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1078-0947

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## Discrete & Continuous Dynamical Systems - A

2014 , Volume 34 , Issue 5

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2014, 34(5): 1701-1745
doi: 10.3934/dcds.2014.34.1701

*+*[Abstract](407)*+*[PDF](620.4KB)**Abstract:**

This review describes reaction-advection-diffusion models for the ecological effects and evolution of dispersal, and mathematical methods for analyzing those models. The topics covered include models for a single species, models for ecological interactions between species, and models for the evolution of dispersal strategies. The models are all set on bounded domains. The mathematical methods include spectral theory, specifically the theory of principal eigenvalues for elliptic operators, maximum principles and comparison theorems, bifurcation theory, and persistence theory.

2014, 34(5): 1747-1774
doi: 10.3934/dcds.2014.34.1747

*+*[Abstract](359)*+*[PDF](556.4KB)**Abstract:**

In this article we consider the following family of nonlinear elliptic problems,

$-\Delta (u^m) - \lambda \frac{u^m}{|x|^2} = |Du|^q + c f(x). $

We will analyze the interaction between the Hardy-Leray potential and the gradient term getting existence and nonexistence results in bounded domains $\Omega\subset\mathbb{R}^N$, $N\ge 3$, containing the pole of the potential.

Recall that $Λ_N = (\frac{N-2}{2})^2$ is the optimal constant in the Hardy-Leray inequality.

1.For $0 < m \le 2$ we prove the existence of a critical exponent $q_+ \le 2$ such that for $q > q_+$, the above equation has no positive distributional solution. If $q < q_+$ we find solutions by using different alternative arguments.

Moreover if $q = q_+ > 1$ we get the following alternative results.

(a) If $m < 2$ and $q=q_+$ there is no solution.

(b) If $m = 2$, then $q_+=2$ for all $\lambda$. We prove that there exists solution if and only if $2\lambda\leq\Lambda_N$ and, moreover, we find infinitely many positive solutions.

2. If $m > 2$ we obtain some partial results on existence and nonexistence.

We emphasize that if $q(\frac{1}{m}-1)<-1$ and $1 < q \le 2$, there exists positive solutions for any $f \in L^1(Ω)$.

2014, 34(5): 1775-1791
doi: 10.3934/dcds.2014.34.1775

*+*[Abstract](236)*+*[PDF](435.5KB)**Abstract:**

We are concerned with travelling wave solutions arising in a reaction diffusion equation with bistable and nonlocal nonlinearity, for which the comparison principle does not hold. Stability of the equilibrium $u\equiv 1$ is not assumed. We construct a travelling wave solution connecting 0 to an unknown steady state, which is "above and away", from the intermediate equilibrium. For focusing kernels we prove that, as expected, the wave connects 0 to 1. Our results also apply readily to the nonlocal ignition case.

2014, 34(5): 1793-1809
doi: 10.3934/dcds.2014.34.1793

*+*[Abstract](223)*+*[PDF](432.4KB)**Abstract:**

In a recent paper, K. Keller has given a characterization of the Kolmogorov-Sinai entropy of a discrete-time measure-preserving dynamical system on the base of an increasing sequence of special partitions. These partitions are constructed from order relations obtained via a given real-valued random vector, which can be interpreted as a collection of observables on the system and is assumed to separate points of it. In the present paper we relax the separation condition in order to generalize the given characterization of Kolmogorov-Sinai entropy, providing a statement on equivalence of $\sigma$-algebras. On its base we show that in the case that a dynamical system is living on an $m$-dimensional smooth manifold and the underlying measure is Lebesgue absolute continuous, the set of smooth random vectors of dimension $n>m$ with given characterization of Kolmogorov-Sinai entropy is large in a certain sense.

2014, 34(5): 1811-1827
doi: 10.3934/dcds.2014.34.1811

*+*[Abstract](260)*+*[PDF](430.0KB)**Abstract:**

Let $\mathcal{L}$ be a $D$-dimensional submanifold of a $2D$ dimensional exact symplectic manifold $(M, \omega)$ and let $f: M\rightarrow M$ be a symplectic diffeomorphism. In this article, we deal with the link between the dynamics $f_{|\mathcal{L}}$ restricted to $\mathcal{L}$ and the geometry of $\mathcal{L}$ (is $\mathcal{L}$ Lagrangian, is it smooth, is it a graph … ?).

