# American Institute of Mathematical Sciences

ISSN:
1078-0947

eISSN:
1553-5231

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## Discrete & Continuous Dynamical Systems - A

May 2019 , Volume 39 , Issue 5

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2019, 39(5): 2325-2342 doi: 10.3934/dcds.2019098 +[Abstract](2238) +[HTML](167) +[PDF](412.62KB)
Abstract:

Let \begin{document}$g$\end{document} be a smooth expanding map of degree \begin{document}$D$\end{document} which maps a circle to itself, where \begin{document}$D$\end{document} is a natural number greater than \begin{document}$1$\end{document}. It is known that the Lyapunov exponent of \begin{document}$g$\end{document} with respect to the unique invariant measure that is absolutely continuous with respect to the Lebesgue measure is positive and less than or equal to \begin{document}$\log D$\end{document} which, in addition, is less than or equal to the Lyapunov exponent of \begin{document}$g$\end{document} with respect to the measure of maximal entropy. Moreover, the equalities can only occur simultaneously. We show that these are the only restrictions on the Lyapunov exponents considered above for smooth expanding maps of degree \begin{document}$D$\end{document} on a circle.

2019, 39(5): 2343-2359 doi: 10.3934/dcds.2019099 +[Abstract](1847) +[HTML](141) +[PDF](488.47KB)
Abstract:

Let \begin{document}$Y$\end{document} be a compact metric space, \begin{document}$G$\end{document} be a group acting by transformations on \begin{document}$Y$\end{document}. For any infinite subset \begin{document}$A\subset Y$\end{document}, we study the density of \begin{document}$gA$\end{document} for \begin{document}$g\in G$\end{document} and quantitative density of the set \begin{document}${\bigcup\limits_{g\in G_n}gA}$\end{document} by the Hausdorff semimetric \begin{document}$d^H$\end{document}, for a family of increasing subsets \begin{document}$G_n\subset G$\end{document}. It is proven that for any integer \begin{document}$n\ge 2$\end{document}, \begin{document}$\epsilon>0$\end{document}, any infinite subset \begin{document}$A\subset \mathbb T^n$\end{document}, there is a \begin{document}$g\in SL(n,\mathbb Z)$\end{document} such that \begin{document}$gA$\end{document} is \begin{document}$\epsilon$\end{document}-dense. We also show that, for any infinite subset \begin{document}$A\subset [0,1]$\end{document}, for a.e. rotation and a.e. 3-IET,

2019, 39(5): 2361-2392 doi: 10.3934/dcds.2019100 +[Abstract](1831) +[HTML](127) +[PDF](533.1KB)
Abstract:

We prove the uniqueness and finite-time existence of bounded-vorticity solutions to the 2D Euler equations having velocity growing slower than the square root of the distance from the origin, obtaining global existence for more slowly growing velocity fields. We also establish continuous dependence on initial data.

2019, 39(5): 2393-2412 doi: 10.3934/dcds.2019101 +[Abstract](1815) +[HTML](140) +[PDF](449.37KB)
Abstract:

We consider the dynamics of smooth covering maps of the circle with a single critical point of order greater than \begin{document}$1$\end{document}. By directly specifying the combinatorics of the critical orbit, we show that for an uncountable number of combinatorial equivalence classes of such maps, there is no periodic attractor nor an ergodic absolutely continuous invariant probability measure.

2019, 39(5): 2413-2435 doi: 10.3934/dcds.2019102 +[Abstract](1882) +[HTML](130) +[PDF](477.41KB)
Abstract:

In the article [12] it is shown that time delay induces self-excited vibrations in a one dimensional damped wave equation. Here we generalize this result for higher spatial dimensions. We prove the existence of branches of nontrivial time periodic solutions for spatial dimensions \begin{document}$d\ge 2$\end{document}. For \begin{document}$d> 2$\end{document}, the bifurcating periodic solutions have a fixed spatial frequency vector, which is the solution of a certain Diophantine equation. The case \begin{document}$d = 2$\end{document} must be treated separately from the others. In particular, it is shown that an arbitrary number of symmetry breaking orbitally distinct time periodic solutions exist, provided \begin{document}$d$\end{document} is big enough, with respect to the symmetric group action. The direction of bifurcation is also obtained.

