# American Institute of Mathematical Sciences

ISSN:
1078-0947

eISSN:
1553-5231

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## Discrete & Continuous Dynamical Systems - A

February 2019 , Volume 39 , Issue 2

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2019, 39(2): 667-682 doi: 10.3934/dcds.2019027 +[Abstract](3200) +[HTML](170) +[PDF](672.91KB)
Abstract:

Let \begin{document}$X$\end{document} be the full shift on two symbols. The lexicographic order induces a partial order known as first-order stochastic dominance on the collection \begin{document}${\mathcal{M}}_{X}$\end{document} of its shift-invariant probability measures. We present a study of the fine structure of this dominance order, denoted by \begin{document}$\prec$\end{document}, and give criteria for establishing comparability or incomparability between measures in \begin{document}${\mathcal{M}}_{X}$\end{document}. The criteria also give an insight to the complicated combinatorics of orbits in the shift. As a by-product, we give a direct proof that Sturmian measures are totally ordered with respect to \begin{document}$\prec$\end{document}.

2019, 39(2): 683-706 doi: 10.3934/dcds.2019028 +[Abstract](3657) +[HTML](237) +[PDF](565.34KB)
Abstract:

We study the asymptotic behavior of a class of non-autonomous non-local fractional stochastic parabolic equation driven by multiplicative white noise on the entire space \begin{document}$\mathbb{R}^n$\end{document}. We first prove the pathwise well-posedness of the equation and define a continuous non-autonomous cocycle in \begin{document}$L^2({\mathbb{R}} ^n)$\end{document}. We then prove the existence and uniqueness of tempered pullback attractors for the cocycle under certain dissipative conditions. The periodicity of the tempered attractors is also proved when the deterministic non-autonomous external terms are periodic in time. The pullback asymptotic compactness of the cocycle in \begin{document}$L^2({\mathbb{R}} ^n)$\end{document} is established by the uniform estimates on the tails of solutions for sufficiently large space and time variables.

2019, 39(2): 707-727 doi: 10.3934/dcds.2019029 +[Abstract](3487) +[HTML](187) +[PDF](1249.97KB)
Abstract:

S-systems are simple examples of power-law dynamical systems (polynomial systems with real exponents). For planar S-systems, we study global stability of the unique positive equilibrium and solve the center problem. Further, we construct a planar S-system with two limit cycles.

2019, 39(2): 729-746 doi: 10.3934/dcds.2019030 +[Abstract](3602) +[HTML](170) +[PDF](448.85KB)
Abstract:

We show that a continuous abelian action (in particular \begin{document}$\mathbb{R}^{d}$\end{document}) on a compact metric space equipped with an invariant ergodic measure has discrete spectrum if and only it is \begin{document}$μ-$\end{document}mean equicontinuous (proven for \begin{document}$\mathbb{Z}^{d}$\end{document} in [14]). In order to do this we introduce mean equicontinuity and mean sensitivity with respect to a function. We study this notion in the topological and measure theoretic setting. In the measure theoretic case we characterize almost periodic functions with these concepts and in the topological case we show that weakly almost periodic functions are mean equicontinuous (the converse does not hold). We compare our results with some results in the theory of Delone dynamical systems and quasicrystals.

2019, 39(2): 747-769 doi: 10.3934/dcds.2019031 +[Abstract](3072) +[HTML](142) +[PDF](458.09KB)
Abstract:

We compute the limit of the free energy

of the mean field generated by the independent Brownian particles \begin{document}$\{B_j(s)\}$\end{document} interacting through the non-negative definite function \begin{document}$\gamma(\cdot)$\end{document}. Our main theorem is relevant to the high moment asymptotics for the parabolic Anderson models with Gaussian noise that is white in time, white or colored in space. Our approach makes a novel connection to the celebrated Donsker-Varadhan's large deviation principle for the i.i.d. random variables in infinite dimensional spaces. As an application of our main theorem, we provide a probabilistic treatment to the Hartree's theory on the asymptotics for the ground state energy of bosonic quantum system.

