# American Institute of Mathematical Sciences

ISSN:
1531-3492

eISSN:
1553-524X

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## Discrete & Continuous Dynamical Systems - B

March 2021 , Volume 26 , Issue 3

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2021, 26(3): 1243-1272 doi: 10.3934/dcdsb.2020161 +[Abstract](1216) +[HTML](338) +[PDF](460.17KB)
Abstract:

In this paper, we consider the initial-boundary value problem to the non-isothermal incompressible liquid crystal system with both variable density and temperature. Global well-posedness of strong solutions is established for initial data being small perturbation around the equilibrium state. As the tools in the proof, we establish the maximal regularities of the linear Stokes equations and parabolic equations with variable coefficients and a rigid lemma for harmonic maps on bounded domains. This paper also generalizes the result in [5] to the inhomogeneous case.

2021, 26(3): 1273-1289 doi: 10.3934/dcdsb.2020162 +[Abstract](1645) +[HTML](373) +[PDF](704.87KB)
Abstract:

A reaction-diffusion predator-prey system with prey-taxis and predator-taxis describes the spatial interaction and random movement of predator and prey species, as well as the spatial movement of predators pursuing prey and prey evading predators. The spatial pattern formation induced by the prey-taxis and predator-taxis is characterized by the Turing type linear instability of homogeneous state and bifurcation theory. It is shown that both attractive prey-taxis and repulsive predator-taxis compress the spatial patterns, while repulsive prey-taxis and attractive predator-taxis help to generate spatial patterns. Our results are applied to the Holling-Tanner predator-prey model to demonstrate the pattern formation mechanism.

2021, 26(3): 1291-1303 doi: 10.3934/dcdsb.2020163 +[Abstract](1207) +[HTML](298) +[PDF](313.53KB)
Abstract:

This paper deals with the 3D incompressible Navier-Stokes equations with density-dependent viscosity in the whole space. The global well-posedness and exponential decay of strong solutions is established in the vacuum cases, provided the assumption that the bound of density is suitably small, which extends the results of [Nonlinear Anal. Real World Appl., 46:58-81, 2019] to the global one. However, it's entirely different from the recent work [arxiv: 1709.05608v1, 2017] and [J. Math. Fluid Mech., 15:747-758, 2013], there is not any smallness condition on the velocity.

2021, 26(3): 1305-1335 doi: 10.3934/dcdsb.2020164 +[Abstract](1194) +[HTML](299) +[PDF](459.03KB)
Abstract:

In this paper we study the rate of convergence to the complex balanced equilibrium for some chemical reaction-diffusion systems with boundary equilibria. We first analyze a three-species system with boundary equilibria in some stoichiometric classes, and whose right hand side is bounded above by a quadratic nonlinearity in the positive orthant. We prove similar results on the convergence to the positive equilibrium for a fairly general two-species reversible reaction-diffusion network with boundary equilibria.

2021, 26(3): 1337-1357 doi: 10.3934/dcdsb.2020165 +[Abstract](955) +[HTML](310) +[PDF](410.52KB)
Abstract:

We solve a conjecture on multiple nondegenerate steady states, and prove bistability for sequestration networks. More specifically, we prove that for any odd number of species, and for any production factor, the fully open extension of a sequestration network admits three nondegenerate positive steady states, two of which are locally asymptotically stable. In addition, we provide a non-empty open set in the parameter space where a sequestration network admits bistability, and we present a procedure for computing a witness for bistability.

2021, 26(3): 1359-1403 doi: 10.3934/dcdsb.2020166 +[Abstract](1169) +[HTML](302) +[PDF](2711.0KB)
Abstract:

We investigate the main phase space properties of the QR-flow when restricted to upper Hessenberg matrices. A complete description of the linear behavior of the equilibrium matrices is given. The main result classifies the possible \begin{document}$\alpha$\end{document}- and \begin{document}$\omega$\end{document}-limits of the orbits for this system. Furthermore, we characterize the set of initial matrices for which there is convergence towards an equilibrium matrix. Several numerical examples show the different limit behavior of the orbits and illustrate the theory.