2019, 39(5): 2437-2454 doi: 10.3934/dcds.2019103 +[Abstract](1724) +[HTML](127) +[PDF](480.31KB)
Abstract:

The Helmholtz–Hodge decomposition (HHD) is applied to the construction of Lyapunov functions. It is shown that if a stability condition is satisfied, such a decomposition can be chosen so that its potential function is a Lyapunov function. In connection with the Lyapunov function, vector fields with strictly orthogonal HHD are analyzed. It is shown that they are a generalization of gradient vector fields and have similar properties. Finally, to examine the limitations of the proposed method, planar vector fields are analyzed.

2019, 39(5): 2455-2471 doi: 10.3934/dcds.2019104 +[Abstract](2247) +[HTML](139) +[PDF](470.15KB)
Abstract:

We would like to present a general principle for the shrinking target problem in a topological dynamical system. More precisely, let \begin{document}$(X, d)$\end{document} be a compact metric space and \begin{document}$T:X\to X$\end{document} a continuous transformation on \begin{document}$X$\end{document}. For any integer valued sequence \begin{document}$\{a_n\}$\end{document} and \begin{document}$y\in X$\end{document}, define

the set of points whose orbit can well approximate a given point infinitely often, where \begin{document}$B_n(x, r)$\end{document} denotes the Bowen-ball. It is shown that

if the system \begin{document}$(X, T)$\end{document} has the specification property. Here \begin{document}$h_{\text {top}}$\end{document} denotes the topological entropy. An example is also given to indicate that the specification property required in the above result cannot be weakened even to almost specification.

2019, 39(5): 2473-2510 doi: 10.3934/dcds.2019105 +[Abstract](2046) +[HTML](144) +[PDF](607.29KB)
Abstract:

In this paper we consider a free boundary tumor model with angiogenesis. The model consists of a reaction-diffusion equation describing the concentration of nutrients \begin{document}$\sigma$\end{document} and an elliptic equation describing the distribution of the internal pressure \begin{document}$p$\end{document}. The vasculature supplies nutrients to the tumor, so that \begin{document}$\frac{\partial\sigma}{\partial \mathbf{n}}+\beta(\sigma-\bar{\sigma}) = 0$\end{document} holds on the boundary, where a positive constant \begin{document}$\beta$\end{document} is the rate of nutrient supply to the tumor and \begin{document}$\bar{\sigma}$\end{document} is the nutrient concentration outside the tumor. The tumor cells proliferate at a rate \begin{document}$\mu$\end{document}. If \begin{document}$0<\widetilde{\sigma}<\overline{\sigma}$\end{document}, where \begin{document}$\widetilde{\sigma}$\end{document} is a threshold concentration for proliferating, then there exists a unique radially symmetric stationary solution \begin{document}$(\sigma_S(r), p_S(r), R_S)$\end{document}. In this paper, we found a function \begin{document}$\mu^\ast = \mu^\ast(R_S)$\end{document} such that if \begin{document}$\mu<\mu^\ast$\end{document} then the radially symmetric stationary solution is linearly stable with respect to non-radially symmetric perturbations, whereas if \begin{document}$\mu>\mu^\ast$\end{document} then the radially symmetric stationary solution is linearly unstable.

