# American Institute of Mathematical Sciences

ISSN:
1078-0947

eISSN:
1553-5231

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## Discrete & Continuous Dynamical Systems - A

January 2020 , Volume 40 , Issue 1

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2020, 40(1): 1-32 doi: 10.3934/dcds.2020001 +[Abstract](1169) +[HTML](112) +[PDF](572.12KB)
Abstract:

We consider the resonant system of amplitude equations for the conformally invariant cubic wave equation on the three-sphere. Using the local bifurcation theory, we characterize all stationary states that bifurcate from the first two eigenmodes. Thanks to the variational formulation of the resonant system and energy conservation, we also determine variational characterization and stability of the bifurcating states. For the lowest eigenmode, we obtain two orbitally stable families of the bifurcating stationary states: one is a constrained maximizer of energy and the other one is a local constrained minimizer of the energy, where the constraints are due to other conserved quantities of the resonant system. For the second eigenmode, we obtain two local constrained minimizers of the energy, which are also orbitally stable in the time evolution. All other bifurcating states are saddle points of energy under these constraints and their stability in the time evolution is unknown.

2020, 40(1): 33-46 doi: 10.3934/dcds.2020002 +[Abstract](1107) +[HTML](109) +[PDF](7082.74KB)
Abstract:

Hereditarily non uniformly perfect (HNUP) sets were introduced by Stankewitz, Sugawa, and Sumi in [19] who gave several examples of such sets based on Cantor set-like constructions using nested intervals. We exhibit a class of examples in non-autonomous iteration where one considers compositions of polynomials from a sequence which is in general allowed to vary. In particular, we give a sharp criterion for when Julia sets from our class will be HNUP and we show that the maximum possible Hausdorff dimension of \begin{document}$1$\end{document} for these Julia sets can be attained. The proof of the latter considers the Julia set as the limit set of a non-autonomous conformal iterated function system and we calculate the Hausdorff dimension using a version of Bowen's formula given in the paper by Rempe-Gillen and Urbánski [15].

2020, 40(1): 47-80 doi: 10.3934/dcds.2020003 +[Abstract](2115) +[HTML](129) +[PDF](581.28KB)
Abstract:

We prove the unconditional uniqueness of solutions to the derivative nonlinear Schrödinger equation (DNLS) in an almost end-point regularity. To this purpose, we employ the normal form method and we transform (a gauge-equivalent) DNLS into a new equation (the so-called normal form equation) for which nonlinear estimates can be easily established in \begin{document}$H^s({\mathbb{R}})$\end{document}, \begin{document}$s>\frac12$\end{document}, without appealing to an auxiliary function space. Also, we prove that low-regularity solutions of DNLS satisfy the normal form equation and this is done by means of estimates in the \begin{document}$H^{s-1}({\mathbb{R}})$\end{document}-norm.

2020, 40(1): 81-105 doi: 10.3934/dcds.2020004 +[Abstract](1805) +[HTML](101) +[PDF](404.02KB)
Abstract:

Let \begin{document}$\mathcal{F}$\end{document} be a random partially hyperbolic dynamical system generated by random compositions of a set of \begin{document}$C^2$\end{document}-diffeomorphisms. For the unstable foliation, the corresponding local unstable measure-theoretic entropy, local unstable topological entropy and local unstable pressure via the dynamics of \begin{document}$\mathcal{F}$\end{document} along the unstable foliation are introduced and investigated. And variational principles for local unstable entropy and local unstable pressure are obtained respectively.

