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Discrete and Continuous Dynamical Systems - B

October 2022 , Volume 27 , Issue 10

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A delayed dynamical model for COVID-19 therapy with defective interfering particles and artificial antibodies
Yanfei Zhao and Yepeng Xing
2022, 27(10): 5367-5387 doi: 10.3934/dcdsb.2021278 +[Abstract](807) +[HTML](237) +[PDF](756.12KB)
Abstract:

In this paper, we use delay differential equations to propose a mathematical model for COVID-19 therapy with both defective interfering particles and artificial antibodies. For this model, the basic reproduction number \begin{document}$ \mathcal{R}_0 $\end{document} is given and its threshold properties are discussed. When \begin{document}$ \mathcal{R}_0<1 $\end{document}, the disease-free equilibrium \begin{document}$ E_0 $\end{document} is globally asymptotically stable. When \begin{document}$ \mathcal{R}_0>1 $\end{document}, \begin{document}$ E_0 $\end{document} becomes unstable and the infectious equilibrium without defective interfering particles \begin{document}$ E_1 $\end{document} comes into existence. There exists a positive constant \begin{document}$ R_1 $\end{document} such that \begin{document}$ E_1 $\end{document} is globally asymptotically stable when \begin{document}$ R_1<1<\mathcal{R}_0 $\end{document}. Further, when \begin{document}$ R_1>1 $\end{document}, \begin{document}$ E_1 $\end{document} loses its stability and infectious equilibrium with defective interfering particles \begin{document}$ E_2 $\end{document} occurs. There exists a constant \begin{document}$ R_2 $\end{document} such that \begin{document}$ E_2 $\end{document} is asymptotically stable without time delay if \begin{document}$ 1<R_1<\mathcal{R}_0<R_2 $\end{document} and it loses its stability via Hopf bifurcation as the time delay increases. Numerical simulation is also presented to demonstrate the applicability of the theoretical predictions.

Nonstationary homoclinic orbit for an infinite-dimensional fractional reaction-diffusion system
Peng Chen, Linfeng Mei and Xianhua Tang
2022, 27(10): 5389-5409 doi: 10.3934/dcdsb.2021279 +[Abstract](620) +[HTML](217) +[PDF](393.99KB)
Abstract:

This paper study nonstationary homoclinic-type solutions for a fractional reaction-diffusion system with asymptotically linear and local super linear nonlinearity. The resulting problem engages two major difficulties: one is that the associated functional is strongly indefinite, the second lies in verifying the link geometry and showing the boundedness of Cerami sequences when the nonlinearity is not super quadratic at infinity globally. These enable us to develop a direct approach and new tricks to overcome the difficulties. We establish the existence of homoclinic orbit under some weak assumptions on nonlinearity.

Exponential decay for quasilinear parabolic equations in any dimension
Jian-Wen Sun and Seonghak Kim
2022, 27(10): 5411-5418 doi: 10.3934/dcdsb.2021280 +[Abstract](701) +[HTML](226) +[PDF](262.75KB)
Abstract:

We estimate decay rates of solutions to the initial-boundary value problem for a class of quasilinear parabolic equations in any dimension. Such decay rates depend only on the constitutive relations, spatial domain, and range of the initial function.

Stabilizing multiple equilibria and cycles with noisy prediction-based control
Elena Braverman and Alexandra Rodkina
2022, 27(10): 5419-5446 doi: 10.3934/dcdsb.2021281 +[Abstract](556) +[HTML](325) +[PDF](1149.93KB)
Abstract:

Pulse stabilization of cycles with Prediction-Based Control including noise and stochastic stabilization of maps with multiple equilibrium points is analyzed for continuous but, generally, non-smooth maps. Sufficient conditions of global stabilization are obtained. Introduction of noise can relax restrictions on the control intensity. We estimate how the control can be decreased with noise and verify it numerically.

On weak martingale solutions to a stochastic Allen-Cahn-Navier-Stokes model with inertial effects
T. Tachim Medjo
2022, 27(10): 5447-5485 doi: 10.3934/dcdsb.2021282 +[Abstract](710) +[HTML](234) +[PDF](489.82KB)
Abstract:

We consider a stochastic Allen-Cahn-Navier-Stokes equations with inertial effects in a bounded domain \begin{document}$ D\subset\mathbb{R}^{d} $\end{document}, \begin{document}$ d = 2, 3 $\end{document}, driven by a multiplicative noise. The existence of a global weak martingale solution is proved under non-Lipschitz assumptions on the coefficients. The construction of the solution is based on the Faedo-Galerkin approximation, compactness method and the Skorokhod representation theorem.

