# American Institute of Mathematical Sciences

ISSN:
1078-0947

eISSN:
1553-5231

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## Discrete & Continuous Dynamical Systems - A

November 2020 , Volume 40 , Issue 11

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2020, 40(11): 6089-6114 doi: 10.3934/dcds.2020271 +[Abstract](657) +[HTML](113) +[PDF](459.02KB)
Abstract:

In this paper, we consider a class of self-similar sets, denoted by \begin{document}$\mathcal{A}$\end{document}, and investigate the set of points in the self-similar sets having unique codings. We call such set the univoque set and denote it by \begin{document}$U_1$\end{document}. We analyze the isomorphism and bi-Lipschitz equivalence between the univoque sets. The main result of this paper, in terms of the dimension of \begin{document}$U_1$\end{document}, is to give several equivalent conditions which describe that the closure of two univoque sets, under the lazy maps, are measure theoretically isomorphic with respect to the unique measure of maximal entropy. Moreover, we prove, under the condition \begin{document}$U_1$\end{document} is closed, that isomorphism and bi-Lipschitz equivalence between the univoque sets have resonant phenomenon.

2020, 40(11): 6115-6133 doi: 10.3934/dcds.2020272 +[Abstract](616) +[HTML](114) +[PDF](346.23KB)
Abstract:

A global existence theorem on weak solutions is shown for the continuous coagulation equation with collisional breakage under certain classes of unbounded collision kernel and distribution functions. The model describes the dynamics of particle growth when binary collisions occur to form either a single particle via coalescence or two/more particles via breakup with possible transfer of mass. Each of these processes may take place with a suitably assigned probability depending on the volume of particles participating in the collision.

2020, 40(11): 6135-6158 doi: 10.3934/dcds.2020273 +[Abstract](566) +[HTML](97) +[PDF](410.55KB)
Abstract:

We study a generalized Frenkel-Kontorova model. Using minimal and Birkhoff solutions as building blocks, we construct a lot of homoclinic solutions and heteroclinic solutions for this generalized Frenkel-Kontorova model under gap conditions. These new solutions are not minimal and Birkhoff any more. We use constrained minimization method to prove our results.

2020, 40(11): 6159-6177 doi: 10.3934/dcds.2020274 +[Abstract](545) +[HTML](107) +[PDF](514.06KB)
Abstract:

The bidomain equations driven by stochastic forces and subject to nonlinearities of FitzHugh-Nagumo or Allen-Cahn type are considered for the first time. It is shown that this set of equations admits a global weak solution as well as a stationary solution, which generates a uniquely determined invariant measure.

2020, 40(11): 6179-6200 doi: 10.3934/dcds.2020275 +[Abstract](906) +[HTML](163) +[PDF](433.64KB)
Abstract:

We study equivariant Gromov-Hausdorff distances for general actions which are not necessarily isometric as Fukaya introduced. We prove that if an action is expansive and has the pseudo-orbit tracing property then it is stable under our adapted equivariant Gromov-Hausdorff topology. Finally, using Lott and Villani's ideas of optimal transport, we investigate equivariant Gromov-Hausdorff convergence for actions of locally compact amenable groups on Wasserstein spaces.

2020, 40(11): 6201-6246 doi: 10.3934/dcds.2020276 +[Abstract](564) +[HTML](105) +[PDF](535.69KB)
Abstract:

In this paper, we prove the existence of local stable and unstable invariant manifolds for a class of random differential equations driven by nonlinear colored noise defined in a fractional power of a separable Banach space. In the case of linear noise, we show the pathwise convergence of these random invariant manifolds as well as invariant foliations as the correlation time of the colored noise approaches zero.

2020, 40(11): 6247-6274 doi: 10.3934/dcds.2020277 +[Abstract](578) +[HTML](100) +[PDF](535.97KB)
Abstract:

We consider the logarithmic Schrödinger equation in the focusing regime. For this equation, Gaussian initial data remains Gaussian. In particular, the Gausson - a time-independent Gaussian function - is an orbitally stable solution. In the general case in dimension \begin{document}$d = 1$\end{document}, the solution with Gaussian initial data is periodic, and we compute some approximations of the period in the case of small and large oscillations, showing that the period can be as large as wanted for the latter. The main result of this article is a principle of nonlinear superposition: starting from an initial data made of the sum of several standing Gaussian functions far from each other, the solution remains close (in \begin{document}$L^2$\end{document}) to the sum of the corresponding Gaussian solutions for a long time, in square of the distance between the Gaussian functions.