2020, 40(1): 107-131 doi: 10.3934/dcds.2020005 +[Abstract](1461) +[HTML](101) +[PDF](456.82KB)
Abstract:

We examine regularity of the extremal solution of nonlinear nonlocal eigenvalue problem

with an integro-differential operator, including the fractional Laplacian, of the form

when \begin{document}$J$\end{document} is a nonnegative measurable even jump kernel. In particular, we consider jump kernels of the form of \begin{document}$J(y) = \frac{a(y/|y|)}{|y|^{n+2s}}$\end{document} where \begin{document}$s\in (0,1)$\end{document} and \begin{document}$a$\end{document} is any nonnegative even measurable function in \begin{document}$L^1(\mathbb {S}^{n-1})$\end{document} that satisfies ellipticity assumptions. We first establish stability inequalities for minimal solutions of the above system for a general nonlinearity and a general kernel. Then, we prove regularity of the extremal solution in dimensions \begin{document}$n < 10s$\end{document} and \begin{document}$n<2s+\frac{4s}{p\mp 1}[p+\sqrt{p(p\mp1)}]$\end{document} for the Gelfand and Lane-Emden systems when \begin{document}$p>1$\end{document} (with positive and negative exponents), respectively. When \begin{document}$s\to 1$\end{document}, these dimensions are optimal. However, for the case of \begin{document}$s\in(0,1)$\end{document} getting the optimal dimension remains as an open problem. Moreover, for general nonlinearities, we consider gradient systems and we establish regularity of the extremal solution in dimensions \begin{document}$n<4s$\end{document}. As far as we know, this is the first regularity result on the extremal solution of nonlocal system of equations.

2020, 40(1): 133-151 doi: 10.3934/dcds.2020006 +[Abstract](1217) +[HTML](98) +[PDF](385.42KB)
Abstract:

We deal with a multiparameter Dirichlet system having the form

where \begin{document}$\mathcal M$\end{document} stands for the mean curvature operator in Minkowski space

\begin{document}$\Omega$\end{document} is a general bounded regular domain in \begin{document}$\mathbb{R}^N$\end{document} and the continuous functions \begin{document}$f_1,f_2$\end{document} satisfy some sign and quasi-monotonicity conditions. Among others, these type of nonlinearities, include the Lane-Emden ones. For such a system we show the existence of a hyperbola like curve which separates the first quadrant in two disjoint sets, an open one \begin{document}$\mathcal{O}_0$\end{document} and a closed one \begin{document}$\mathcal{F}$\end{document}, such that the system has zero or at least one strictly positive solution, according to \begin{document}$(\lambda_1, \lambda_2)\in \mathcal{O}_0$\end{document} or \begin{document}$(\lambda_1, \lambda_2)\in \mathcal{F}$\end{document}. Moreover, we show that inside of \begin{document}$\mathcal{F}$\end{document} there exists an infinite rectangle in which the parameters being, the system has at least two strictly positive solutions. Our approach relies on a lower and upper solutions method - which we develop here, together with topological degree type arguments. In a sense, our results extend to non-radial systems some recent existence/non-existence and multiplicity results obtained in the radial case.

2020, 40(1): 153-187 doi: 10.3934/dcds.2020007 +[Abstract](1129) +[HTML](109) +[PDF](369.27KB)
Abstract:

In the present paper, we show that an analogous N-barrier maximum principle (see [3,7,5]) remains true for lattice systems. This extends the results in [3,7,5] from continuous equations to discrete equations. In order to overcome the difficulty induced by a discretized version of the classical diffusion in the lattice systems, we propose a more delicate construction of the N-barrier which is appropriate for the proof of the N-barrier maximum principle for lattice systems. As an application of the discrete N-barrier maximum principle, we study a coexistence problem of three species arising from biology, and show that the three species cannot coexist under certain conditions.

2020, 40(1): 189-206 doi: 10.3934/dcds.2020008 +[Abstract](1130) +[HTML](109) +[PDF](381.62KB)
Abstract:

We continue and supplement studies from [G. Karch and X. Zheng, Discrete Contin. Dyn. Syst. 35 (2015), 3039-3057] on solutions to the three dimensional incompressible Navier-Stokes system which are regular outside a curve in \begin{document}$\big(\gamma(t), t\big)\in \mathbb{R}^3\times [0, \infty)$\end{document} and singular on it. We revisit some of the existence results as well as some of the asymptotic estimates obtained in that work in order prove that those solutions belongs to the space \begin{document}$C\big([0, \infty), L^2( \mathbb{R}^3)^3\big)$\end{document}.