We prove different kinds of results.

1. for $D=3$, we prove that is $\mathcal{L}$ if a torus that carries some characteristic loop, then either $\mathcal{L}$ is Lagrangian or $f_{|\mathcal{L}}$ can not be minimal (i.e. all the orbits are dense) with $(f^k_{|\mathcal{L}})$ equilipschitz;

2. for a Tonelli Hamiltonian of $T^*\mathbb{T}^3$, we give an example of an invariant submanifold $\mathcal{L}$ with no conjugate points that is not Lagrangian and such that for every $f:T^*\mathbb{T}^3\rightarrow T^*\mathbb{T}^3$ symplectic, if $f(\mathcal{L})=\mathcal{L}$, then $\mathcal{L}$ is not minimal;

3. with some hypothesis for the restricted dynamics, we prove that some invariant Lipschitz $D$-dimensional submanifolds of Tonelli Hamiltonian flows are in fact Lagrangian, $C^1$ and graphs;

4.we give similar results for $C^1$ submanifolds with weaker dynamical assumptions.

2014, 34(5): 1829-1840
doi: 10.3934/dcds.2014.34.1829

*+*[Abstract](205)*+*[PDF](386.8KB)**Abstract:**

In this paper we study a quasilinear elliptic problem whose functional satisfies a weak version of the well known Palais-Smale condition. An existence result is proved under general assumptions on the nonlinearities.

2014, 34(5): 1841-1872
doi: 10.3934/dcds.2014.34.1841

*+*[Abstract](369)*+*[PDF](893.4KB)**Abstract:**

We give estimates on the Hausdorff dimension of the levels sets of the Lyapunov exponent for the geodesic flow of a compact rank 1 surface. We show that the level sets of points with small (but non-zero) exponents has full Hausdorff dimension, but carries small topological entropy.

2014, 34(5): 1873-1878
doi: 10.3934/dcds.2014.34.1873

*+*[Abstract](234)*+*[PDF](394.9KB)**Abstract:**

Let $\mathfrak{S}$ be a compact, connected surface and $H \in C^2(T^* \mathfrak{S})$ a Tonelli Hamiltonian. This note extends V. V. Kozlov's result on the Euler characteristic of $\mathfrak{S}$ when $H$ is real-analytically integrable, using a definition of topologically-tame integrability called semisimplicity. Theorem: If $H$ is $2$-semisimple, then $\mathfrak{S}$ has non-negative Euler characteristic; if $H$ is $1$-semisimple, then $\mathfrak{S}$ has positive Euler characteristic.

2014, 34(5): 1879-1904
doi: 10.3934/dcds.2014.34.1879

*+*[Abstract](236)*+*[PDF](482.7KB)**Abstract:**

In this paper, we consider the positive solutions of the following weighted integral system involving Wolff potential in $R^{n}$: $$ \left\{ \begin{array}{ll} u(x) = R_1(x)W_{\beta,\gamma}(\frac{v^q}{|y|^{\sigma}})(x), \\ v(x) = R_2(x)W_{\beta,\gamma}(\frac{u^p}{|y|^{\sigma}})(x). (0.1) \end{array} \right. $$ This system is helpful to understand some nonlinear PDEs and other nonlinear problems. Different from the case of $\sigma=0$, it is difficult to handle the properties of the solutions since there is singularity at origin. First, we overcome this difficulty by modifying and refining the new method which was introduced to explore the integrability result establishes by Ma, Chen and Li, and obtain an optimal integrability. Second, we use the method of moving planes to prove the radial symmetry for the positive solutions of (0.1) when $R_{1}(x)\equiv R_{2}(x)\equiv 1$. Based on these results, by intricate analytical techniques, we obtain the estimate of the decay rates of those solutions near infinity.

