of view on bifurcations which naturally allows to see the set of bifurcations as the projection of a Julia set of a complex dynamical system in dimension three. We expect our approach to be extendable to other holomorphic families of dynamical systems.
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How to identify a hyperbolic set as a blender Discrete Contin. Dyn. Syst. A (IF 1.338) Pub Date : 2020-08-03 Stefanie Hittmeyer; Bernd Krauskopf; Hinke M. Osinga; Katsutoshi Shinohara
A blender is a hyperbolic set with a stable or unstable invariant manifold that behaves as a geometric object of a dimension larger than that of the respective manifold itself. Blenders have been constructed in diffeomorphisms with a phase space of dimension at least three. We consider here the question of how one can identify, characterize and also visualize the underlying hyperbolic set of a given
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Automatic sequences are orthogonal to aperiodic multiplicative functions Discrete Contin. Dyn. Syst. A (IF 1.338) Pub Date : 2020-07-11 Mariusz Lemańczyk; Clemens Müllner
Given a finite alphabet $ \mathbb{A} $
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Finding polynomial roots by dynamical systems – A case study Discrete Contin. Dyn. Syst. A (IF 1.338) Pub Date : 2020-07-11 Sergey Shemyakov; Roman Chernov; Dzmitry Rumiantsau; Dierk Schleicher; Simon Schmitt; Anton Shemyakov
We investigate two well known dynamical systems that are designed to find roots of univariate polynomials by iteration: the methods known by Newton and by Ehrlich–Aberth. Both are known to have found all roots of high degree polynomials with good complexity. Our goal is to determine in which cases which of the two algorithms is more efficient. We come to the conclusion that Newton is faster when the
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Multiple ergodic averages for tempered functions Discrete Contin. Dyn. Syst. A (IF 1.338) Pub Date : 2020-08-11 Andreas Koutsogiannis
Following Frantzikinakis' approach on averages for Hardy field functions of different growth, we add to the topic by studying the corresponding averages for tempered functions, a class which also contains functions that oscillate and is in general more restrictive to deal with. Our main result is the existence and the explicit expression of the $ L^2 $-norm limit of the aforementioned averages, which
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Construction of a solitary wave solution of the nonlinear focusing schrödinger equation outside a strictly convex obstacle in the \begin{document}$ L^2 $\end{document}-supercritical case Discrete Contin. Dyn. Syst. A (IF 1.338) Pub Date : 2020-08-11 Oussama Landoulsi
We consider the focusing $ L^2 $-supercritical Schrödinger equation in the exterior of a smooth, compact, strictly convex obstacle $ \Theta \subset \mathbb{R}^3 $. We construct a solution behaving asymptotically as a solitary wave on $ \mathbb{R}^3, $ for large times. When the velocity of the solitary wave is high, the existence of such a solution can be proved by a classical fixed point argument.
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Large time behavior of exchange-driven growth Discrete Contin. Dyn. Syst. A (IF 1.338) Pub Date : 2020-08-11 Emre Esentürk; Juan Velazquez
Exchange-driven growth (EDG) is a model in which pairs of clusters interact by exchanging single unit with a rate given by a kernel $ K(j, k) $. Despite EDG model's common use in the applied sciences, its rigorous mathematical treatment is very recent. In this article we study the large time behaviour of EDG equations. We show two sets of results depending on the properties of the kernel $ (i) $ $
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Attainability property for a probabilistic target in wasserstein spaces Discrete Contin. Dyn. Syst. A (IF 1.338) Pub Date : 2020-08-11 Giulia Cavagnari; Antonio Marigonda
In this paper we establish an attainability result for the minimum time function of a control problem in the space of probability measures endowed with Wasserstein distance. The dynamics is provided by a suitable controlled continuity equation, where we impose a nonlocal nonholonomic constraint on the driving vector field, which is assumed to be a Borel selection of a given set-valued map. This model
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Asymptotic dynamics of a system of conservation laws from chemotaxis Discrete Contin. Dyn. Syst. A (IF 1.338) Pub Date : 2020-08-11 Neng Zhu; Zhengrong Liu; Fang Wang; Kun Zhao
This paper is devoted to the analytical study of the long-time asymptotic behavior of solutions to the Cauchy problem of a system of conservation laws in one space dimension, which is derived from a repulsive chemotaxis model with singular sensitivity and nonlinear chemical production rate. Assuming the $ H^2 $-norm of the initial perturbation around a constant ground state is finite and using energy
