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Discrete & Continuous Dynamical Systems

March 2021 , Volume 41 , Issue 3

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Properties of the minimizers for a constrained minimization problem arising in Kirchhoff equation
Helin Guo and Huan-Song Zhou
2021, 41(3): 1023-1050 doi: 10.3934/dcds.2020308 +[Abstract](617) +[HTML](226) +[PDF](488.47KB)
Abstract:

Let \begin{document}$ a>0,b>0 $\end{document} and \begin{document}$ V(x)\geq0 $\end{document} be a coercive function in \begin{document}$ \mathbb R^2 $\end{document}. We study the following constrained minimization problem on a suitable weighted Sobolev space \begin{document}$ \mathcal{H} $\end{document}:

where \begin{document}$ E_{a}^{b}(u) $\end{document} is a Kirchhoff type energy functional defined on \begin{document}$ \mathcal{H} $\end{document} by

It is known that, for some \begin{document}$ a^{\ast}>0 $\end{document}, \begin{document}$ e_{a}(b) $\end{document} has no minimizer if \begin{document}$ b = 0 $\end{document} and \begin{document}$ a\geq a^{\ast} $\end{document}, but \begin{document}$ e_{a}(b) $\end{document} has always a minimizer for any \begin{document}$ a\geq0 $\end{document} if \begin{document}$ b>0 $\end{document}. The aim of this paper is to investigate the limit behaviors of the minimizers of \begin{document}$ e_{a}(b) $\end{document} as \begin{document}$ b\rightarrow0^{+} $\end{document}. Moreover, the uniqueness of the minimizers of \begin{document}$ e_{a}(b) $\end{document} is also discussed for \begin{document}$ b $\end{document} close to 0.

Pomeau-Manneville maps are global-local mixing
Claudio Bonanno and Marco Lenci
2021, 41(3): 1051-1069 doi: 10.3934/dcds.2020309 +[Abstract](665) +[HTML](247) +[PDF](397.64KB)
Abstract:

We prove that a large class of expanding maps of the unit interval with a \begin{document}$ C^2 $\end{document}-regular indifferent fixed point in 0 and full increasing branches are global-local mixing. This class includes the standard Pomeau-Manneville maps \begin{document}$ T(x) = x + x^{p+1} $\end{document} mod 1 (\begin{document}$ p \ge 1 $\end{document}), the Liverani-Saussol-Vaienti maps (with index \begin{document}$ p \ge 1 $\end{document}) and many generalizations thereof.

Existence uniqueness and regularity theory for elliptic equations with complex-valued potentials
Tuoc Phan, Grozdena Todorova and Borislav Yordanov
2021, 41(3): 1071-1099 doi: 10.3934/dcds.2020310 +[Abstract](736) +[HTML](301) +[PDF](449.94KB)
Abstract:

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. We particularly show that the non-zero imaginary parts of the potentials are the main mechanisms that control the solutions. Our results can be considered as limiting absorption principle for Schrödinger operators with measurable coefficients and they could be useful in applications. The approach is based on the perturbation technique that freezes the potentials. The results of the paper not only generalize known results but also provide a key ingredient for the study of \begin{document}$ L^p $\end{document}-diffusion phenomena for dissipative wave equations.

Solution of the 33rd Palis-Pugh problem for gradient-like diffeomorphisms of a two-dimensional sphere
Elena Nozdrinova and Olga Pochinka
2021, 41(3): 1101-1131 doi: 10.3934/dcds.2020311 +[Abstract](644) +[HTML](292) +[PDF](24094.18KB)
Abstract:

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.

Long time behavior of the fractional Korteweg-de Vries equation with cubic nonlinearity
Jean-Claude Saut and Yuexun Wang
2021, 41(3): 1133-1155 doi: 10.3934/dcds.2020312 +[Abstract](645) +[HTML](244) +[PDF](381.97KB)
Abstract:

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.