2020, 40(1): 207-231 doi: 10.3934/dcds.2020009 +[Abstract](1363) +[HTML](107) +[PDF](434.38KB)
Abstract:

We establish a generalized Perron-Frobenius theorem, based on a combinatorial criterion which entails the existence of an eigenvector for any nonlinear order-preserving and positively homogeneous map \begin{document}$f$\end{document} acting on the open orthant \begin{document}$\mathbb{R}_{ >0}^n$\end{document}. This criterion involves dominions, i.e., sets of states that can be made invariant by one player in a two-person game that only depends on the behavior of \begin{document}$f$\end{document} "at infinity". In this way, we characterize the situation in which for all \begin{document}$\alpha, \beta > 0$\end{document}, the "slice space" \begin{document}$\mathcal{S}_\alpha^\beta : = \{ x \in \mathbb{R}_{ >0}^n \mid \alpha x \leqslant f(x) \leqslant \beta x \}$\end{document} is bounded in Hilbert's projective metric, or, equivalently, for all uniform perturbations \begin{document}$g$\end{document} of \begin{document}$f$\end{document}, all the orbits of \begin{document}$g$\end{document} are bounded in Hilbert's projective metric. This solves a problem raised by Gaubert and Gunawardena (Trans. AMS, 2004). We also show that the uniqueness of an eigenvector is characterized by a dominion condition, involving a different game depending now on the local behavior of \begin{document}$f$\end{document} near an eigenvector. We show that the dominion conditions can be verified by directed hypergraph methods. We finally illustrate these results by considering specific classes of nonlinear maps, including Shapley operators, generalized means and nonnegative tensors.

2020, 40(1): 233-266 doi: 10.3934/dcds.2020010 +[Abstract](1144) +[HTML](105) +[PDF](432.07KB)
Abstract:

This paper deals with the derivation of entropy solutions to Cauchy problems for a class of scalar conservation laws with space-density depending fluxes from systems of deterministic particles of follow-the-leader type. We consider fluxes which are product of a function of the density \begin{document}$v(\rho)$\end{document} and a function of the space variable \begin{document}$\phi(x)$\end{document}. We cover four distinct cases in terms of the sign of \begin{document}$\phi$\end{document}, including cases in which the latter is not constant. The convergence result relies on a local maximum principle and on a uniform \begin{document}$BV$\end{document} estimate for the approximating density.

2020, 40(1): 267-318 doi: 10.3934/dcds.2020011 +[Abstract](1172) +[HTML](154) +[PDF](626.09KB)
Abstract:

We consider the probabilistic Cauchy problem for the Benjamin-Bona-Mahony equation (BBM) on the one-dimensional torus \begin{document}$\mathbb{T}$\end{document} with initial data below \begin{document}$L^{2}( \mathbb{T})$\end{document}. With respect to random initial data of strictly negative Sobolev regularity, we prove that BBM is almost surely globally well-posed. The argument employs the \begin{document}$I$\end{document}-method to obtain an a priori bound on the growth of the 'residual' part of the solution. We then discuss the stability properties of the solution map in the deterministically ill-posed regime.

2020, 40(1): 319-329 doi: 10.3934/dcds.2020012 +[Abstract](1125) +[HTML](107) +[PDF](348.08KB)
Abstract:

Let \begin{document}$f:X\to X$\end{document} be a continuous map on a compact metric space with finite topological entropy. Further, we assume that the entropy map \begin{document}$\mu\mapsto h_\mu(f)$\end{document} is upper semi-continuous. It is well-known that this implies the continuity of the localized entropy function of a given continuous potential \begin{document}$\phi:X\to {\mathbb R}$\end{document}. In this note we show that this result does not carry over to the case of higher-dimensional potentials \begin{document}$\Phi:X\to {\mathbb R}^m$\end{document}. Namely, we construct for a shift map \begin{document}$f$\end{document} a \begin{document}$2$\end{document}-dimensional Lipschitz continuous potential \begin{document}$\Phi$\end{document} with a discontinuous localized entropy function.