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Well-posedness of some non-linear stable driven SDEs Discrete Contin. Dyn. Syst. A (IF 1.338) Pub Date : 2020-08-11 Noufel Frikha; Valentin Konakov; Stéphane Menozzi
We prove the well-posedness of some non-linear stochastic differential equations in the sense of McKean-Vlasov driven by non-degenerate symmetric $ \alpha $-stable Lévy processes with values in $ {{{\mathbb R}}}^d $ under some mild Hölder regularity assumptions on the drift and diffusion coefficients with respect to both space and measure variables. The methodology developed here allows to consider
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A generalization of the Babbage functional equation Discrete Contin. Dyn. Syst. A (IF 1.338) Pub Date : 2020-08-11 Marc Homs-Dones
A recent refinement of Kerékjártó's Theorem has shown that in $ \mathbb R $ and $ \mathbb R^2 $ all $ \mathcal C^l $–solutions of the functional equation $ f^n = \text{Id} $ are $ \mathcal C^l $–linearizable, where $ l\in \{0,1,\dots \infty\} $. When $ l\geq 1 $, in the real line we prove that the same result holds for solutions of $ f^n = f $, while we can only get a local version of it in the plane
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Low Mach number limit for the compressible inertial Qian-Sheng model of liquid crystals: Convergence for classical solutions Discrete Contin. Dyn. Syst. A (IF 1.338) Pub Date : 2020-08-11 Yi-Long Luo; Yangjun Ma
In this paper we study the incompressible limit of the compressible inertial Qian-Sheng model for liquid crystal flow. We first derive the uniform energy estimates on the Mach number $ \epsilon $ for both the compressible system and its differential system with respect to time under uniformly in $ \epsilon $ small initial data. Then, based on these uniform estimates, we pass to the limit in the compressible
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Geometric singular perturbation analysis of Degasperis-Procesi equation with distributed delay Discrete Contin. Dyn. Syst. A (IF 1.338) Pub Date : 2020-08-11 Feifei Cheng; Ji Li
In this paper we consider the Degasperis-Procesi equation, which is an approximation to the incompressible Euler equation in shallow water regime. First we provide the existence of solitary wave solutions for the original DP equation and the general theory of geometric singular perturbation. Then we prove the existence of solitary wave solutions for the equation with a special local delay convolution
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Properties of the minimizers for a constrained minimization problem arising in Kirchhoff equation Discrete Contin. Dyn. Syst. A (IF 1.338) Pub Date : 2020-08-10 Helin Guo; Huan-Song Zhou
Let $ a>0,b>0 $
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Pomeau-Manneville maps are global-local mixing Discrete Contin. Dyn. Syst. A (IF 1.338) Pub Date : 2020-08-10 Claudio Bonanno; Marco Lenci
We prove that a large class of expanding maps of the unit interval with a $ C^2 $-regular indifferent fixed point in 0 and full increasing branches are global-local mixing. This class includes the standard Pomeau-Manneville maps $ T(x) = x + x^{p+1} $ mod 1 ($ p \ge 1 $), the Liverani-Saussol-Vaienti maps (with index $ p \ge 1 $) and many generalizations thereof.
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Existence uniqueness and regularity theory for elliptic equations with complex-valued potentials Discrete Contin. Dyn. Syst. A (IF 1.338) Pub Date : 2020-08-10 Tuoc Phan; Grozdena Todorova; Borislav Yordanov
This paper studies second order elliptic equations in both divergence and non-divergence forms with measurable complex valued principle coefficients and measurable complex valued potentials. The PDE operators can be considered as generalized Schrödinger operators. Under some sufficient conditions, we prove existence, uniqueness, and regularity estimates in Sobolev spaces for solutions to the equations
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Solution of the 33rd Palis-Pugh problem for gradient-like diffeomorphisms of a two-dimensional sphere Discrete Contin. Dyn. Syst. A (IF 1.338) Pub Date : 2020-08-10 Elena Nozdrinova; Olga Pochinka
In the present paper, a solution to the 33rd Palis-Pugh problem for gradient-like diffeomorphisms of a two-dimensional sphere is obtained. It is precisely shown that with respect to the stable isotopic connectedness relation there exists countable many of equivalence classes of such systems. 43 words.
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Long time behavior of the fractional Korteweg-de Vries equation with cubic nonlinearity Discrete Contin. Dyn. Syst. A (IF 1.338) Pub Date : 2020-08-10 Jean-Claude Saut; Yuexun Wang
We prove global existence and modified scattering for the solutions of the Cauchy problem to the fractional Korteweg-de Vries equation with cubic nonlinearity for small, smooth and localized initial data.