Periodic solutions and Hyers-Ulam stability of atmospheric Ekman flows
Yi Guan, Michal Fečkan and Jinrong Wang
2021, 41(3): 1157-1176 doi: 10.3934/dcds.2020313 +[Abstract](597) +[HTML](234) +[PDF](409.32KB)
Abstract:

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 and show the Hyers-Ulam stability results for the nonhomogeneous systems under the mild conditions via the matrix theory. Further, we consider the nonhomogeneous systems with varying eddy viscosity coefficient and study systems with piecewise constants, systems with small oscillations, systems with rapidly varying coefficients and systems with slowly varying coefficients and give more continued results.

Multiple ergodic averages for tempered functions
Andreas Koutsogiannis
2021, 41(3): 1177-1205 doi: 10.3934/dcds.2020314 +[Abstract](497) +[HTML](244) +[PDF](508.9KB)
Abstract:

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 \begin{document}$ L^2 $\end{document}-norm limit of the aforementioned averages, which turns out, as in the Hardy field case, to be the "expected" one. The main ingredients are the use of, the now classical, PET induction (introduced by Bergelson), covering a more general case, namely a "nice" class of tempered functions (developed by Chu-Frantzikinakis-Host for polynomials and Frantzikinakis for Hardy field functions) and some equidistribution results on nilmanifolds (analogous to the ones of Frantzikinakis' for the Hardy field case).

Asymptotic stability in a chemotaxis-competition system with indirect signal production
Pan Zheng
2021, 41(3): 1207-1223 doi: 10.3934/dcds.2020315 +[Abstract](527) +[HTML](221) +[PDF](408.59KB)
Abstract:

This paper deals with a fully parabolic inter-species chemotaxis-competition system with indirect signal production

under zero Neumann boundary conditions in a smooth bounded domain \begin{document}$ \Omega\subset \mathbb{R}^{N} $\end{document} (\begin{document}$ N\geq 1 $\end{document}), where \begin{document}$ d_{u}>0, d_{v}>0 $\end{document} and \begin{document}$ d_{w}>0 $\end{document} are the diffusion coefficients, \begin{document}$ \chi\in \mathbb{R} $\end{document} is the chemotactic coefficient, \begin{document}$ \mu_{1}>0 $\end{document} and \begin{document}$ \mu_{2}>0 $\end{document} are the population growth rates, \begin{document}$ a_{1}>0, a_{2}>0 $\end{document} denote the strength coefficients of competition, and \begin{document}$ \lambda $\end{document} and \begin{document}$ \alpha $\end{document} describe the rates of signal degradation and production, respectively. Global boundedness of solutions to the above system with \begin{document}$ \chi>0 $\end{document} was established by Tello and Wrzosek in [J. Math. Anal. Appl. 459 (2018) 1233-1250]. The main purpose of the paper is further to give the long-time asymptotic behavior of global bounded solutions, which could not be derived in the previous work.

On asymptotic stability of ground states of some systems of nonlinear Schrödinger equations
Andrew Comech and Scipio Cuccagna
2021, 41(3): 1225-1270 doi: 10.3934/dcds.2020316 +[Abstract](574) +[HTML](213) +[PDF](627.46KB)
Abstract:

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.

On entropy of $ \Phi $-irregular and $ \Phi $-level sets in maps with the shadowing property
Magdalena Foryś-Krawiec, Jiří Kupka, Piotr Oprocha and Xueting Tian
2021, 41(3): 1271-1296 doi: 10.3934/dcds.2020317 +[Abstract](608) +[HTML](215) +[PDF](468.9KB)
Abstract:

We study the properties of \begin{document}$ \Phi $\end{document}-irregular sets (sets of points for which the Birkhoff average diverges) in dynamical systems with the shadowing property. We estimate the topological entropy of \begin{document}$ \Phi $\end{document}-irregular set in terms of entropy on chain recurrent classes and prove that \begin{document}$ \Phi $\end{document}-irregular sets of full entropy are typical. We also consider \begin{document}$ \Phi $\end{document}-level sets (sets of points whose Birkhoff average is in a specified interval), relating entropy they carry with the entropy of some ergodic measures. Finally, we study the problem of large deviations considering the level sets with respect to reference measures.