2014, 34(5): 1905-1931
doi: 10.3934/dcds.2014.34.1905

*+*[Abstract](222)*+*[PDF](531.9KB)**Abstract:**

We consider the dissipative heat flow and conservative Gross-Pitaevskii dynamics associated with the Ginzburg-Landau energy \begin{equation*} E_\varepsilon(u) = \int_{\mathcal M} \frac{|\nabla_g u|^2}{2} + \frac{(1-|u|^2)^2}{4\varepsilon^2} dv_g \end{equation*} posed on a Riemannian $2$-manifold $\mathcal{M}$ endowed with a metric $g$. In the $ε \to 0$ limit, we show the vortices of the solutions to these two problems evolve according to the gradient flow and Hamiltonian point-vortex flow respectively, associated with the renormalized energy on $\mathcal{M}.$ For the heat flow, we then specialize to the case where $\mathcal{M}=S^2$ and study the limiting system of ODE's and establish an annihilation result. Finally, for the Ginzburg-Landau heat flow on $S^2$, we derive some weighted energy identities.

2014, 34(5): 1933-1949
doi: 10.3934/dcds.2014.34.1933

*+*[Abstract](289)*+*[PDF](484.9KB)**Abstract:**

In this paper we study unimodal maps on the closed unit interval, which have a stable period 3 orbit and an unstable period 3 orbit, and give conditions under which all points in the open unit interval are either asymptotic to the stable period 3 orbit or land after a finite time on an invariant Cantor set $\Lambda$ on which the dynamics is conjugate to a subshift of finite type and is, in fact, chaotic. For the particular value of $\mu=3.839$, Devaney [3], following ideas of Smale and Williams, shows that the logistic map $f(x)=\mu x(1-x)$ has this property. In this case the stable and unstable period 3 orbits appear when $\mu=\mu_0=1+\sqrt{8}$. We use our theorem to show that the property holds for all values of $\mu>\mu_0$ for which the stable period 3 orbit persists.

2014, 34(5): 1951-1959
doi: 10.3934/dcds.2014.34.1951

*+*[Abstract](227)*+*[PDF](350.8KB)**Abstract:**

Hardy-Littlewood-Sobolev (HLS) Inequality fails in the ``critical'' case: $μ=n$. However, for discrete HLS, we can derive a finite form of HLS inequality with logarithm correction for a critical case: $μ=n$ and $p=q$, by limiting the inequality on a finite domain. The best constant in the inequality and its corresponding solution, the optimizer, are studied. First, we obtain a sharp estimate for the best constant. Then for the optimizer, we prove the uniqueness and a symmetry property. This is achieved by proving that the corresponding Euler-Lagrange equation has a unique nontrivial nonnegative critical point. Also, by using a discrete version of maximum principle, we prove certain monotonicity of this optimizer.

2014, 34(5): 1961-1993
doi: 10.3934/dcds.2014.34.1961

*+*[Abstract](221)*+*[PDF](584.5KB)**Abstract:**

For the $L^2$ supercritical generalized Korteweg-de Vries equation, we proved in [2] the existence and uniqueness of an $N$-parameter family of $N$-solitons. Recall that, for any $N$ given solitons, we call $N$-soliton a solution of the equation which behaves as the sum of these $N$ solitons asymptotically as $t \to +\infty$. In the present paper, we also construct an $N$-parameter family of $N$-solitons for the supercritical nonlinear Schrödinger equation in dimension $1$. Nevertheless, we do not obtain any classification result; but recall that, even in subcritical and critical cases, no general uniqueness result has been proved yet.

2014, 34(5): 1995-2011
doi: 10.3934/dcds.2014.34.1995

*+*[Abstract](228)*+*[PDF](401.7KB)**Abstract:**

In this paper, we prove the convergence of the vortex blob method for a family of nonlinear evolutionary partial differential equations (PDEs), the so-called b-equation. This kind of PDEs, including the Camassa-Holm equation and the Degasperis-Procesi equation, has many applications in diverse scientific fields. Our convergence analysis also provides a proof for the existence of the global weak solution to the b-equation when the initial data is a nonnegative Radon measure with compact support.

2014, 34(5): 2013-2036
doi: 10.3934/dcds.2014.34.2013

*+*[Abstract](248)*+*[PDF](511.0KB)**Abstract:**

We consider skew-products of quadratic maps over certain Misiurewicz-Thurston maps and study their statistical properties. We prove that, when the coupling function is a polynomial of odd degree, such a system admits two positive Lyapunov exponents almost everywhere and a unique absolutely continuous invariant probability measure.