2019, 39(5): 2511-2553 doi: 10.3934/dcds.2019106 +[Abstract](1724) +[HTML](116) +[PDF](674.99KB)
Abstract:

Dynamical systems on the interval \begin{document}$[0, 1]$\end{document}, satisfying the Thaler's condition, have been extensively studied. In this paper we consider invariant density and statistical properties of non-uniformly expanding dynamical systems on \begin{document}${\Bbb{R}}^d$\end{document} (\begin{document}$d \geq 1$\end{document}). We present a critical regular condition that is a supplement and a development of the Thaler's condition, and it is very closely related to Lamperti's criterion. Under this new condition, we offer a method for studying the dynamical systems. A continuity description of the invariant density is presented; and a convergence theorem for iterations of Perron-Frobenius operator is set up. Furthermore, we establish a more exact result for one-dimensional systems.

2019, 39(5): 2555-2579 doi: 10.3934/dcds.2019107 +[Abstract](1767) +[HTML](155) +[PDF](442.4KB)
Abstract:

In this paper we consider the non-autonomous quasilinear elliptic problem

where \begin{document}$f:\mathbb{R}\to[0,\infty)$\end{document} is a nonnegative \begin{document}$C^1-$\end{document}function with \begin{document}$f(0) = 0$\end{document}, \begin{document}$f(U) = 0$\end{document} for some \begin{document}$U>0$\end{document}, and \begin{document}$f$\end{document} is superlinear in \begin{document}$0$\end{document} and in \begin{document}$U$\end{document}. Assuming subcriticality either in \begin{document}$U$\end{document} or at infinity we study existence and multiplicity of positive radial solutions with respect to the parameter \begin{document}$\lambda$\end{document}. In addition, we study the bifurcation diagrams with respect to the maximum over the eventual solutions as the parameter \begin{document}$\lambda$\end{document} varies in the positive halfline.

2019, 39(5): 2581-2612 doi: 10.3934/dcds.2019108 +[Abstract](1529) +[HTML](121) +[PDF](704.0KB)
Abstract:

We study properties of arithmetic sets coming from multiplicative number theory and obtain applications in the theory of uniform distribution and ergodic theory. Our main theorem is a generalization of Kátai's orthogonality criterion. Here is a special case of this theorem:

Theorem. Let \begin{document}$a\colon \mathbb{N} \to \mathbb{C}$\end{document} be a bounded sequence satisfying

Then for any multiplicative function \begin{document}$f$\end{document} and any \begin{document}$z\in \mathbb{C}$\end{document} the indicator function of the level set \begin{document}$E = \{n\in \mathbb{N} :f(n) = z\}$\end{document} satisfies

With the help of this theorem one can show that if \begin{document}$E = \{n_1<n_2<\ldots\}$\end{document} is a level set of a multiplicative function having positive density, then for a large class of sufficiently smooth functions $h\colon(0, \infty)\to \mathbb{R}$ the sequence $(h(n_j))_{j\in \mathbb{N} }$ is uniformly distributed $\bmod 1$. This class of functions $h(t)$ includes: all polynomials $p(t) = a_kt^k+\ldots+a_1t+a_0$ such that at least one of the coefficients $a_1, a_2, \ldots, a_k$ is irrational, $t^c$ for any $c > 0$ with $c\notin \mathbb{N}$, $\log^r(t)$ for any $r > 2$, $\log(\Gamma(t))$, $t\log(t)$, and $\frac{t}{\log t}$. The uniform distribution results, in turn, allow us to obtain new examples of ergodic sequences, i.e. sequences along which the ergodic theorem holds.

2019, 39(5): 2613-2636 doi: 10.3934/dcds.2019109 +[Abstract](2133) +[HTML](141) +[PDF](511.57KB)
Abstract:

Existence and uniqueness of positive radial solution \begin{document}$u_p$\end{document} of the Navier boundary value problem:

where \begin{document}$B \subset \mathbb{R} ^N \; (N \geq 5)$\end{document} is the unit ball and \begin{document}$p>\frac{N+4}{N-4}$\end{document}, are obtained. Meanwhile, the asymptotic behavior as \begin{document}$p \to \infty$\end{document} of \begin{document}$u_p$\end{document} is studied. We also find the conditions such that \begin{document}$u_p$\end{document} is non-degenerate.