2021, 26(3): 1405-1446 doi: 10.3934/dcdsb.2020167 +[Abstract](1210) +[HTML](288) +[PDF](418.83KB)
Abstract:

A new explicit Milstein-type scheme for SDE driven by Lévy noise is proposed where both drift and diffusion coefficients are allowed to grow super-linearly. The strong rate of convergence (in \begin{document}$\mathcal{L}^2$\end{document}-sense) is shown to be arbitrarily close to one which is consistent with the corresponding result on the classical Milstein scheme obtained for coefficients satisfying global Lipschitz conditions.

2021, 26(3): 1447-1468 doi: 10.3934/dcdsb.2020168 +[Abstract](1197) +[HTML](351) +[PDF](426.34KB)
Abstract:

In this article we generalize the convolution quadrature (CQ) method, which aims at approximating the fractional calculus, to the case for the distributed order calculus. Our method is a natural expansion that the approximation formulas, convergence results and correction technique reduce to the cases for the CQ method if the weight function \begin{document}$\mu(\alpha)$\end{document} is defined by \begin{document}$\delta(\alpha-\alpha_0)$\end{document}. Further, we explore a new structure of the solution of an ODE with the distributed order fractional derivative, which differs from those of the solutions of traditional fractional ODEs, and propose a new correction technique for this new structure to restore the optimal convergence rate. Numerical tests with smooth and nonsmooth solutions confirm our theoretical results and the efficiency of our correction technique.

2021, 26(3): 1469-1497 doi: 10.3934/dcdsb.2020169 +[Abstract](1018) +[HTML](294) +[PDF](563.69KB)
Abstract:

In this paper we consider a selection-mutation model with an advection term formulated on the space of finite signed measures on \begin{document}$\mathbb{R}^d$\end{document}. The selection-mutation kernel is described by a family of measures which allows the study of continuous and discrete kernels under the same setting. We rescale the selection-mutation kernel to obtain a diffusively rescaled selection-mutation model. We prove that if the rescaled selection-mutation kernel converges to a pure selection kernel then the solution of the diffusively rescaled model converges to a solution of an advection-diffusion equation.

2021, 26(3): 1499-1529 doi: 10.3934/dcdsb.2020170 +[Abstract](1581) +[HTML](342) +[PDF](451.85KB)
Abstract:

In the last years, several authors studied a class of continuous-time semi-Markov processes obtained by time-changing Markov processes by hitting times of independent subordinators. Such processes are governed by integro-differential convolution equations of generalized fractional type. The aim of this paper is to develop a discrete-time counterpart of such a theory and to show relationships and differences with respect to the continuous time case. We present a class of discrete-time semi-Markov chains which can be constructed as time-changed Markov chains and we obtain the related governing convolution type equations. Such processes converge weakly to those in continuous time under suitable scaling limits.

2021, 26(3): 1531-1547 doi: 10.3934/dcdsb.2020171 +[Abstract](1144) +[HTML](339) +[PDF](351.11KB)
Abstract:

This paper investigates the Cauchy problem to a class of stochastic non-autonomous evolution equations of parabolic type governed by noncompact evolution families in Hilbert spaces. Combining the theory of evolution families, the fixed point theorem with respect to convex-power condensing operator and a new estimation technique of the measure of noncompactness, we established some new existence results of mild solutions under the situation that the nonlinear function satisfy some appropriate local growth condition and a noncompactness measure condition. Our results generalize and improve some previous results on this topic, since the strong restriction on the constants in the condition of noncompactness measure is completely deleted, and also the condition of uniformly continuity of the nonlinearity is not required. At last, as samples of applications, we consider the Cauchy problem to a class of stochastic non-autonomous partial differential equation of parabolic type.