2014, 34(5): 2037-2060
doi: 10.3934/dcds.2014.34.2037

*+*[Abstract](296)*+*[PDF](498.1KB)**Abstract:**

We consider a parametric nonlinear Dirichlet equation driven by the sum of a $p$-Laplacian and a $q$-Laplacian ($1 < q < p < +\infty$, $p ≥ 2$) and with a Carathéodory reaction which at $\pm\infty$ is resonant with respect to the principal eigenvalue $\widehat{\lambda}_1(p) > 0$ of $(-\Delta_p, W^{1,p}_0(\Omega))$. Using critical point theory, truncation and comparison techniques and critical groups (Morse theory), we show that for all small values of the parameter $\lambda>0$, the problem has at least five nontrivial solutions, four of constant sign (two positive and two negative) and the fifth nodal (sign-changing).

2014, 34(5): 2061-2068
doi: 10.3934/dcds.2014.34.2061

*+*[Abstract](318)*+*[PDF](377.0KB)**Abstract:**

The Cauchy problem for the Zakharov-Kuznetsov equation is shown to be locally well-posed in $H^s(\mathbb{R}^2)$ for all $s>\frac{1}{2}$ by using the Fourier restriction norm method and bilinear refinements of Strichartz type inequalities.

2014, 34(5): 2069-2090
doi: 10.3934/dcds.2014.34.2069

*+*[Abstract](195)*+*[PDF](464.5KB)**Abstract:**

The asymptotic behavior of solutions to an anisotropic crystalline motion is investigated. In this motion, a solution polygon changes the shape by a power of crystalline curvature in its normal direction and develops singularity in a finite time. At the final time, two types of singularity appear: one is a single point-extinction and the other is degenerate pinching. We will discuss the latter case of singularity and show the exact blow-up rate for a fast blow-up or a type II blow-up solution which arises in an equivalent blow-up problem.

2014, 34(5): 2091-2104
doi: 10.3934/dcds.2014.34.2091

*+*[Abstract](257)*+*[PDF](411.2KB)**Abstract:**

We consider real analytic Hamiltonians on $\mathbb{R}^n \times \mathbb{R}^n$ whose flow depends linearly on time. Trivial examples are Hamiltonians $H(q,p)$ that do not depend on the coordinate $q\in \mathbb{R}^n$. By a theorem of Moser [11], every polynomial Hamiltonian of degree $3$ reduces to such a $q$-independent Hamiltonian via a linear symplectic change of variables. We show that such a reduction is impossible, in general, for polynomials of degree $4$ or higher. But we give a condition that implies linear-symplectic conjugacy to another simple class of Hamiltonians. The condition is shown to hold for all nondegenerate Hamiltonians that are homogeneous of degree $4$.

2014, 34(5): 2105-2133
doi: 10.3934/dcds.2014.34.2105

*+*[Abstract](284)*+*[PDF](562.7KB)**Abstract:**

We study an optimal boundary control problem (OCP) associated to a linear elliptic equation $-\mathrm{div}\,\left(\nabla y+A(x)\nabla y\right)=f$. The characteristic feature of this equation is the fact that the matrix $A(x)=[a_{ij}(x)]_{i,j=1,\dots,N}$ is skew-symmetric, $a_{ij}(x)=-a_{ji}(x)$, measurable, and belongs to $L^2$-space (rather than $L^\infty$). In spite of the fact that the equations of this type can exhibit non-uniqueness of weak solutions--- namely, they have approximable solutions as well as another type of weak solutions that can not be obtained through an approximation of matrix $A$, the corresponding OCP is well-possed and admits a unique solution. At the same time, an optimal solution to such problem can inherit a singular character of the original matrix $A$. We indicate two types of optimal solutions to the above problem: the so-called variational and non-variational solutions, and show that each of that optimal solutions can be attainable by solutions of special optimal boundary control problems.