2020, 40(1): 331-373 doi: 10.3934/dcds.2020013 +[Abstract](1409) +[HTML](107) +[PDF](517.57KB)
Abstract:

In this paper, we explore the metastable states of nonlinear Klein-Gordon equations with potentials. These states come from instability of a bound state under a nonlinear Fermi's golden rule. In [16], Soffer and Weinstein studied the instability mechanism and obtained an anomalously slow-decaying rate \begin{document}$1/(1+t)^{ \frac14}$\end{document}. Here we develop a new method to study the evolution of \begin{document}$L^2_x$\end{document} norm of solutions to Klein-Gordon equations. With this method, we prove a \begin{document}$H^1$\end{document} scattering result for Klein-Gordon equations with metastable states. By exploring the oscillations, with a dynamical system approach we also find a more robust and more intuitive way to derive the sharp decay rate \begin{document}$1/(1+t)^{ \frac14}$\end{document}.

2020, 40(1): 375-394 doi: 10.3934/dcds.2020014 +[Abstract](1139) +[HTML](94) +[PDF](474.13KB)
Abstract:

This paper is devoted to an extension of rigidity results for nonlinear differential equations, based on carré du champ methods, in the one-dimensional periodic case. The main result is an interpolation inequality with non-trivial explicit estimates of the constants in \begin{document}$\mathrm W^{1,p}( {\mathbb S}^1)$\end{document} with \begin{document}$p\ge2$\end{document}. Mostly for numerical reasons, we relate our estimates with issues concerning periodic dynamical systems. Our interpolation inequalities have a dual formulation in terms of generalized spectral estimates of Keller-Lieb-Thirring type, where the differential operator is now a \begin{document}$p$\end{document}-Laplacian type operator. It is remarkable that the carré du champ method adapts to such a nonlinear framework, but significant changes have to be done and, for instance, the underlying parabolic equation has a nonlocal term whenever \begin{document}$p\neq2$\end{document}.

2020, 40(1): 395-421 doi: 10.3934/dcds.2020015 +[Abstract](1153) +[HTML](106) +[PDF](679.78KB)
Abstract:

We investigate a distinguished low Mach and Rossby - high Reynolds and Péclet number singular limit in the complete Navier-Stokes-Fourier system towards a strong solution of a geostrophic system of equations. The limit is effectuated in the context of weak solutions with ill prepared initial data. The main tool in the proof is based on the relative energy method.

2020, 40(1): 423-439 doi: 10.3934/dcds.2020016 +[Abstract](1141) +[HTML](107) +[PDF](371.21KB)
Abstract:

In this paper we study spatial analyticity of solutions to the defocusing nonlinear Schrödinger equations \begin{document}$iu_t + \Delta u = |u|^{p-1}u$\end{document}, given initial data which is analytic with fixed radius. It is shown that the uniform radius of spatial analyticity of solutions at later time \begin{document}$t$\end{document} cannot decay faster than \begin{document}$1/|t|$\end{document} as \begin{document}$|t|\rightarrow\infty$\end{document}. This extends the previous work of Tesfahun [19] for the cubic case \begin{document}$p = 3$\end{document} to the cases where \begin{document}$p$\end{document} is any odd integer greater than \begin{document}$3$\end{document}.

2020, 40(1): 441-465 doi: 10.3934/dcds.2020017 +[Abstract](1195) +[HTML](106) +[PDF](553.15KB)
Abstract:

We present an example of a \begin{document}$C^1$\end{document} Anosov diffeomorphism of a two-torus with a physical measure such that its basin has full Lebesgue measure and its support is a horseshoe of zero measure.