2019, 39(2): 771-801 doi: 10.3934/dcds.2019032 +[Abstract](3415) +[HTML](215) +[PDF](505.24KB)
Abstract:

In this paper, we study a final value problem for a reaction-diffusion system with time and space dependent diffusion coefficients. In general, the inverse problem of identifying the initial data is not well-posed, and herein the Hadamard-instability occurs. Applying a new version of a modified quasi-reversibility method, we propose a stable approximate (regularized) problem. The existence, uniqueness and stability of the corresponding regularized problem are obtained. Furthermore, we also investigate the error estimate and show that the approximate solution converges to the exact solution in \begin{document}$L_2$\end{document} and \begin{document}$\stackrel{0}{H_1}$\end{document} norms. Our method can be applied to some concrete models that arise in biology, chemical engineering, etc.

2019, 39(2): 803-817 doi: 10.3934/dcds.2019033 +[Abstract](3187) +[HTML](168) +[PDF](483.98KB)
Abstract:

In this paper we prove that the heat kernel \begin{document}$k$\end{document} associated to the operator \begin{document}$A: = (1+|x|^\alpha)\Delta +b|x|^{\alpha-1}\frac{x}{|x|}\cdot\nabla -|x|^\beta$\end{document} satisfies

for \begin{document}$t>0,\,|x|,\,|y|\ge 1$\end{document}, where \begin{document}$b\in\mathbb{R}$\end{document}, \begin{document}$c_1,\,c_2$\end{document} are positive constants, \begin{document}$\lambda_0$\end{document} is the largest eigenvalue of the operator \begin{document}$A$\end{document}, and \begin{document}$\gamma = \frac{\beta-\alpha+2}{\beta+\alpha-2}$\end{document}, in the case where \begin{document}$N>2,\,\alpha>2$\end{document} and \begin{document}$\beta>\alpha -2$\end{document}. The proof is based on the relationship between the log-Sobolev inequality and the ultracontractivity of a suitable semigroup in a weighted space.

2019, 39(2): 819-839 doi: 10.3934/dcds.2019034 +[Abstract](2867) +[HTML](155) +[PDF](492.51KB)
Abstract:

In this paper, we study the self-dual Einstein-Maxwell-Higgs equation on compact surfaces. The solution structure depends on the parameter \begin{document}$\varepsilon$\end{document} appearing in the equation. We find an upper bound \begin{document}$\varepsilon _c$\end{document} of \begin{document}$\varepsilon$\end{document} for the existence of solutions. By using the topological degree theory, we prove that there exist at least two solutions for \begin{document}$0<\varepsilon <\varepsilon _c$\end{document}. We also study the asymptotic behavior of solutions as \begin{document}$\varepsilon \to 0$\end{document}.

2019, 39(2): 841-862 doi: 10.3934/dcds.2019035 +[Abstract](2937) +[HTML](161) +[PDF](491.16KB)
Abstract:

For an \begin{document}$N$ \end{document}-body system of linear Schrödinger equation with space dependent interaction between particles, one would expect that the corresponding one body equation, arising as a mean field approximation, would have a space dependent nonlinearity. With such motivation we consider the following model of a nonlinear reduced Schrödinger equation with space dependent nonlinearity

where \begin{document}$V(x) = -χ_{[-1,1]} (x)$ \end{document} is minus the characteristic function of the interval \begin{document}$[-1,1]$ \end{document} and where \begin{document}$h'$ \end{document} is any continuous strictly increasing function. In this article, for any negative value of \begin{document}$λ$ \end{document} we present the construction and analysis of the infinitely many solutions of this equation, which are localized in space and hence correspond to bound-states of the associated time-dependent version of the equation.

2019, 39(2): 863-880 doi: 10.3934/dcds.2019036 +[Abstract](3132) +[HTML](168) +[PDF](439.61KB)
Abstract:

We will give a new proof of a recent result of P. Daskalopoulos, G. Huisken and J.R. King ([2] and reference [7] of [2]) on the existence of self-similar solution of the inverse mean curvature flow which is the graph of a radially symmetric solution in \begin{document}$\mathbb{R}^n$ \end{document}, \begin{document}$n≥ 2$ \end{document}, of the form \begin{document}$u(x,t) = e^{λ t}f(e^{-λ t} x)$ \end{document} for any constants \begin{document}$λ>\frac{1}{n-1}$ \end{document} and \begin{document}$μ < 0$ \end{document} such that \begin{document}$f(0) = μ$ \end{document}. More precisely we will give a new proof of the existence of a unique radially symmetric solution \begin{document}$f$ \end{document} of the equation \begin{document}$\text{div}\,\left(\frac{\nabla f}{\sqrt{1+|\nabla f|^2}} \right) = \frac{1}{λ}·\frac{\sqrt{1+|\nabla f|^2}}{x·\nabla f-f}$ \end{document} in \begin{document}$\mathbb{R}^n$ \end{document}, \begin{document}$f(0) = μ$ \end{document}, for any \begin{document}$λ>\frac{1}{n-1}$ \end{document} and \begin{document}$μ < 0$ \end{document}, which satisfies \begin{document}$f_r(r)>0$ \end{document}, \begin{document}$f_{rr}(r)>0$ \end{document} and \begin{document}$rf_r(r)>f(r)$ \end{document} for all \begin{document}$r>0$ \end{document}. We will also prove that \begin{document}$\lim_{r\to∞}\frac{rf_r(r)}{f(r)} = \frac{λ (n-1)}{λ (n-1)-1}$ \end{document}.

2019, 39(2): 881-903 doi: 10.3934/dcds.2019037 +[Abstract](2836) +[HTML](144) +[PDF](552.64KB)
Abstract:

We study the problem of rigidity of closures of totally geodesic plane immersions in geometrically finite manifolds containing rank 1 cusps. We show that the key notion of K-thick recurrence of horocycles fails generically in this setting. This property played a key role in the recent breakthroughs of McMullen, Mohammadi and Oh. Nonetheless, in the setting of geometrically finite groups whose limit sets are circle packings, we derive 2 density criteria for non-closed geodesic plane immersions, and show that closed immersions give rise to surfaces with finitely generated fundamental groups. We also obtain results on the existence and isolation of proper closed immersions of elementary surfaces.

2019, 39(2): 905-926 doi: 10.3934/dcds.2019038 +[Abstract](2911) +[HTML](145) +[PDF](552.57KB)
Abstract:

We consider the following Liouville-type PDE, which is related to stationary solutions of the Keller-Segel's model for chemotaxis:

where $\Omega \subset {\mathbb{R}^2}$ is a smooth bounded domain and $\beta, ρ$ are real parameters. We prove existence of solutions under some algebraic conditions involving $\beta, ρ$. In particular, if $\Omega$ is not simply connected, then we can find solution for a generic choice of the parameters. We use variational and Morse-theoretical methods.

2019, 39(2): 927-958 doi: 10.3934/dcds.2019039 +[Abstract](2768) +[HTML](148) +[PDF](578.16KB)
Abstract:

In order to adapt to the differentiable setting a formula for linear response proved by Pollicott and Vytnova in the analytic setting, we show a result of parameter regularity of dynamical determinants of expanding maps of the circle. Linear response can then be expressed in terms of periodic points of the perturbed dynamics.

2019, 39(2): 959-993 doi: 10.3934/dcds.2019040 +[Abstract](3166) +[HTML](159) +[PDF](535.22KB)
Abstract:

This paper extends the definition of Bowen topological entropy of subsets to Pesin-Pitskel topological pressure for the continuous action of amenable groups on a compact metric space. We introduce the local measure theoretic pressure of subsets and investigate the relation between local measure theoretic pressure of Borel probability measures and Pesin-Pitskel topological pressure on an arbitrary subset of a compact metric space.

2019, 39(2): 995-1017 doi: 10.3934/dcds.2019041 +[Abstract](3626) +[HTML](163) +[PDF](500.53KB)
Abstract:

In this paper we introduce the topological entropy and lower and upper capacity topological entropies of a free semigroup action, which extends the notion of the topological entropy of a free semigroup action defined by Bufetov [10], by using the Carathéodory-Pesin structure (C-P structure). We provide some properties of these notions and give three main results. The first is the relationship between the upper capacity topological entropy of a skew-product transformation and the upper capacity topological entropy of a free semigroup action with respect to arbitrary subset. The second are a lower and a upper estimations of the topological entropy of a free semigroup action by local entropies. The third is that for any free semigroup action with $m$ generators of Lipschitz maps, topological entropy for any subset is upper bounded by the Hausdorff dimension of the subset multiplied by the maximum logarithm of the Lipschitz constants. The results of this paper generalize results of Bufetov [10], Ma et al. [26], and Misiurewicz [27].