2014, 34(5): 2135-2171
doi: 10.3934/dcds.2014.34.2135

*+*[Abstract](316)*+*[PDF](665.8KB)**Abstract:**

We study the Stokes initial boundary value problem, in $(0,T) \times Ω$, where $Ω \subseteq \mathbb{R}^n$, $n\geq3$, is an exterior domain, assuming that the initial data belongs to $L^\infty(Ω)$ and has null divergence in weak sense. We prove the maximum modulus theorem for the corresponding solutions. Crucial for the proof of this result is the analogous one proved by Abe-Giga for bounded domains. Our proof is developed by duality arguments and employing the semigroup properties of the resolving operator defined on $L^1(Ω)$. Our results are similar to the ones proved by Solonnikov by means of the potential theory.

2014, 34(5): 2173-2241
doi: 10.3934/dcds.2014.34.2173

*+*[Abstract](329)*+*[PDF](743.6KB)**Abstract:**

We construct cross sections for the geodesic flow on the orbifolds $\Gamma $\$ \mathbb{H}$ which are tailor-made for the requirements of transfer operator approaches to Maass cusp forms and Selberg zeta functions. Here, $\mathbb{H}$ denotes the hyperbolic plane and $\Gamma$ is a nonuniform geometrically finite Fuchsian group (not necessarily a lattice, not necessarily arithmetic) which satisfies an additional condition of geometric nature. The construction of the cross sections is uniform, geometric, explicit and algorithmic.

2014, 34(5): 2243-2259
doi: 10.3934/dcds.2014.34.2243

*+*[Abstract](193)*+*[PDF](434.6KB)**Abstract:**

We consider the compressible barotropic Navier-Stokes-Korteweg system with friction in this paper. The global solutions and optimal convergence rates are obtained by pure energy method provided the initial perturbation around a constant state is small enough. In particular, the decay rates of the higher-order spatial derivatives of the solution are obtained. Our proof is based on a family of scaled energy estimates and interpolations among them without linear decay analysis.

2014, 34(5): 2261-2281
doi: 10.3934/dcds.2014.34.2261

*+*[Abstract](337)*+*[PDF](459.2KB)**Abstract:**

The paper concerns a scattering problem of the wave equation of a Hartree type with small initial data with fast decay. The equation is \[ \partial_t^2 u - \Delta u = V_1(x)u+ (V_2\ast |u|^{p-1})u , \qquad t\in {\bf R}, \; x \in {\bf R}^3, \] where $p\ge 3, \; V_1(x)=O(|x|^{-\gamma_1})$ with $\gamma_1>0$ as $|x|\to\infty, \; V_2(x) = \pm |x|^{-\gamma_2}$ with $\gamma_2>0$. We prove the existence of scattering operators under almost optimal conditions on the potentials and initial data in terms of decay, using pointwise estimates. Our result generalizes the one by [14, 15] for the case $p=3$.

2014, 34(5): 2283-2305
doi: 10.3934/dcds.2014.34.2283

*+*[Abstract](190)*+*[PDF](443.3KB)**Abstract:**

We study the dynamics of polynomial skew products on $\mathbb{C}^2$. By using suitable weights, we prove the existence of several types of Green functions. Largely, continuity and plurisubharmonicity follow. Moreover, it relates to the dynamics of the rational extensions to weighted projective spaces.

2014, 34(5): 2307-2314
doi: 10.3934/dcds.2014.34.2307

*+*[Abstract](231)*+*[PDF](356.9KB)**Abstract:**

Interval translation maps (ITMs) are a non-invertible generalization of interval exchange transformations (IETs). The dynamics of finite type ITMs is similar to IETs, while infinite type ITMs are known to exhibit new interesting effects. In this paper, we prove the finiteness conjecture for the ITMs of three intervals. Namely, the subset of ITMs of finite type contains an open, dense, and full Lebesgue measure subset of the space of ITMs of three intervals. For this, we show that any ITM of three intervals can be reduced either to a rotation or to a double rotation.

2014, 34(5): 2315-2332
doi: 10.3934/dcds.2014.34.2315

*+*[Abstract](287)*+*[PDF](438.1KB)**Abstract:**

Without constructing any measure and using properties of Markov partition, this paper provides a direct proof of dimension estimates for any subset of a limit set of a Markov construction. Furthermore, this paper investigate the dimensions of asymptotically conformal repellers. And the dimension spectrum of the level sets of nonadditive potentials on asymptotically conformal repellers are also obtained.