2020, 40(1): 467-489 doi: 10.3934/dcds.2020018 +[Abstract](1186) +[HTML](114) +[PDF](378.26KB)
Abstract:

This paper is concerned with the existence/nonexistence of positive solutions of a weighted Hardy-Littlewood-Sobolev type integral system. Such a system is related to the extremal functions of the weighted Hardy-Littlewood-Sobolev inequality. The Serrin-type condition is critical for existence of positive solutions in \begin{document}$L_{loc}^\infty(R^n \setminus \{0\})$\end{document}. When the Serrin-type condition does not hold, we prove the nonexistence by an iteration process. In addition, we find three pairs of radial solutions when the Serrin-type condition holds. One is singular, and the other two are integrable in \begin{document}$R^n$\end{document} and decaying fast and slowly respectively.

2020, 40(1): 491-508 doi: 10.3934/dcds.2020019 +[Abstract](1672) +[HTML](132) +[PDF](358.76KB)
Abstract:

We investigate the relations between Bowen and packing invariance pressures and measure-theoretical lower and upper invariance pressures for invariant partitions of a controlled invariant set respectively. We mainly show that Bowen and packing invariance pressures can be determined via the local lower and upper invariance pressures of probability measures, which are analogues of Billingsley's Theorem for the Hausdorff dimension; and give variational principles between Bowen and packing invariance pressures and measure-theoretical lower and upper invariance pressures under some technical assumptions.

2020, 40(1): 509-528 doi: 10.3934/dcds.2020020 +[Abstract](1057) +[HTML](117) +[PDF](387.21KB)
Abstract:

In this paper we investigate conditions for maximal regularity of Volterra equations defined on the Lebesgue space of sequences \begin{document}$\ell_p(\mathbb{Z})$\end{document} by using Blünck's theorem on the equivalence between operator-valued \begin{document}$\ell_p$\end{document}-multipliers and the notion of \begin{document}$R$\end{document}-boundedness. We show sufficient conditions for maximal \begin{document}$\ell_p-\ell_q$\end{document} regularity of solutions of such problems solely in terms of the data. We also explain the significance of kernel sequences in the theory of viscoelasticity, establishing a new and surprising connection with schemes of approximation of fractional models.

2020, 40(1): 529-548 doi: 10.3934/dcds.2020021 +[Abstract](1330) +[HTML](99) +[PDF](381.9KB)
Abstract:

We study integrability –in the sense of admitting recursion operators– of two nonlinear equations which are known to possess compacton solutions: the \begin{document}$K(m, n)$\end{document} equation introduced by Rosenau and Hyman

and the CSS equation introduced by Coooper, Shepard, and Sodano,

We obtain a full classification of integrable \begin{document}$K(m, n)$\end{document} and CSS equations; we present their recursion operators, and we prove that all of them are related (via nonlocal transformations) to the Korteweg-de Vries equation. As an application, we construct isochronous hierarchies of equations associated to the integrable cases of CSS.

2020, 40(1): 549-577 doi: 10.3934/dcds.2020022 +[Abstract](1229) +[HTML](110) +[PDF](414.22KB)
Abstract:

We investigate qualitative properties of positive singular solutions of some elliptic systems in bounded and unbounded domains. We deduce symmetry and monotonicity properties via the moving plane procedure. Moreover, in the unbounded case, we study some cooperative elliptic systems involving critical nonlinearities in \begin{document}${\mathbb{R}}^n$\end{document}.