Blow-up and bounded solutions for a semilinear parabolic problem in a saturable medium
Juliana Fernandes and Liliane Maia
2021, 41(3): 1297-1318 doi: 10.3934/dcds.2020318 +[Abstract](557) +[HTML](218) +[PDF](411.25KB)
Abstract:

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 tools are the comparison principle and variational methods. In particular, the Nehari manifold is used to separate the phase space into regions of initial data where uniform boundedness or grow-up behavior of the semiflow may occur. Additionally, some attention is paid to initial data at high energy level.

Entropy production in random billiards
Timothy Chumley and Renato Feres
2021, 41(3): 1319-1346 doi: 10.3934/dcds.2020319 +[Abstract](488) +[HTML](211) +[PDF](907.57KB)
Abstract:

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 the boundary of the domain. We derive a formula for entropy production rate that applies to a general class of random billiard systems. This formula establishes a relation between the purely mathematical concept of entropy production rate and textbook thermodynamic entropy, recovering in particular Clausius' formulation of the second law of thermodynamics. We also study an explicit class of examples whose reflection operator, referred to as the Maxwell-Smoluchowski thermostat, models systems with boundary thermostats kept at possibly different temperatures. We prove that, under certain mild regularity conditions, the class of models are uniformly ergodic Markov chains and derive formulas for the stationary distribution and entropy production rate in terms of geometric and thermodynamic parameters.

Gromov-Hausdorff stability for group actions
Meihua Dong, Keonhee Lee and Carlos Morales
2021, 41(3): 1347-1357 doi: 10.3934/dcds.2020320 +[Abstract](514) +[HTML](196) +[PDF](282.35KB)
Abstract:

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 isometric conjugacy.

Sharp regularity for degenerate obstacle type problems: A geometric approach
João Vitor da Silva and Hernán Vivas
2021, 41(3): 1359-1385 doi: 10.3934/dcds.2020321 +[Abstract](469) +[HTML](189) +[PDF](515.65KB)
Abstract:

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

with \begin{document}$ \gamma>0 $\end{document}, \begin{document}$ \phi \in C^{1, \alpha}(B_1) $\end{document} for some \begin{document}$ \alpha\in(0,1] $\end{document}, a continuous boundary datum \begin{document}$ g $\end{document} and \begin{document}$ f\in L^\infty(B_1)\cap C^0(B_1) $\end{document} and prove that they are \begin{document}$ C^{1,\beta}(B_{1/2}) $\end{document} (and in particular at free boundary points) where \begin{document}$ \beta = \min\left\{\alpha, \frac{1}{\gamma+1}\right\} $\end{document}. Moreover, we achieve such a feature by using a recently developed geometric approach which is a novelty for these types of free boundary problems. Furthermore, under a natural non-degeneracy assumption on the obstacle, we prove that the free boundary \begin{document}$ \partial\{u>\phi\} $\end{document} has Hausdorff dimension less than \begin{document}$ n $\end{document} (and in particular zero Lebesgue measure). Our results are new even for degenerate problems such as

Global large solutions and optimal time-decay estimates to the Korteweg system
Xiaoping Zhai and Yongsheng Li
2021, 41(3): 1387-1413 doi: 10.3934/dcds.2020322 +[Abstract](537) +[HTML](207) +[PDF](448.24KB)
Abstract:

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 pure energy argument (independent of spectral analysis). In contrast to the compressible Navier-Stokes system, the time-decay estimates are more accurate owing to smoothing effect from the Korteweg tensor.

A sharp scattering threshold level for mass-subcritical nonlinear Schrödinger system
Masaru Hamano and Satoshi Masaki
2021, 41(3): 1415-1447 doi: 10.3934/dcds.2020323 +[Abstract](554) +[HTML](236) +[PDF](509.61KB)
Abstract:

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 solution, the zero solution. By the structure of the nonlinearity we are dealing with, the system admits a scattering solution which is a pair of the zero function and a linear Schrödinger flow. Taking this fact into account, we introduce a new optimizing quantity and give a sharp scattering criterion in terms of it.