2020, 40(11): 6275-6288 doi: 10.3934/dcds.2020279 +[Abstract](536) +[HTML](89) +[PDF](407.18KB)
Abstract:

We obtain some new Morawetz estimates for the Klein-Gordon flow of the form

where \begin{document}$\sigma, s\geq0$\end{document} and \begin{document}$\alpha>0$\end{document}. The conventional approaches to Morawetz estimates with \begin{document}$|x|^{-\alpha}$\end{document} are no longer available in the case of time-dependent weights \begin{document}$|(x, t)|^{-\alpha}$\end{document}. Here we instead apply the Littlewood-Paley theory with Muckenhoupt \begin{document}$A_2$\end{document} weights to frequency localized estimates thereof that are obtained by making use of the bilinear interpolation between their bilinear form estimates which need to carefully analyze some relevant oscillatory integrals according to the different scaling of \begin{document}$\sqrt{1-\Delta}$\end{document} for low and high frequencies.

2020, 40(11): 6289-6307 doi: 10.3934/dcds.2020280 +[Abstract](494) +[HTML](95) +[PDF](395.72KB)
Abstract:

In this paper we study the initial boundary value problem for the system \begin{document}$-\mbox{div}\left[(I+\mathbf{m} \mathbf{m}^T)\nabla p\right] = s(x), \ \ \mathbf{m}_t-\alpha^2\Delta\mathbf{m}+|\mathbf{m}|^{2(\gamma-1)}\mathbf{m} = \beta^2(\mathbf{m}\cdot\nabla p)\nabla p$\end{document} in two space dimensions. This problem has been proposed as a continuum model for biological transportation networks. The mathematical challenge is due to the presence of cubic nonlinearities, also known as trilinear forms, in the system. We obtain a weak solution \begin{document}$(\mathbf{m}, p)$\end{document} with both \begin{document}$|\nabla p|$\end{document} and \begin{document}$|\nabla\mathbf{m}|$\end{document} being bounded. The result immediately triggers a bootstrap argument which can yield higher regularity for the weak solution. This is achieved by deriving an equation for \begin{document}$v\equiv(I+\mathbf{m} \mathbf{m}^T)\nabla p\cdot\nabla p$\end{document}, and then suitably applying the De Giorge iteration method to the equation.

2020, 40(11): 6309-6330 doi: 10.3934/dcds.2020281 +[Abstract](522) +[HTML](101) +[PDF](1063.31KB)
Abstract:

As a natural counterpart to Nakada's \begin{document}$\alpha$\end{document}-continued fraction maps, we study a one-parameter family of continued fraction transformations with an indifferent fixed point. We prove that matching holds for Lebesgue almost every parameter in this family and that the exceptional set has Hausdorff dimension 1. Due to this matching property, we can construct a planar version of the natural extension for a large part of the parameter space. We use this to obtain an explicit expression for the density of the unique infinite \begin{document}$\sigma$\end{document}-finite absolutely continuous invariant measure and to compute the Krengel entropy, return sequence and wandering rate of the corresponding maps.

2020, 40(11): 6331-6350 doi: 10.3934/dcds.2020282 +[Abstract](617) +[HTML](90) +[PDF](390.8KB)
Abstract:

Transfer operators and Ruelle zeta functions for super-continuous functions on one-sided topological Markov shifts are considered. For every super-continuous function, we construct a Banach space on which the associated transfer operator is compact. Using this Banach space, we establish the trace formula and spectral representation of Ruelle zeta functions for a certain class of super-continuous functions. Our results include, as a special case, the classical trace formula and spectral representation for the class of locally constant functions.

2020, 40(11): 6351-6378 doi: 10.3934/dcds.2020283 +[Abstract](505) +[HTML](97) +[PDF](458.39KB)
Abstract:

We study the dynamics and energy exchanges between a linear oscillator and a nonlinear interaction state for the one dimensional, quintic nonlinear Schrödinger equation. Grébert and Thomann [9] proved that there exist solutions with initial data built on four Fourier modes, that confirm the conservative exchange of wave energy. Captured multi resonance in multiple Fourier modes, we simulate a similar energy exchange in long-period waves.

2020, 40(11): 6379-6409 doi: 10.3934/dcds.2020284 +[Abstract](574) +[HTML](97) +[PDF](397.6KB)
Abstract:

In this paper, we consider the chemotaxis–shallow water system in a bounded domain \begin{document}$\Omega\subset\mathbb{R}^2$\end{document}. By energy method, we establish the global existence of strong solution with small initial perturbation and obtain the exponential decaying rate of the solution. We divide the bounded domain into interior domain and the domain up to the boundary. In the interior domain, the problem is treated like the Cauchy problem. In the domain up to the boundary, the tangential and normal directions are treated differently. We use different method to get the estimates for the tangential and normal directions.