2020, 40(1): 579-596 doi: 10.3934/dcds.2020023 +[Abstract](1059) +[HTML](102) +[PDF](363.97KB)
Abstract:

In this paper, we consider the augmented Hessian equations \begin{document}$S_k^{\frac{1}{k}}[D^2u+\sigma(x)I] = f(u)$\end{document} in \begin{document}$\mathbb{R}^{n}$\end{document} or \begin{document}$\mathbb{R}^{n}_+$\end{document}. We first give the necessary and sufficient condition of the existence of classical subsolutions to the equations in \begin{document}$\mathbb{R}^{n}$\end{document} for \begin{document}$\sigma(x) = \alpha$\end{document}, which is an extended Keller-Osserman condition. Then we obtain the nonexistence of positive viscosity subsolutions of the equations in \begin{document}$\mathbb{R}^{n}$\end{document} or \begin{document}$\mathbb{R}^{n}_+$\end{document} for \begin{document}$f(u) = u^p$\end{document} with \begin{document}$p>1$\end{document}.

2020, 40(1): 597-608 doi: 10.3934/dcds.2020024 +[Abstract](1463) +[HTML](103) +[PDF](331.49KB)
Abstract:

We prove weighted \begin{document}$L^2$\end{document} estimates for the Klein-Gordon equation perturbed with singular potentials such as the inverse-square potential. We then deduce the well-posedness of the Cauchy problem for this equation with small perturbations, and go on to discuss local smoothing and Strichartz estimates which improve previously known ones.

2020, 40(1): 609-634 doi: 10.3934/dcds.2020025 +[Abstract](1290) +[HTML](92) +[PDF](460.81KB)
Abstract:

This paper is concerned with the initial-boundary value problem for the classical Keller–Segel system

in a bounded domain \begin{document}$\Omega\subset\mathbb{R}^d$\end{document} with \begin{document}$d\geq2$\end{document} under homogeneous Neumann boundary conditions, where \begin{document}$\gamma\geq0$\end{document}. We study the existence of non-trivial global classical solutions near the spatially homogeneous equilibria \begin{document}$\rho = c\equiv\mathcal{M}>0$\end{document} with \begin{document}$\mathcal{M}$\end{document} being any given large constant which is an open problem proposed in [2,p. 1687]. More precisely, we prove that if \begin{document}$0<\mathcal{M}<1+\lambda_1$\end{document} with \begin{document}$\lambda_1$\end{document} being the first positive eigenvalue of the Neumann Laplacian operator, one can find \begin{document}$\varepsilon_0>0$\end{document} such that for all suitable regular initial data \begin{document}$(\rho_0,\gamma c_0)$\end{document} satisfying

and

problem (1) possesses a unique global classical solution which is bounded and converges to the trivial state \begin{document}$(\mathcal{M},\mathcal{M})$\end{document} exponentially as time goes to infinity. The key step of our proof lies in deriving certain delicate \begin{document}$L^p-L^q$\end{document} decay estimates for the semigroup associated with the corresponding linearized system of (1) around the constant steady states. It is well-known that classical solution to system (1) may blow up in finite or infinite time when the conserved total mass \begin{document}$m\triangleq\int_\Omega \rho_0 dx$\end{document} exceeds some threshold number if \begin{document}$d = 2$\end{document} or for arbitrarily small mass if \begin{document}$d\geq3$\end{document}. In contrast, our results indicates that non-trivial classical solutions starting from initial data satisfying (2)-(3) with arbitrarily large total mass \begin{document}$m$\end{document} exists globally provided that \begin{document}$|\Omega|$\end{document} is large enough such that \begin{document}$m<(1+\lambda_1)|\Omega|$\end{document}.

2020, 40(1): 635-682 doi: 10.3934/dcds.2020026 +[Abstract](1460) +[HTML](117) +[PDF](919.16KB)
Abstract:

We consider a nonlocal family of Gross–Pitaevskii equations with nonzero conditions at infinity in dimension one. We provide conditions on the nonlocal interaction such that there is a branch of traveling waves solutions with nonvanishing conditions at infinity. Moreover, we show that the branch is orbitally stable. In this manner, this result generalizes known properties for the contact interaction given by a Dirac delta function. Our proof relies on the minimization of the energy at fixed momentum. As a by-product of our analysis, we provide a simple condition to ensure that the solution to the Cauchy problem is global in time.

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