2021, 26(3): 1549-1563 doi: 10.3934/dcdsb.2020172 +[Abstract](1160) +[HTML](350) +[PDF](399.14KB)
Abstract:

For FitzHugh-Nagumo lattice dynamical systems (LDSs) many authors studied the existence of global attractors for deterministic systems [4,34,41,43] and the existence of global random attractors for stochastic systems [23,24,27,48,49], where for non-autonomous cases, the nonlinear parts are considered of the form \begin{document}$f\left( u\right)$\end{document}. Here we study the existence of the uniform global attractor for a new family of non-autonomous FitzHugh-Nagumo LDSs with nonlinear parts of the form \begin{document}$f\left( u,t\right)$\end{document}, where we introduce a suitable Banach space of functions \begin{document}$W$\end{document} and we assume that \begin{document}$f$\end{document} is an element of the hull of an almost periodic function \begin{document}$f_{0}\left( \cdot ,t\right)$\end{document} with values in \begin{document}$W$\end{document}.

2021, 26(3): 1565-1577 doi: 10.3934/dcdsb.2020173 +[Abstract](1128) +[HTML](323) +[PDF](564.62KB)
Abstract:

A van der Pol damped SD oscillator, which was proposed by Ruilan Tian, Qingjie Cao and Shaopu Yang (2010, Nonlinear Dynamics, 59, 19-27), is studied. By improving the criterion function of determining the lowest upper bound of the number of zeros of Abelian Integrals, we show that the number of zeros of Abelian integrals of this SD oscillator is two which is sharp.

2021, 26(3): 1579-1613 doi: 10.3934/dcdsb.2020174 +[Abstract](1282) +[HTML](345) +[PDF](526.85KB)
Abstract:

We study a terminal value parabolic system with nonlinear-nonlocal diffusions. Firstly, we consider the issue of existence and ill-posed property of a solution. Then we introduce two regularization methods to solve the system in which the diffusion coefficients are globally Lipschitz or locally Lipschitz under some a priori assumptions on the sought solutions. The existence, uniqueness and regularity of solutions of the regularized problem are obtained. Furthermore, The error estimates show that the approximate solution converges to the exact solution in \begin{document}$L^2$\end{document} norm and also in \begin{document}$H^1$\end{document} norm.

2021, 26(3): 1615-1626 doi: 10.3934/dcdsb.2020175 +[Abstract](1017) +[HTML](299) +[PDF](340.13KB)
Abstract:

The current paper is devoted to the stochastic damped Ostrovsky equation driven by white noise. By establishing the uniform estimates for the solution in \begin{document}$H^1$\end{document} norm, we prove the global well-posedness and the existence of invariant measure for stochastic damped Ostrovsky equation with random initial value. Moreover, we obtain the ergodicity of stochastic damped Ostrovsky equation with deterministic initial conditions.

2021, 26(3): 1627-1652 doi: 10.3934/dcdsb.2020176 +[Abstract](929) +[HTML](280) +[PDF](420.71KB)
Abstract:

In this paper, we consider a class of second-order Hamiltonian systems of the form

where \begin{document}$L:R\rightarrow R^{N^2}$\end{document} and \begin{document}$W \in C^1(R\times R^N, R)$\end{document} are asymptotically periodic in \begin{document}$t$\end{document} at infinity. Under the reformative perturbation conditions and weaker superquadratic conditions on the nonlinearity, the existence of a ground state homoclinic orbit is established. The main tools employed here are the local mountain pass theorem and the concentration-compactness principle.

2021, 26(3): 1653-1673 doi: 10.3934/dcdsb.2020177 +[Abstract](1006) +[HTML](275) +[PDF](846.13KB)
Abstract:

In this paper we study global dynamic aspects of the quadratic system

where \begin{document}$(x,y,z) \in \mathbb R^3$\end{document} and \begin{document}$\alpha, \beta \in[0,1]$\end{document} are two parameters. It contains the Sprott B and the Sprott C systems at the two extremes of its parameter spectrum and we call it Sprott BC system. Here we present the complete description of its singularities and we show that this system passes through a Hopf bifurcation at \begin{document}$\alpha = 0$\end{document}. Using the Poincaré compactification of a polynomial vector field in \begin{document}$\mathbb R^3$\end{document} we give a complete description of its dynamic on the Poincaré sphere at infinity. We also show that such a system does not admit a polynomial first integral, nor algebraic invariant surfaces, neither Darboux first integral.