2020, 40(11): 6411-6440 doi: 10.3934/dcds.2020285 +[Abstract](504) +[HTML](92) +[PDF](431.5KB)
Abstract:

We are concerned with the inverse problem of identifying magnetic anomalies with varying parameters beneath the Earth using geomagnetic monitoring. Observations of the change in Earth's magnetic field–the secular variation–provide information about the anomalies as well as their variations. In this paper, we rigorously establish the unique recovery results for this magnetic anomaly detection problem. We show that one can uniquely recover the locations, the variation parameters including the growth or decaying rates as well as their material parameters of the anomalies. This paper extends the existing results in [9] by two of the authors to the more practical and challenging scenario with varying anomalies.

2020, 40(11): 6441-6471 doi: 10.3934/dcds.2020286 +[Abstract](603) +[HTML](111) +[PDF](493.59KB)
Abstract:

We consider the Cauchy problem for focusing nonlinear Schrödinger equation

where \begin{document}$N\geq 1$\end{document}, \begin{document}$\alpha>\frac{4}{N}$\end{document} and \begin{document}$\alpha <\frac{4}{N-2}$\end{document} if \begin{document}$N\geq 3$\end{document}. We give a criterion for energy scattering for the equation that covers well-known scattering results below, at and above the mass and energy ground state threshold. The proof is based on a recent argument of Dodson-Murphy [Math. Res. Lett. 25(6):1805-1825, 2018] using the interaction Morawetz estimate.

2020, 40(11): 6473-6506 doi: 10.3934/dcds.2020287 +[Abstract](541) +[HTML](87) +[PDF](499.52KB)
Abstract:

The contribution of this paper will be focused on the global existence and uniqueness topic in three-dimensional case of the axisymmetric viscous Boussinesq system in critical Lebesgue spaces. We aim at deriving analogous results for the classical two-dimensional and three-dimensional axisymmetric Navier-Stokes equations recently obtained in [19,20]. Roughly speaking, we show essentially that if the initial data \begin{document}$(v_0, \rho_0)$\end{document} is axisymmetric and \begin{document}$(\omega_0, \rho_0)$\end{document} belongs to the critical space \begin{document}$L^1(\Omega)\times L^1( \mathbb{R}^3)$\end{document}, with \begin{document}$\omega_0$\end{document} is the initial vorticity associated to \begin{document}$v_0$\end{document} and \begin{document}$\Omega = \{(r, z)\in \mathbb{R}^2:r>0\}$\end{document}, then the viscous Boussinesq system has a unique global solution.

2020, 40(11): 6507-6527 doi: 10.3934/dcds.2020288 +[Abstract](554) +[HTML](103) +[PDF](353.46KB)
Abstract:

Studied herein is the local well-posedness of solutions with the low regularity in the periodic setting for a class of one-dimensional generalized Rotation-Camassa-Holm equation, which is considered as an asymptotic model to describe the propagation of shallow-water waves in the equatorial region with the weak Coriolis effect due to the Earth's rotation. With the aid of the semigroup approach and a refined viscosity technique, the local existence, uniqueness and continuity of periodic solutions in the spatial space \begin{document}$C^1$\end{document} is established based on the local structure of the dynamics along the characteristics.

2020, 40(11): 6529-6546 doi: 10.3934/dcds.2020289 +[Abstract](556) +[HTML](90) +[PDF](351.82KB)
Abstract:

We study the heat equation in the exterior of the unit ball with a linear dynamical boundary condition. Our main aim is to find upper and lower bounds for the rate of convergence to solutions of the Laplace equation with the same dynamical boundary condition as the diffusion coefficient tends to infinity.

2020, 40(11): 6547-6573 doi: 10.3934/dcds.2020290 +[Abstract](634) +[HTML](140) +[PDF](476.51KB)
Abstract:

Previously in [14], we considered a diffusive logistic equation with two parameters, \begin{document}$r(x)$\end{document} – intrinsic growth rate and \begin{document}$K(x)$\end{document} – carrying capacity. We investigated and compared two special cases of the way in which \begin{document}$r(x)$\end{document} and \begin{document}$K(x)$\end{document} are related for both the logistic equations and the corresponding Lotka-Volterra competition-diffusion systems. In this paper, we continue to study the Lotka-Volterra competition-diffusion system with general intrinsic growth rates and carrying capacities for two competing species in heterogeneous environments. We establish the main result that determines the global dynamics of the system under a general criterion. Furthermore, when the ratios of the intrinsic growth rate to the carrying capacity for each species are proportional — such ratios can also be interpreted as the competition coefficients — this criterion reduces to what we obtained in [18]. We also study the detailed dynamics in terms of dispersal rates for such general case. On the other hand, when the two ratios are not proportional, our results in [14] show that the criterion in [18] cannot be fully recovered as counterexamples exist. This indicates the importance and subtleties of the roles of heterogeneous competition coefficients in the dynamics of the Lotka-Volterra competition-diffusion systems. Our results apply to competition-diffusion-advection systems as well. (See Corollary 5.1 in the last section.)

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