Mean-square random invariant manifolds for stochastic differential equations
Bixiang Wang
2021, 41(3): 1449-1468 doi: 10.3934/dcds.2020324 +[Abstract](564) +[HTML](191) +[PDF](378.36KB)
Abstract:

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 involving the conditional expectation with respect to a filtration. The existence of mean-square random stable invariant sets is also established but the existence of mean-square random stable invariant manifolds remains open.

Global graph of metric entropy on expanding Blaschke products
Yunping Jiang
2021, 41(3): 1469-1482 doi: 10.3934/dcds.2020325 +[Abstract](491) +[HTML](208) +[PDF](343.97KB)
Abstract:

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 map on the real line. Thus we can give a new explanation of Boole's formula discovered more than one hundred and fifty years ago. We modify the first smooth path to get a second smooth path in the space of expanding Blaschke products. The second smooth path tends to a totally degenerate map. We see that the first and second smooth paths have completely different asymptotic behaviors near the boundary of the space of expanding Blaschke products. However, they represent the same smooth path in the space of all smooth conjugacy classes of expanding Blaschke products. We use this to give a complete description of the global graph of the metric entropy on the space of expanding Blaschke products. We prove that the global graph looks like a bell. It is the first result to show a global picture of the metric entropy on a space of hyperbolic dynamical systems. We apply our results to the measure-theoretic entropy of a quadratic polynomial with respect to its Gibbs measure on its Julia set. We prove that the measure-theoretic entropy on the main cardioid of the Mandelbrot set is a real analytic function and asymptotically zero near the boundary.

Existence of nodal solutions for the sublinear Moore-Nehari differential equation
Ryuji Kajikiya
2021, 41(3): 1483-1506 doi: 10.3934/dcds.2020326 +[Abstract](469) +[HTML](202) +[PDF](398.79KB)
Abstract:

We study the existence of symmetric and asymmetric nodal solutions for the sublinear Moore-Nehari differential equation, \begin{document}$ u''+h(x, \lambda)|u|^{p-1}u = 0 $\end{document} in \begin{document}$ (-1, 1) $\end{document} with \begin{document}$ u(-1) = u(1) = 0 $\end{document}, where \begin{document}$ 0<p<1 $\end{document}, \begin{document}$ h(x, \lambda) = 0 $\end{document} for \begin{document}$ |x|<\lambda $\end{document}, \begin{document}$ h(x, \lambda) = 1 $\end{document} for \begin{document}$ \lambda\leq |x|\leq 1 $\end{document} and \begin{document}$ \lambda\in (0, 1) $\end{document} is a parameter. We call a solution \begin{document}$ u $\end{document} symmetric if it is even or odd. For an integer \begin{document}$ n\geq 0 $\end{document}, we call a solution \begin{document}$ u $\end{document} an \begin{document}$ n $\end{document}-nodal solution if it has exactly \begin{document}$ n $\end{document} zeros in \begin{document}$ (-1, 1) $\end{document}. For each integer \begin{document}$ n\geq 0 $\end{document} and any \begin{document}$ \lambda\in (0, 1) $\end{document}, we prove that the equation has a unique \begin{document}$ n $\end{document}-nodal symmetric solution with \begin{document}$ u'(-1)>0 $\end{document}. For integers \begin{document}$ m, n \geq 0 $\end{document}, we call a solution \begin{document}$ u $\end{document} an \begin{document}$ (m, n) $\end{document}-solution if it has exactly \begin{document}$ m $\end{document} zeros in \begin{document}$ (-1, 0) $\end{document} and exactly \begin{document}$ n $\end{document} zeros in \begin{document}$ (0, 1) $\end{document}. We show the existence of an \begin{document}$ (m, n) $\end{document}-solution for each \begin{document}$ m, n $\end{document} and prove that any \begin{document}$ (m, m) $\end{document}-solution is symmetric.

Failure of scattering to solitary waves for long-range nonlinear Schrödinger equations
Jason Murphy and Kenji Nakanishi
2021, 41(3): 1507-1517 doi: 10.3934/dcds.2020328 +[Abstract](506) +[HTML](185) +[PDF](378.94KB)
Abstract:

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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