2019, 39(2): 1019-1032 doi: 10.3934/dcds.2019042 +[Abstract](3120) +[HTML](154) +[PDF](442.59KB)
Abstract:

We prove that the average Lyapunov exponents of asymptotically additive functions have the intermediate value property provided the dynamical system has the periodic gluing orbit property. To be precise, consider a continuous map \begin{document}$f \colon X\rightarrow X$\end{document} over a compact metric space \begin{document}$X$\end{document} and an asymptotically additive sequence of functions \begin{document}$\Phi = \{\phi_n\colon X\rightarrow \mathbb{R}\}_{n\geq 1}$\end{document}. If \begin{document}$f$\end{document} has the periodic gluing orbit property, then for any constant \begin{document}$a$\end{document} satisfying

where \begin{document}$\chi_\Phi(\mu) = \liminf_{n\rightarrow \infty}\int\frac1n\phi_n d\mu$\end{document}, and the infimum and supremum are taken over the set of all \begin{document}$f$\end{document}-invariant probability measures, there is an ergodic measure \begin{document}$\mu_a\in \mathcal M_{inv} (f,X)$\end{document} such that \begin{document}$\chi_\Phi(\mu_a) = a$\end{document} and \begin{document}${\rm{supp}}(\mu)=X.$\end{document}

2019, 39(2): 1033-1048 doi: 10.3934/dcds.2019043 +[Abstract](2540) +[HTML](143) +[PDF](465.9KB)
Abstract:

Recently a generalization of shifts of finite type to the infinite alphabet case was proposed, in connection with the theory of ultragraph C*-algebras. In this work we characterize the class of continuous shift commuting maps between these spaces. In particular, we prove a Curtis-Hedlund-Lyndon type theorem and use it to completely characterize continuous, shift commuting, length preserving maps in terms of generalized sliding block codes.

2019, 39(2): 1049-1070 doi: 10.3934/dcds.2019044 +[Abstract](2824) +[HTML](139) +[PDF](569.02KB)
Abstract:

We are considering partially hyperbolic diffeomorphims of the torus, with \begin{document}${\rm{dim}}(E^c) > 1.$\end{document} We prove, under some conditions, that if the all center Lyapunov exponents of the linearization \begin{document}$A,$\end{document} of a DA-diffeomorphism \begin{document}$f,$\end{document} are positive and the center foliation of \begin{document}$f$\end{document} is absolutely continuous, then the sum of the center Lyapunov exponents of \begin{document}$f$\end{document} is bounded by the sum of the center Lyapunov exponents of \begin{document}$A.$\end{document} After, we construct a \begin{document}$C^1-$\end{document}open class of volume preserving DA-diffeomorphisms, far from Anosov diffeomorphisms, with non compact pathological two dimensional center foliation. Indeed, each \begin{document}$f$\end{document} in this open set satisfies the previously established hypothesis, but the sum of the center Lyapunov exponents of \begin{document}$f$\end{document} is greater than the corresponding sum with respect to its linearization. It allows to conclude that the center foliation of \begin{document}$f$\end{document} is non absolutely continuous. We still build an example of a DA-diffeomorphism, such that the disintegration of volume along the two dimensional, non compact center foliation is neither Lebesgue nor atomic.

2019, 39(2): 1071-1099 doi: 10.3934/dcds.2019045 +[Abstract](4183) +[HTML](188) +[PDF](1482.69KB)
Abstract:

It is well known that the Leslie-Gower prey-predator model (without Allee effect) has a unique globally asymptotically stable positive equilibrium point, thus there is no Hopf bifurcation branching from positive equilibrium point. In this paper we study the Leslie-Gower prey-predator model with strong Allee effect in prey, and perform a detailed Hopf bifurcation analysis to both the ODE and PDE models, and derive conditions for determining the steady-state bifurcation of PDE model. Moreover, by the center manifold theory and the normal form method, the direction and stability of Hopf bifurcation solutions are established. Finally, some numerical simulations are presented. Apparently, Allee effect changes the topology structure of the original Leslie-Gower model.