2014, 34(5): 2333-2357
doi: 10.3934/dcds.2014.34.2333

*+*[Abstract](292)*+*[PDF](475.6KB)**Abstract:**

We study positive solutions of the equation $ε^2 \Delta u - u + u^\frac{n+2}{n-2} = 0$ where $ε >0$ is small, with Neumann boundary condition in a unit ball $B\subset\mathbb R^3$. We prove the existence of solutions with multiple interior bubbles near the center and a boundary layer. The method may also be used to the case $n=4$, $5$ and get the analogous results.

2014, 34(5): 2359-2388
doi: 10.3934/dcds.2014.34.2359

*+*[Abstract](281)*+*[PDF](516.2KB)**Abstract:**

By a perturbation approach, we construct traveling solitary solutions with various vortex structures(vortex pairs, vortex rings) for Klein-Gordon equation with Ginzburg-Landau nonlinearities.

2014, 34(5): 2389-2403
doi: 10.3934/dcds.2014.34.2389

*+*[Abstract](269)*+*[PDF](402.1KB)**Abstract:**

In this paper, we investigate the well-posedness of the Maxwell-Chern-Simons-Higgs system in the Lorenz gauge. In particular, we prove that the system is globally wellposed in the energy space. As an application, we prove that the solution of the Maxwell-Chern-Simons-Higgs system converges to that of Maxwell-Higgs system in $H^s\times H^{s-1}$($s\geq1$) as the Chern-Simons coupling constant $\kappa\rightarrow0$.

2014, 34(5): 2405-2450
doi: 10.3934/dcds.2014.34.2405

*+*[Abstract](231)*+*[PDF](583.2KB)**Abstract:**

Consider the following nonlinear scalar integral differential equation arising from delayed synaptically coupled neuronal networks \begin{eqnarray*} \frac{\partial u}{\partial t}+f(u) &=&\alpha\int^{\infty}_0\xi(c)\left[\int_{\mathbb R}K(x-y)H\left(u\left(y,t-\frac1c|x-y|\right)-\theta\right){\rm d}y\right]{\rm d}c\\ &+&\beta\int^{\infty}_0\eta(\tau)\left[\int_{\mathbb R}W(x-y)H(u(y,t-\tau)-\Theta){\rm d}y\right]{\rm d}\tau. \end{eqnarray*} This model equation generalizes many important nonlinear scalar integral differential equations arising from synaptically coupled neuronal networks. The kernel functions $K$ and $W$ represent synaptic couplings between neurons in synaptically coupled neuronal networks. The synaptic couplings can be very general, including not only pure excitations (modeled with nonnegative kernel functions), but also lateral inhibitions (modeled with Mexican hat kernel functions) and lateral excitations (modeled with upside down Mexican hat kernel functions). In this nonlinear scalar integral differential equation, $u=u(x,t)$ stands for the membrane potential of a neuron at position $x$ and time $t$. The integrals represent nonlocal spatio-temporal interactions between neurons.

We have accomplished the existence and stability of three traveling wave fronts $u(x,t)=U_k(x+\mu_kt)$ of the nonlinear scalar integral differential equation in an earlier work [42], where $\mu_k$ denotes the wave speed and $z=x+\mu_kt$ denotes the moving coordinate, $k=1,2,3$. In this paper, we will investigate how the neurobiological mechanisms represented by the synaptic couplings $(K,W)$, by the probability density functions $(\xi,\eta)$, by the synaptic rate constants $(\alpha,\beta)$ and by the firing thresholds $(\theta,\Theta)$ influence the wave speeds $\mu_k$ of the traveling wave fronts. We will define several speed index functions and use rigorous mathematical analysis to investigate the influence of the neurobiological mechanisms on the wave speeds. In particular, we will compare wave speeds of the traveling wave fronts of the nonlinear scalar integral differential equation with different synaptic couplings and with different probability density functions; we will accomplish new asymptotic behaviors of the wave speeds; we will compare wave speeds of traveling wave fronts of many reduced forms of nonlinear scalar integral differential equations of the above model equation; we will establish new estimates of the wave speeds. All these will greatly improve results obtained in previous work [38], [40] and [41].

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