# American Institute of Mathematical Sciences

ISSN:
1078-0947

eISSN:
1553-5231

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## Discrete and Continuous Dynamical Systems

March 2022 , Volume 42 , Issue 3

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2022, 42(3): 1039-1065 doi: 10.3934/dcds.2021146 +[Abstract](581) +[HTML](179) +[PDF](470.99KB)
Abstract:

It is established existence of solutions for subcritical and critical nonlinearities considering a fourth-order elliptic problem defined in the whole space \begin{document}$\mathbb{R}^N$\end{document}. The work is devoted to study a class of potentials and nonlinearities which can be periodic or asymptotically periodic. Here we consider a general fourth-order elliptic problem where the principal part is given by \begin{document}$\alpha \Delta^2 u + \beta \Delta u + V(x)u$\end{document} where \begin{document}$\alpha > 0, \beta \in \mathbb{R}$\end{document} and \begin{document}$V: \mathbb{R}^N \rightarrow \mathbb{R}$\end{document} is a continuous potential. Hence our main contribution is to consider general fourth-order elliptic problems taking into account the cases where \begin{document}$\beta$\end{document} is negative, zero or positive. In order to do that we employ some fine estimates proving the compactness for the associated energy functional.

2022, 42(3): 1067-1103 doi: 10.3934/dcds.2021147 +[Abstract](684) +[HTML](187) +[PDF](570.01KB)
Abstract:

The Zakharov system in dimension \begin{document}$d\leqslant 3$\end{document} is shown to be locally well-posed in Sobolev spaces \begin{document}$H^s \times H^l$\end{document}, extending the previously known result. We construct new solution spaces by modifying the \begin{document}$X^{s,b}$\end{document} spaces, specifically by introducing temporal weights. We use the contraction mapping principle to prove local well-posedness in the same. The result obtained is sharp up to endpoints.

2022, 42(3): 1105-1126 doi: 10.3934/dcds.2021148 +[Abstract](495) +[HTML](178) +[PDF](502.68KB)
Abstract:

A zero-dimensional (resp. symbolic) flow is a suspension flow over a zero-dimensional system (resp. a subshift). We show that any topological flow admits a principal extension by a zero-dimensional flow. Following [6] we deduce that any topological flow admits an extension by a symbolic flow if and only if its time-\begin{document}$t$\end{document} map admits an extension by a subshift for any \begin{document}$t\neq 0$\end{document}. Moreover the existence of such an extension is preserved under orbit equivalence for regular topological flows, but this property does not hold for singular flows. Finally we investigate symbolic extensions for singular suspension flows. In particular, the suspension flow over the full shift on \begin{document}$\{0,1\}^{\mathbb Z}$\end{document} with a roof function \begin{document}$f$\end{document} vanishing at the zero sequence \begin{document}$0^\infty$\end{document} admits a principal symbolic extension or not depending on the smoothness of \begin{document}$f$\end{document} at \begin{document}$0^\infty$\end{document}.

2022, 42(3): 1127-1162 doi: 10.3934/dcds.2021149 +[Abstract](779) +[HTML](244) +[PDF](485.18KB)
Abstract:

We study a class of free boundary systems with nonlocal diffusion, which are natural extensions of the corresponding free boundary problems of reaction diffusion systems. As before the free boundary represents the spreading front of the species, but here the population dispersal is described by "nonlocal diffusion" instead of "local diffusion". We prove that such a nonlocal diffusion problem with free boundary has a unique global solution, and for models with Lotka-Volterra type competition or predator-prey growth terms, we show that a spreading-vanishing dichotomy holds, and obtain criteria for spreading and vanishing; moreover, for the weak competition case and for the weak predation case, we can determine the long-time asymptotic limit of the solution when spreading happens. Compared with the single species free boundary model with nonlocal diffusion considered recently in [7], and the two species cases with local diffusion extensively studied in the literature, the situation considered in this paper involves several new difficulties, which are overcome by the use of some new techniques.