2019, 39(2): 1101-1115 doi: 10.3934/dcds.2019046 +[Abstract](3647) +[HTML](171) +[PDF](415.13KB)
Abstract:

The goal of the work is to verify the fractional Leibniz rule for Dirichlet Laplacian in the exterior domain of a compact set. The key point is the proof of gradient estimates for the Dirichlet problem of the heat equation in the exterior domain. Our results describe the time decay rates of the derivatives of solutions to the Dirichlet problem.

2019, 39(2): 1117-1133 doi: 10.3934/dcds.2019047 +[Abstract](3449) +[HTML](158) +[PDF](498.14KB)
Abstract:

In this paper, we consider the global strong solutions to the Cauchy problem of the compressible Navier-Stokes equations in two spatial dimensions with vacuum as far field density. It is proved that the strong solutions exist globally if the density is bounded above. Furthermore, we show that if the solutions of the two-dimensional (2D) viscous compressible flows blow up, then the mass of the compressible fluid will concentrate on some points in finite time.

2019, 39(2): 1135-1155 doi: 10.3934/dcds.2019048 +[Abstract](2816) +[HTML](155) +[PDF](513.56KB)
Abstract:

In this paper, we prove the existence of extremal functions for the best constant of embedding from anisotropic space, allowing some of the Sobolev exponents to be equal to \begin{document}$1$\end{document}. We prove also that the extremal functions satisfy a partial differential equation involving the \begin{document}$1$\end{document} Laplacian.

2019, 39(2): 1157-1170 doi: 10.3934/dcds.2019049 +[Abstract](2706) +[HTML](148) +[PDF](397.43KB)
Abstract:

We give necessary and sufficient conditions for which the elliptic equation

has nontrivial bounded solutions.

2019, 39(2): 1171-1183 doi: 10.3934/dcds.2019050 +[Abstract](3590) +[HTML](182) +[PDF](441.39KB)
Abstract:

We consider the nonlinear Schrödinger equation

for \begin{document}$H^1$\end{document}-subcritical or critical nonlinearities: \begin{document}$(N-2) α ≤ 4$\end{document}. Under the additional technical assumptions \begin{document}$α≥ 2$\end{document} (and thus \begin{document}$N≤4$\end{document}), we construct \begin{document}$H^1$\end{document} solutions that blow up in finite time with explicit blow-up profiles and blow-up rates. In particular, blowup can occur at any given finite set of points of \begin{document}$\mathbb{R}^N$\end{document}.

The construction involves explicit functions \begin{document}$U$\end{document}, solutions of the ordinary differential equation \begin{document}$U_t=|U|^α U$\end{document}. In the simplest case, \begin{document}$U(t,x)=(|x|^k-α t)^{-\frac 1α}$\end{document} for \begin{document}$t<0$\end{document}, \begin{document}$x∈ \mathbb{R}^N$\end{document}. For \begin{document}$k$\end{document} sufficiently large, \begin{document}$U$\end{document} satisfies \begin{document}$|Δ U|\ll U_t$\end{document} close to the blow-up point \begin{document}$(t,x)=(0,0)$\end{document}, so that it is a suitable approximate solution of the problem. To construct an actual solution u close to U, we use energy estimates and a compactness argument.

2019, 39(2): 1185-1203 doi: 10.3934/dcds.2019051 +[Abstract](3773) +[HTML](194) +[PDF](518.18KB)
Abstract:

In this paper, we study the initial-boundary value problem for infinitely degenerate semilinear pseudo-parabolic equations with logarithmic nonlinearity \begin{document}$u_t-\triangle_{X} u_t-\triangle_{X} u=u\log|u|$\end{document}, where \begin{document}$X=(X_1, X_2, ··· , X_m)$\end{document} is an infinitely degenerate system of vector fields, and \begin{document}$\triangle_{X}:=\sum^{m}_{j=1}X^{2}_{j}$\end{document} is an infinitely degenerate elliptic operator. Using potential well method, we first prove the invariance of some sets and vacuum isolating of solutions. Then, by the Galerkin approximation technique, the logarithmic Sobolev inequality and Poincaré inequality, we obtain the global existence and blow-up at \begin{document}$+∞$\end{document} of solutions with low initial energy or critical initial energy, and discuss the asymptotic behavior of the solutions.

2019  Impact Factor: 1.338

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