2019, 39(5): 2637-2659 doi: 10.3934/dcds.2019110 +[Abstract](2121) +[HTML](206) +[PDF](576.41KB)
Abstract:

In this paper, we establish the existence of a positive solution to

under certain conditions on \begin{document}$k,f$\end{document} and \begin{document}$H,$\end{document} using viscosity sub-and supersolution method. The main feature of this problem is that it has singularity as well as a superlinear growth in the gradient term. We use Hopf-Cole transformation to handle the superlinear gradient term and an approximation method combined with suitable stability result for viscosity solution to outfit the singular nonlinearity. This work extends and complements the recent works on elliptic equations involving singular as well as superlinear gradient nonlinearities.

2019, 39(5): 2661-2678 doi: 10.3934/dcds.2019111 +[Abstract](2248) +[HTML](134) +[PDF](440.88KB)
Abstract:

We study the initial value problem for the half Ginzburg-Landau-Kuramoto (hGLK) equation with the second order elliptic operator having rough coefficients and potential type perturbation. The blow-up of solutions for hGLK equation with non-positive nonlinearity is shown by an ODE argument. The key tools in the proof are appropriate commutator estimates and the essential self-adjointness of the symmetric uniformly elliptic operator with rough metric and potential type perturbation.

2019, 39(5): 2679-2708 doi: 10.3934/dcds.2019112 +[Abstract](1881) +[HTML](124) +[PDF](560.99KB)
Abstract:

We develop an operator-theoretical method for the analysis on well posedness of partial differential-difference equations that can be modeled in the form

where \begin{document}$A$\end{document} is a closed linear operator defined on a Banach space \begin{document}$X$\end{document}. Our ideas are inspired on the Poisson distribution as a tool to sampling fractional differential operators into fractional differences. Using our abstract approach, we are able to show existence and uniqueness of solutions for the problem (*) on a distinguished class of weighted Lebesgue spaces of sequences, under mild conditions on sequences of strongly continuous families of bounded operators generated by \begin{document}$A,$\end{document} and natural restrictions on the nonlinearity \begin{document}$f$\end{document}. Finally we present some original examples to illustrate our results.

2019, 39(5): 2709-2730 doi: 10.3934/dcds.2019113 +[Abstract](1836) +[HTML](116) +[PDF](472.12KB)
Abstract:

In this paper, we prove global well-posedness of smooth solutions to the two-dimensional incompressible Boussinesq equations with only a velocity damping term when the initial data is close to an nontrivial equilibrium state \begin{document}$(0, x_2)$\end{document}. As a by-product, under this equilibrium state, our result gives a positive answer to the question proposed by [1] (see P.3597).

2019, 39(5): 2731-2742 doi: 10.3934/dcds.2019114 +[Abstract](1783) +[HTML](120) +[PDF](414.69KB)
Abstract:

In this paper we consider positive supersolutions of the nonlinear elliptic equation

where \begin{document}$0\le p<1$\end{document}, \begin{document}$\Omega$\end{document} is an arbitrary domain (bounded or unbounded) in \begin{document}${\mathbb{R}}^N$\end{document} (\begin{document}$N\ge 2$\end{document}), \begin{document}$f: [0, a_{f}) \rightarrow {\mathbb{R}}_{+}$\end{document} \begin{document}$(0 < a_{f} \leq +\infty)$\end{document} is a non-decreasing continuous function and \begin{document}$\rho: \Omega \rightarrow \mathbb{R}$\end{document} is a positive function. Using the maximum principle we give explicit estimates on positive supersolutions \begin{document}$u$\end{document} at each point \begin{document}$x\in\Omega$\end{document} where \begin{document}$\nabla u\not\equiv0$\end{document} in a neighborhood of \begin{document}$x$\end{document}. As applications, we discuss the dead core set of supersolutions on bounded domains, and also obtain Liouville type results in unbounded domains \begin{document}$\Omega$\end{document} with the property that \begin{document}$\sup_{x\in\Omega}dist (x, \partial\Omega) = \infty$\end{document}. In particular when \begin{document}$\rho(x) = |x|^\beta$\end{document} (\begin{document}$\beta\in {\mathbb{R}}$\end{document}) and \begin{document}$f(u) = u^q$\end{document} with \begin{document}$q+p>1$\end{document} then every positive supersolution in an exterior domain is eventually constant if