2021, 26(3): 1675-1710 doi: 10.3934/dcdsb.2020178 +[Abstract](980) +[HTML](296) +[PDF](561.19KB)
Abstract:

In this work we estimate the convergence rate for time stepping schemes applied to nonlocal dynamic fracture modeling. Here we use the nonlocal formulation given by the bond based peridynamic equation of motion. We begin by establishing the existence of \begin{document}$H^2$\end{document} peridynamic solutions over any finite time interval. For this model the gradients can become large and steep slopes appear and localize when the non-locality of the model tends to zero. In this treatment spatial approximation by finite elements are used. We consider the central-difference scheme for time discretization and linear finite elements for discretization in the spatial variable. The fully discrete scheme is shown to converge to the actual \begin{document}$H^2$\end{document} solution in the mean square norm at the rate \begin{document}$C_t\Delta t +C_s h^2/\epsilon^2$\end{document}. Here \begin{document}$h$\end{document} is the mesh size, \begin{document}$\epsilon$\end{document} is the length scale of nonlocal interaction and \begin{document}$\Delta t$\end{document} is the time step. The constants \begin{document}$C_t$\end{document} and \begin{document}$C_s$\end{document} are independent of \begin{document}$\Delta t$\end{document}, and \begin{document}$h$\end{document}. In the absence of nonlinearity a CFL like condition for the energy stability of the central difference time discretization scheme is developed. As an example we consider Plexiglass and compute constants in the a-priori error bound.

2021, 26(3): 1711-1722 doi: 10.3934/dcdsb.2020179 +[Abstract](1052) +[HTML](286) +[PDF](594.33KB)
Abstract:

In this paper, an anisotropic bilinear finite element method is constructed for the elliptic boundary layer optimal control problems. Supercloseness properties of the numerical state and numerical adjoint state in a \begin{document}$\epsilon$\end{document}-norm are established on anisotropic meshes. Moreover, an interpolation type post-processed solution is shown to be superconvergent of order \begin{document}$O(N^{-2})$\end{document}, where the total number of nodes is of \begin{document}$O(N^2)$\end{document}. Finally, numerical results are provided to verify the theoretical analysis.

2021, 26(3): 1723-1747 doi: 10.3934/dcdsb.2020180 +[Abstract](1320) +[HTML](284) +[PDF](453.02KB)
Abstract:

In this paper, we study some existence results for fractional differential equations subject to some kind of initial conditions. First, we focus on the linear problem and we give an explicit form of solutions by reduction to an integral problem. We analyze some properties of the solutions to the linear problem in terms of its coefficients. Then we provide examples of application of the mentioned properties. Secondly, with the help of this theory, we study the nonlinear problem subject to initial value conditions. By using the upper and lower solutions method and the monotone iterative algorithm, we show the existence and localization of solutions to the nonlinear fractional differential equation.

2021, 26(3): 1749-1762 doi: 10.3934/dcdsb.2020318 +[Abstract](1330) +[HTML](161) +[PDF](306.96KB)
Abstract:

This paper is devoted to study of time-fractional elliptic equations driven by a multiplicative noise. By combining the eigenfunction expansion method for symmetry elliptic operators, the variation of constant formula for strong solutions to scalar stochastic fractional differential equations, Ito's formula and establishing a new weighted norm associated with a Lyapunov–Perron operator defined from this representation of solutions, we show the asymptotic behaviour of solutions to these systems in the mean square sense. As a consequence, we also prove existence, uniqueness and the convergence rate of their solutions.

2020 Impact Factor: 1.327
5 Year Impact Factor: 1.492
2020 CiteScore: 2.2