2022, 42(3): 1163-1184 doi: 10.3934/dcds.2021150 +[Abstract](426) +[HTML](145) +[PDF](449.04KB)
Abstract:

A minimal system \begin{document}$(X,T)$\end{document} is topologically mildly mixing if for all non-empty open subsets \begin{document}$U,V$\end{document}, \begin{document}$\{n\in {\mathbb Z}: U\cap T^{-n}V\neq \emptyset\}$\end{document} is an IP\begin{document}$^*$\end{document}-set. In this paper we show that if a minimal system is topologically mildly mixing, then it is mild mixing of all orders along polynomials. That is, suppose that \begin{document}$(X,T)$\end{document} is a topologically mildly mixing minimal system, \begin{document}$d\in {\mathbb N}$\end{document}, \begin{document}$p_1(n),\ldots, p_d(n)$\end{document} are integral polynomials with no \begin{document}$p_i$\end{document} and no \begin{document}$p_i-p_j$\end{document} constant, \begin{document}$1\le i\neq j\le d$\end{document}. Then for all non-empty open subsets \begin{document}$U , V_1, \ldots, V_d$\end{document}, \begin{document}$\{n\in {\mathbb Z}: U\cap T^{-p_1(n) }V_1\cap T^{-p_2(n)}V_2\cap \ldots \cap T^{-p_d(n) }V_d \neq \emptyset \}$\end{document} is an IP\begin{document}$^*$\end{document}-set. We also give the corresponding theorem for systems under abelian group actions.

2022, 42(3): 1185-1200 doi: 10.3934/dcds.2021151 +[Abstract](477) +[HTML](139) +[PDF](1188.63KB)
Abstract:

For the evolution Navier-Stokes equations in bounded 3D domains, it is well-known that the uniqueness of a solution is related to the existence of a regular solution. They may be obtained under suitable assumptions on the data and smoothness assumptions on the domain (at least \begin{document}$C^{2,1}$\end{document}). With a symmetrization technique, we prove these results in the case of Navier boundary conditions in a wide class of merely Lipschitz domains of physical interest, that we call sectors.

2022, 42(3): 1201-1223 doi: 10.3934/dcds.2021152 +[Abstract](532) +[HTML](139) +[PDF](395.8KB)
Abstract:

In this paper, we establish global \begin{document}$C^2$\end{document} a priori estimates for solutions to the uniformly parabolic equations with Neumann boundary condition on the smooth bounded domain in \begin{document}$\mathbb R^n$\end{document} by a blow-up argument. As a corollary, we obtain that the solutions converge to ones which move by translation. This generalizes the viscosity results derived before by Da Lio.

2022, 42(3): 1225-1241 doi: 10.3934/dcds.2021153 +[Abstract](484) +[HTML](151) +[PDF](390.69KB)
Abstract:

In this article, we consider a counting problem for orbits of hyperbolic rational maps on the Riemann sphere, where constraints are placed on the multipliers of orbits. Using arguments from work of Dolgopyat, we consider varying and potentially shrinking intervals, and obtain a result which resembles a local central limit theorem for the logarithm of the absolute value of the multiplier and an equidistribution theorem for the holonomies.

2022, 42(3): 1243-1316 doi: 10.3934/dcds.2021154 +[Abstract](458) +[HTML](186) +[PDF](807.43KB)
Abstract:

This is the first part of a series devoting to the generalizations and applications of common theorems in variational bifurcation theory. Using parameterized versions of splitting theorems in Morse theory we generalize some famous bifurcation theorems for potential operators by weakening standard assumptions on the differentiability of the involved functionals, which opens up a way of bifurcation studies for quasi-linear elliptic boundary value problems.

2022, 42(3): 1317-1368 doi: 10.3934/dcds.2021155 +[Abstract](418) +[HTML](147) +[PDF](684.27KB)
Abstract:

This is the second part of a series devoting to the generalizations and applications of common theorems in variational bifurcation theory. Using abstract theorems in the first part we obtain many new bifurcation results for quasi-linear elliptic boundary value problems of higher order.