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Periodic solutions and Hyers-Ulam stability of atmospheric Ekman flows Discrete Contin. Dyn. Syst. A (IF 1.338) Pub Date : 2020-08-10 Yi Guan; Michal Fečkan; Jinrong Wang
In this paper, we study the classical problem of the wind in the steady atmospheric Ekman layer with constant eddy viscosity. Different from the well-known homogeneous system in [14,20], we retain the turbulent fluxes and establish a new nonhomogeneous system of first order differential equations involving a term with the horizontal dependent. We present the existence and uniqueness of periodic solutions
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Maximal factors of order \begin{document}$ d $\end{document} of dynamical cubespaces Discrete Contin. Dyn. Syst. A (IF 1.338) Pub Date : 2020-08-10 Jiahao Qiu; Jianjie Zhao
For a dynamical system $ (X, T) $, $ l\in\mathbb{N} $ and $ x\in X $, let $ \mathbf{Q}^{[l]}(X) $ and $ \overline{\mathcal{F}^{[l]}}(x^{[l]}) $ be the orbit closures of the diagonal point $ x^{[l]} $ under the parallelepipeds group $ \mathcal{G}^{[l]} $ and the face group $ \mathcal{F}^{[l]} $ actions respectively. In this paper, it is shown that for a minimal system $ (X, T) $ and every $ l\in \mathbb{N}
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\begin{document}$ N- $\end{document}Laplacian problems with critical double exponential nonlinearities Discrete Contin. Dyn. Syst. A (IF 1.338) Pub Date : 2020-08-10 Shengbing Deng; Tingxi Hu; Chun-Lei Tang
In this paper, we prove the existence of a nontrivial solution for the following boundary value problem $ \left\{ {\begin{array}{*{20}{l}}{ - {\rm{div}}(\omega (x)|\nabla u(x){|^{N - 2}}\nabla u(x)) = f(x,u),\;\;\quad }&{\;\;\;\;\;{\rm{in}}\;B;}\\{u = 0,\;\;\quad }&{\;\;\;\;\;{\rm{on}}\;\partial B,}\end{array}} \right.{\rm{ }}\;\;\;\;\;\;\;\left( 1 \right)$
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Recoding Lie algebraic subshifts Discrete Contin. Dyn. Syst. A (IF 1.338) Pub Date : 2020-08-10 Ville Salo; Ilkka Törmä
We study internal Lie algebras in the category of subshifts on a fixed group – or Lie algebraic subshifts for short. We show that if the acting group is virtually polycyclic and the underlying vector space has dense homoclinic points, such subshifts can be recoded to have a cellwise Lie bracket. On the other hand there exist Lie algebraic subshifts (on any finitely-generated non-torsion group) with
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Singular solutions of a Lane-Emden system Discrete Contin. Dyn. Syst. A (IF 1.338) Pub Date : 2020-08-03 Craig Cowan; Abdolrahman Razani
In this work we consider the existence of positive singular solutions $ \begin{equation} \left\{ \begin{array}{lcl} \hfill -\Delta u_1 & = & \lambda_1 | \nabla u_2|^p \qquad \mbox{ in } \Omega, \\ \hfill -\Delta u_2 & = & \lambda_2 | \nabla u_1|^q \qquad \mbox{ in } \Omega, \\ \hfill u_1 = u_2 & = & 0 \hfill \mbox{ on } \partial \Omega, \end{array}\right. \end{equation} $
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Uniform stability estimate for the Vlasov-Poisson-Boltzmann system Discrete Contin. Dyn. Syst. A (IF 1.338) Pub Date : 2020-08-03 Hao Wang
This paper is concerned with the uniform stability estimate to the Cauchy problem of the Vlasov-Poisson-Boltzmann system. Our analysis is based on compensating function introduced by Kawashima and the standard energy method.