2019, 39(5): 2743-2761 doi: 10.3934/dcds.2019115 +[Abstract](2107) +[HTML](122) +[PDF](400.88KB)
Abstract:

In this paper, we examine the notion of topological stability and its relation to the shadowing properties in zero-dimensional spaces. Several counter-examples on the topological stability and the shadowing properties are given. Also, we prove that any topologically stable (in a modified sense) homeomorphism of a Cantor space exhibits only simple typical dynamics.

2019, 39(5): 2763-2783 doi: 10.3934/dcds.2019116 +[Abstract](1855) +[HTML](119) +[PDF](501.74KB)
Abstract:

For \begin{document}$p\geq 2$\end{document}, we prove local wellposedness for the nonlinear Schrödinger equation \begin{document}$(i\partial _t + \Delta)u = \pm|u|^pu$\end{document} on \begin{document}$\mathbb{T}^3$\end{document} with initial data in \begin{document}$H^{s_c}(\mathbb{T}^3)$\end{document}, where \begin{document}$\mathbb{T}^3$\end{document} is a rectangular irrational \begin{document}$3$\end{document}-torus and \begin{document}$s_c = \frac{3}{2} - \frac{2}{p}$\end{document} is the scaling-critical regularity. This extends work of earlier authors on the local Cauchy theory for NLS on \begin{document}$\mathbb{T}^3$\end{document} with power nonlinearities where \begin{document}$p$\end{document} is an even integer.

2019, 39(5): 2785-2805 doi: 10.3934/dcds.2019117 +[Abstract](1765) +[HTML](138) +[PDF](463.44KB)
Abstract:

Recently it has been found that for a stochastic linear-quadratic optimal control problem (LQ problem, for short) in a finite horizon, open-loop solvability is strictly weaker than closed-loop solvability which is equivalent to the regular solvability of the corresponding Riccati equation. Therefore, when an LQ problem is merely open-loop solvable not closed-loop solvable, which is possible, the usual Riccati equation approach will fail to produce a state feedback representation of open-loop optimal controls. The objective of this paper is to introduce and investigate the notion of weak closed-loop optimal strategy for LQ problems so that its existence is equivalent to the open-loop solvability of the LQ problem. Moreover, there is at least one open-loop optimal control admitting a state feedback representation. Finally, we present an example to illustrate the procedure for finding weak closed-loop optimal strategies.

2019, 39(5): 2807-2875 doi: 10.3934/dcds.2019118 +[Abstract](1908) +[HTML](149) +[PDF](1427.07KB)
Abstract:

In this paper, we consider stationary solutions of the following one-dimensional Schnakenberg model with heterogeneity:

We concentrate on the case that \begin{document}$d, c, D>0$\end{document} are given constants, \begin{document}$g(x)$\end{document} is a given symmetric function, namely \begin{document}$g(x) = g(-x)$\end{document}, and \begin{document}$\varepsilon>0$\end{document} is sufficiently small and are interested in the effect of the heterogeneity \begin{document}$g(x)$\end{document} on the stability. For the case \begin{document}$g(x) = 1$\end{document} and \begin{document}$d = 0$\end{document}, Iron, Wei, and Winter (2004) studied the existence of \begin{document}$N-$\end{document}peaks symmetric stationary solutions and their stability. In this paper, first we construct symmetric one-peak stationary solutions \begin{document}$(u_{\varepsilon}, v_{\varepsilon})$\end{document} by using the contraction mapping principle. Furthermore, we give a linear stability analysis of the solutions \begin{document}$(u_{\varepsilon}, v_{\varepsilon})$\end{document} in details and reveal the effect of heterogeneity on the stability, which is a new phenomenon compared with the case \begin{document}$g(x) = 1$\end{document}.