2022, 42(3): 1369-1401 doi: 10.3934/dcds.2021156 +[Abstract](554) +[HTML](172) +[PDF](845.27KB)
Abstract:

We prove the existence of a bounded positive solution of the following elliptic system involving Schrödinger operators

where \begin{document}$p,q,r,s\geq0$\end{document}, \begin{document}$V_{i}$\end{document} is a nonnegative vanishing potential, and \begin{document}$\rho_{i}$\end{document} has the property \begin{document}$(\mathrm{H})$\end{document} introduced by Brezis and Kamin [4].As in that celebrated work we will prove that for every \begin{document}$R> 0$\end{document} there is a solution \begin{document}$(u_R, v_R)$\end{document} defined on the ball of radius \begin{document}$R$\end{document} centered at the origin. Then, we will show that this sequence of solutions tends to a bounded solution of the previous system when \begin{document}$R$\end{document} tends to infinity. Furthermore, by imposing some restrictions on the powers \begin{document}$p,q,r,s$\end{document} without additional hypotheses on the weights \begin{document}$\rho_{i}$\end{document}, we obtain a second solution using variational methods. In this context we consider two particular cases: a gradient system and a Hamiltonian system.

2022, 42(3): 1403-1414 doi: 10.3934/dcds.2021157 +[Abstract](390) +[HTML](140) +[PDF](329.58KB)
Abstract:

Two Delone sets are bounded distance equivalent to each other if there is a bijection between them such that the distance of corresponding points is uniformly bounded. Bounded distance equivalence is an equivalence relation. We show that the hull of a repetitive Delone set with finite local complexity has either one equivalence class or uncountably many.

2022, 42(3): 1415-1434 doi: 10.3934/dcds.2021158 +[Abstract](438) +[HTML](145) +[PDF](390.58KB)
Abstract:

In this article we consider the following weighted nonlinear eigenvalue problem for the \begin{document}$g-$\end{document}Laplacian

with Dirichlet boundary conditions. Here \begin{document}$w$\end{document} is a suitable weight and \begin{document}$g = G'$\end{document} and \begin{document}$h = H'$\end{document} are appropriated Young functions satisfying the so called \begin{document}$\Delta'$\end{document} condition, which includes for instance logarithmic perturbation of powers and different power behaviors near zero and infinity. We prove several properties on its spectrum, being our main goal to obtain lower bounds of eigenvalues in terms of \begin{document}$G$\end{document}, \begin{document}$H$\end{document}, \begin{document}$w$\end{document} and the normalization \begin{document}$\mu$\end{document} of the corresponding eigenfunctions.

We introduce some new strategies to obtain results that generalize several inequalities from the literature of \begin{document}$p-$\end{document}Laplacian type eigenvalues.

2022, 42(3): 1435-1463 doi: 10.3934/dcds.2021159 +[Abstract](514) +[HTML](164) +[PDF](527.93KB)
Abstract:

In the paper we study what sets can be obtained as \begin{document}$\alpha$\end{document}-limit sets of backward trajectories in graph maps. We show that in the case of mixing maps, all those \begin{document}$\alpha$\end{document}-limit sets are \begin{document}$\omega$\end{document}-limit sets and for all but finitely many points \begin{document}$x$\end{document}, we can obtain every \begin{document}$\omega$\end{document}-limits set as the \begin{document}$\alpha$\end{document}-limit set of a backward trajectory starting in \begin{document}$x$\end{document}. For zero entropy maps, every \begin{document}$\alpha$\end{document}-limit set of a backward trajectory is a minimal set. In the case of maps with positive entropy, we obtain a partial characterization which is very close to complete picture of the possible situations.

2022, 42(3): 1465-1491 doi: 10.3934/dcds.2021160 +[Abstract](557) +[HTML](149) +[PDF](661.43KB)
Abstract:

We prove existence and uniqueness of eternal solutions in self-similar form growing up in time with exponential rate for the weighted reaction-diffusion equation

posed in \begin{document}$\mathbb{R}^N$\end{document}, with \begin{document}$m>1$\end{document}, \begin{document}$0<p<1$\end{document} and the critical value for the weight

Existence and uniqueness of some specific solution holds true when \begin{document}$m+p\geq2$\end{document}. On the contrary, no eternal solution exists if \begin{document}$m+p<2$\end{document}. We also classify exponential self-similar solutions with a different interface behavior when \begin{document}$m+p>2$\end{document}. Some transformations to reaction-convection-diffusion equations and traveling wave solutions are also introduced.