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Asymptotic behavior of minimal solutions of \begin{document}$ -\Delta u = \lambda f(u) $\end{document} as \begin{document}$ \lambda\to-\infty $\end{document} Discrete Contin. Dyn. Syst. A (IF 1.338) Pub Date : 2020-08-03 Luca Battaglia; Francesca Gladiali; Massimo Grossi
We consider the following Dirichlet problem $\left\{ \begin{matrix} -\Delta u=\lambda f(u)\ \ \ \ \text{in}\ \Omega \\ u=0\ \ \ \ \ \ \ \ \ \ \ \ \ \text{on}\ \partial \Omega \\\end{matrix} \right.,\ \ \ \ \ \ \ \left( \mathcal{P}_{f}^{\lambda } \right)$
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Strongly localized semiclassical states for nonlinear Dirac equations Discrete Contin. Dyn. Syst. A (IF 1.338) Pub Date : 2020-08-03 Thomas Bartsch; Tian Xu
We study semiclassical states of the nonlinear Dirac equation $ -i\hbar{\partial}_t\psi = ic\hbar\sum\limits_{k = 1}^3{\alpha}_k{\partial}_k\psi - mc^2{\beta} \psi - M(x)\psi + f(|\psi|)\psi, \quad t\in \mathbb{R}, \ x\in \mathbb{R}^3, $
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Function approximation via the subsampled Poincaré inequality Discrete Contin. Dyn. Syst. A (IF 1.338) Pub Date : 2020-08-03 Yifan Chen; Thomas Y. Hou
Function approximation and recovery via some sampled data have long been studied in a wide array of applied mathematics and statistics fields. Analytic tools, such as the Poincaré inequality, have been handy for estimating the approximation errors in different scales. The purpose of this paper is to study a generalized Poincaré inequality, where the measurement function is of subsampled type, with
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Isomorphism and bi-Lipschitz equivalence between the univoque sets Discrete Contin. Dyn. Syst. A (IF 1.338) Pub Date : 2020-07-27 Kan Jiang; Lifeng Xi; Shengnan Xu; Jinjin Yang
In this paper, we consider a class of self-similar sets, denoted by $ \mathcal{A} $, and investigate the set of points in the self-similar sets having unique codings. We call such set the univoque set and denote it by $ U_1 $. We analyze the isomorphism and bi-Lipschitz equivalence between the univoque sets. The main result of this paper, in terms of the dimension of $ U_1 $, is to give several equivalent
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Weak solutions to the continuous coagulation model with collisional breakage Discrete Contin. Dyn. Syst. A (IF 1.338) Pub Date : 2020-07-27 Prasanta Kumar Barik; Ankik Kumar Giri
A global existence theorem on weak solutions is shown for the continuous coagulation equation with collisional breakage under certain classes of unbounded collision kernel and distribution functions. The model describes the dynamics of particle growth when binary collisions occur to form either a single particle via coalescence or two/more particles via breakup with possible transfer of mass. Each
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Multitransition solutions for a generalized Frenkel-Kontorova model Discrete Contin. Dyn. Syst. A (IF 1.338) Pub Date : 2020-07-27 Wen-Long Li; Xiaojun Cui
We study a generalized Frenkel-Kontorova model. Using minimal and Birkhoff solutions as building blocks, we construct a lot of homoclinic solutions and heteroclinic solutions for this generalized Frenkel-Kontorova model under gap conditions. These new solutions are not minimal and Birkhoff any more. We use constrained minimization method to prove our results.
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On the Bidomain equations driven by stochastic forces Discrete Contin. Dyn. Syst. A (IF 1.338) Pub Date : 2020-07-27 Matthias Hieber; Oleksandr Misiats; Oleksandr Stanzhytskyi
The bidomain equations driven by stochastic forces and subject to nonlinearities of FitzHugh-Nagumo or Allen-Cahn type are considered for the first time. It is shown that this set of equations admits a global weak solution as well as a stationary solution, which generates a uniquely determined invariant measure.