2019, 39(5): 2877-2891 doi: 10.3934/dcds.2019119 +[Abstract](1840) +[HTML](112) +[PDF](413.94KB)
Abstract:

We consider the Cauchy problem defined for a general class of nonlocal wave equations modeling bidirectional wave propagation in a nonlocally and nonlinearly elastic medium whose constitutive equation is given by a convolution integral. We prove a long-time existence result for the nonlocal wave equations with a power-type nonlinearity and a small parameter. As the energy estimates involve a loss of derivatives, we follow the Nash-Moser approach proposed by Alvarez-Samaniego and Lannes. As an application to the long-time existence theorem, we consider the limiting case in which the kernel function is the Dirac measure and the nonlocal equation reduces to the governing equation of one-dimensional classical elasticity theory. The present study also extends our earlier result concerning local well-posedness for smooth kernels to nonsmooth kernels.

2019, 39(5): 2893-2913 doi: 10.3934/dcds.2019120 +[Abstract](2175) +[HTML](113) +[PDF](514.35KB)
Abstract:

We prove convergence of a variational formulation of the BDF2 method applied to the non-linear Fokker-Planck equation. Our approach is inspired by the JKO-method and exploits the differential structure of the underlying \begin{document}$L^2$\end{document}-Wasserstein space. The technique presented here extends and strengthens the results of our own recent work [27] on the BDF2 method for general metric gradient flows in the special case of the non-linear Fokker-Planck equation: firstly, we do not require uniform semi-convexity of the augmented energy functional; secondly, we prove strong instead of merely weak convergence of the time-discrete approximations; thirdly, we directly prove without using the abstract theory of curves of maximal slope that the obtained limit curve is a weak solution of the non-linear Fokker-Planck equation.

2019, 39(5): 2915-2931 doi: 10.3934/dcds.2019121 +[Abstract](1824) +[HTML](157) +[PDF](667.62KB)
Abstract:

In this paper, we reconsider the continuity of the Lyapunov exponents for a class of Schrödinger cocycles with a \begin{document}$C^2$\end{document} cos-type potential and a Diophantine frequency. We propose a new method to prove the Lyapunov exponent is continuous in energies. In particular, a large deviation theorem is not needed in the proof.

2019, 39(5): 2933-2960 doi: 10.3934/dcds.2019122 +[Abstract](1796) +[HTML](149) +[PDF](488.48KB)
Abstract:

In this work we introduce a generalized linear model regulating the spread of population displayed in a \begin{document}$d$\end{document}-dimensional spatial region \begin{document}$\Omega$\end{document} of \begin{document}$\mathbb{R}^{d}$\end{document} constituted by two juxtaposed habitats having a common interface \begin{document}$\Gamma$\end{document}. This model is described by an operator \begin{document}$\mathcal{L}$\end{document} of fourth order combining the Laplace and Biharmonic operators under some natural boundary and transmission conditions. We then invert explicitly this operator in \begin{document}$L^{p}$\end{document}-spaces using the \begin{document}$H^{\infty }$\end{document}-calculus and the Dore-Venni sums theory. This main result will lead us in a later work to study the nature of the semigroup generated by \begin{document}$\mathcal{L}$\end{document} which is important for the study of the complete nonlinear generalized diffusion equation associated to it.

2019, 39(5): 2961-2976 doi: 10.3934/dcds.2019123 +[Abstract](2001) +[HTML](121) +[PDF](495.38KB)
Abstract:

The paper reconsiders the issue of the regularity of the Duhamel part of the solution to the \begin{document}$L^2$\end{document}-critical high-order NLS already studied by the authors in [4]. This model includes the mass critical 4D fourth order NLS. The improvement is due to the use of a more sophisticated space involving a nonlinear term and taking profit of a suitable trilinear Strichartz estimate.

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