2022, 42(3): 1493-1494 doi: 10.3934/dcds.2021161 +[Abstract](482) +[HTML](149) +[PDF](167.8KB)
Abstract:

This paper is an Erratum on "Stability and dynamics of a weak viscoelastic system with memory and nonlinear time-varying delay" (Discrete Continuous Dynamic Systems, 40(3), 2020, 1493-1515).

2022, 42(3): 1495-1533 doi: 10.3934/dcds.2021162 +[Abstract](426) +[HTML](142) +[PDF](508.13KB)
Abstract:

In this paper, we first prove the strong Birkhoff Ergodic Theorem for subharmonic functions with the irrational shift on the Torus. Then, we apply it to the analytic quasi-periodic Jacobi cocycles and show that for suitable frequency and coupling number, if the Lyapunov exponent of these cocycles is positive at one point, then it is positive on an interval centered at this point and Hölder continuous in \begin{document}$E$\end{document} on this interval. What's more, if the coupling number of the potential is large, then the Lyapunov exponent is always positive for all irrational frequencies and Hölder continuous in \begin{document}$E$\end{document} for all finite Liouville frequencies. For the Schrödinger cocycles, a special case of the Jacobi ones, its Lyapunov exponent is also Hölder continuous in the frequency and the lengths of the intervals where the Hölder condition of the Lyapunov exponent holds only depend on the coupling number.

2022, 42(3): 1535-1568 doi: 10.3934/dcds.2021163 +[Abstract](440) +[HTML](136) +[PDF](428.76KB)
Abstract:

In this paper, we are concerned with the long-time asymptotic behavior of the two-dimensional temperature-dependent tropical climate model. More precisely, we obtain the sharp time-decay of the solution of the system with the general initial data belonging to an appropriate Sobolev space with negative indices. In addition, when such condition of the initial data is absent, it is shown that any spatial derivative of the positive integer \begin{document}$k$\end{document}-order of the solution actually decays at least at the rate of \begin{document}$(1+t)^{-\frac{k}{2}}$\end{document}.

2022, 42(3): 1569-1583 doi: 10.3934/dcds.2021164 +[Abstract](378) +[HTML](119) +[PDF](345.88KB)
Abstract:

For area-preserving twist maps on the annulus, we consider the problem on quantitative destruction of invariant circles with a given frequency \begin{document}$\omega$\end{document} of an integrable system by a trigonometric polynomial of degree \begin{document}$N$\end{document} perturbation \begin{document}$R_N$\end{document} with \begin{document}$\|R_N\|_{C^r}<\epsilon$\end{document}. We obtain a relation among \begin{document}$N$\end{document}, \begin{document}$r$\end{document}, \begin{document}$\epsilon$\end{document} and the arithmetic property of \begin{document}$\omega$\end{document}, for which the area-preserving map admit no invariant circles with \begin{document}$\omega$\end{document}.

2022, 42(3): 1585-1598 doi: 10.3934/dcds.2021165 +[Abstract](472) +[HTML](143) +[PDF](344.15KB)
Abstract:

In this paper we study several dynamical properties on uniform spaces. We define expansive flows on uniform spaces and provide some equivalent ways of defining expansivity. We also define the concept of expansive measures for flows on uniform spaces. We prove for flows on compact uniform spaces that every expansive measure vanishes along the orbits and has no singularities in the support. We also prove that every expansive measure for flows on uniform spaces is aperiodic and is expansive with respect to time-\begin{document}$T$\end{document} map. Furthermore we show that every expansive measure for flows on compact uniform spaces maintains expansive under topological equivalence.

2020 Impact Factor: 1.392
5 Year Impact Factor: 1.610
2020 CiteScore: 2.2