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Gromov-Hausdorff distances for dynamical systems Discrete Contin. Dyn. Syst. A (IF 1.338) Pub Date : 2020-07-27 Nhan-Phu Chung
We study equivariant Gromov-Hausdorff distances for general actions which are not necessarily isometric as Fukaya introduced. We prove that if an action is expansive and has the pseudo-orbit tracing property then it is stable under our adapted equivariant Gromov-Hausdorff topology. Finally, using Lott and Villani's ideas of optimal transport, we investigate equivariant Gromov-Hausdorff convergence
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Invariant manifolds and foliations for random differential equations driven by colored noise Discrete Contin. Dyn. Syst. A (IF 1.338) Pub Date : 2020-07-27 Jun Shen; Kening Lu; Bixiang Wang
In this paper, we prove the existence of local stable and unstable invariant manifolds for a class of random differential equations driven by nonlinear colored noise defined in a fractional power of a separable Banach space. In the case of linear noise, we show the pathwise convergence of these random invariant manifolds as well as invariant foliations as the correlation time of the colored noise approaches
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The focusing logarithmic Schrödinger equation: Analysis of breathers and nonlinear superposition Discrete Contin. Dyn. Syst. A (IF 1.338) Pub Date : 2020-07-27 Guillaume Ferriere
We consider the logarithmic Schrödinger equation in the focusing regime. For this equation, Gaussian initial data remains Gaussian. In particular, the Gausson - a time-independent Gaussian function - is an orbitally stable solution. In the general case in dimension $ d = 1 $, the solution with Gaussian initial data is periodic, and we compute some approximations of the period in the case of small and
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On morawetz estimates with time-dependent weights for the klein-gordon equation Discrete Contin. Dyn. Syst. A (IF 1.338) Pub Date : 2020-07-27 Jungkwon Kim; Hyeongjin Lee; Ihyeok Seo; Jihyeon Seok
We obtain some new Morawetz estimates for the Klein-Gordon flow of the form $ \begin{equation*} \big\| |\nabla|^{\sigma} e^{it \sqrt{1-\Delta}}f \big\|_{L^{2}_{x, t}(|(x, t)|^{-\alpha})} \lesssim \| f \|_{H^s} \end{equation*} $
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Global existence of strong solutions to a biological network formulation model in 2+1 dimensions Discrete Contin. Dyn. Syst. A (IF 1.338) Pub Date : 2020-07-27 Xiangsheng Xu
In this paper we study the initial boundary value problem for the system $ -\mbox{div}\left[(I+\mathbf{m} \mathbf{m}^T)\nabla p\right] = s(x), \ \ \mathbf{m}_t-\alpha^2\Delta\mathbf{m}+|\mathbf{m}|^{2(\gamma-1)}\mathbf{m} = \beta^2(\mathbf{m}\cdot\nabla p)\nabla p $ in two space dimensions. This problem has been proposed as a continuum model for biological transportation networks. The mathematical
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Matching for a family of infinite measure continued fraction transformations Discrete Contin. Dyn. Syst. A (IF 1.338) Pub Date : 2020-07-27 Charlene Kalle; Niels Langeveld; Marta Maggioni; Sara Munday
As a natural counterpart to Nakada's $ \alpha $-continued fraction maps, we study a one-parameter family of continued fraction transformations with an indifferent fixed point. We prove that matching holds for Lebesgue almost every parameter in this family and that the exceptional set has Hausdorff dimension 1. Due to this matching property, we can construct a planar version of the natural extension
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Compactness of transfer operators and spectral representation of Ruelle zeta functions for super-continuous functions Discrete Contin. Dyn. Syst. A (IF 1.338) Pub Date : 2020-07-27 Katsukuni Nakagawa
Transfer operators and Ruelle zeta functions for super-continuous functions on one-sided topological Markov shifts are considered. For every super-continuous function, we construct a Banach space on which the associated transfer operator is compact. Using this Banach space, we establish the trace formula and spectral representation of Ruelle zeta functions for a certain class of super-continuous functions
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Energy transfer model and large periodic boundary value problem for the quintic nonlinear Schrödinger equations Discrete Contin. Dyn. Syst. A (IF 1.338) Pub Date : 2020-07-27 Hideo Takaoka
We study the dynamics and energy exchanges between a linear oscillator and a nonlinear interaction state for the one dimensional, quintic nonlinear Schrödinger equation. Grébert and Thomann [9] proved that there exist solutions with initial data built on four Fourier modes, that confirm the conservative exchange of wave energy. Captured multi resonance in multiple Fourier modes, we simulate a similar
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Global existence and large time behavior for the chemotaxis–shallow water system in a bounded domain Discrete Contin. Dyn. Syst. A (IF 1.338) Pub Date : 2020-07-27 Weike Wang; Yucheng Wang
In this paper, we consider the chemotaxis–shallow water system in a bounded domain $ \Omega\subset\mathbb{R}^2 $. By energy method, we establish the global existence of strong solution with small initial perturbation and obtain the exponential decaying rate of the solution. We divide the bounded domain into interior domain and the domain up to the boundary. In the interior domain, the problem is treated
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Identifying varying magnetic anomalies using geomagnetic monitoring Discrete Contin. Dyn. Syst. A (IF 1.338) Pub Date : 2020-07-27 Youjun Deng; Hongyu Liu; Wing-Yan Tsui
We are concerned with the inverse problem of identifying magnetic anomalies with varying parameters beneath the Earth using geomagnetic monitoring. Observations of the change in Earth's magnetic field–the secular variation–provide information about the anomalies as well as their variations. In this paper, we rigorously establish the unique recovery results for this magnetic anomaly detection problem