Exponential stability for a piezoelectric beam with a magnetic effect and past history
Manoel J. Dos Santos, João C. P. Fortes and Marcos L. Cardoso
2022, 27(10): 5487-5501 doi: 10.3934/dcdsb.2021283 +[Abstract](709) +[HTML](263) +[PDF](286.75KB)
Abstract:

Solutions for systems consisting of coupled wave equations, one of them with past history, may present different behaviors due to the type of coupling. In this paper, the issue of exponential stability for a piezoelectric beam with magnetic effect and past history is analyzed. In the work is proved that the past history term acting on the longitudinal motion equation is sufficient to cause the exponential decay of the semigroup associated with the system, independent of any relation involving the model coefficients.

Dynamics for the 3D incompressible Navier-Stokes equations with double time delays and damping
Wei Shi, Xiaona Cui, Xuezhi Li and Xin-Guang Yang
2022, 27(10): 5503-5534 doi: 10.3934/dcdsb.2021284 +[Abstract](603) +[HTML](250) +[PDF](478.35KB)
Abstract:

This paper is concerned with the tempered pullback attractors for 3D incompressible Navier-Stokes model with a double time-delays and a damping term. The delays are in the convective term and external force, which originate from the control in engineer and application. Based on the existence of weak and strong solutions for three dimensional hydrodynamical model with subcritical nonlinearity, we proved the existence of minimal family for pullback attractors with respect to tempered universes for the non-autonomous dynamical systems.

Phenomenologies of intermittent Hall MHD turbulence
Mimi Dai
2022, 27(10): 5535-5560 doi: 10.3934/dcdsb.2021285 +[Abstract](545) +[HTML](205) +[PDF](400.25KB)
Abstract:

We introduce the concept of intermittency dimension for the magnetohydrodynamics (MHD) to quantify the intermittency effect. With dependence on the intermittency dimension, we derive phenomenological laws for intermittent MHD turbulence with and without the Hall effect. In particular, scaling laws of dissipation wavenumber, energy spectra and structure functions are predicted. Moreover, we are able to provide estimates for energy spectra and structure functions which are consistent with the predicted scalings.

Stability of positive steady-state solutions to a time-delayed system with some applications
Shihe Xu, Fangwei Zhang and Meng Bai
2022, 27(10): 5561-5572 doi: 10.3934/dcdsb.2021286 +[Abstract](528) +[HTML](199) +[PDF](333.12KB)
Abstract:

In this paper, we study a general nonlinear retarded system:

\begin{document}$ \begin{equation} y'(t) = a(t)F(y(t),y(t-\tau)), \; \; t\geq 0, \end{equation} $\end{document}

where \begin{document}$ \tau>0 $\end{document} is a constant, \begin{document}$ a(t) $\end{document} is a positive value function defined on \begin{document}$ [0,\infty) $\end{document}, \begin{document}$ F(y,z) $\end{document} is continuous in \begin{document}$ \mathscr{D} = \mathbb{R}_+^2 $\end{document}, where \begin{document}$ \mathbb{R_+} = (0,+\infty) $\end{document}. Sufficient conditions for stability of the unique positive equilibrium are established. Our results show that if \begin{document}$ F_z(y,z)>0 $\end{document} for \begin{document}$ y,z\in \mathbb{R_+} $\end{document}, then the unique positive equilibrium of (1) which denoted by \begin{document}$ \bar{y} $\end{document} is globally stable for any positive initial value and all \begin{document}$ \tau>0 $\end{document}; if \begin{document}$ F(y,z) $\end{document} is decreasing in \begin{document}$ y $\end{document}, then \begin{document}$ \bar{y} $\end{document} is globally stable for small \begin{document}$ \tau $\end{document}. Some applications are given.

Dynamics of a delayed Lotka-Volterra model with two predators competing for one prey
Minzhen Xu and Shangjiang Guo
2022, 27(10): 5573-5595 doi: 10.3934/dcdsb.2021287 +[Abstract](874) +[HTML](272) +[PDF](575.75KB)
Abstract:

In this paper, we study the local dynamics of a class of 3-dimensional Lotka-Volterra systems with a discrete delay. This system describes two predators competing for one prey. Firstly, linear stability and Hopf bifurcation are investigated. Then some regions of attraction for the positive steady state are obtained by means of Liapunov functional in a restricted region. Finally, sufficient and necessary conditions for the principle of competitive exclusion are obtained.

Limiting dynamics for stochastic nonclassical diffusion equations
Peng Gao
2022, 27(10): 5597-5629 doi: 10.3934/dcdsb.2021288 +[Abstract](523) +[HTML](201) +[PDF](531.19KB)
Abstract:

In this paper, we are concerned with the dynamical behavior of the stochastic nonclassical parabolic equation, more precisely, it is shown that the inviscid limits of the stochastic nonclassical diffusion equations reduces to the stochastic heat equations. The key points in the proof of our convergence results are establishing some uniform estimates and the regularity theory for the solutions of the stochastic nonclassical diffusion equations which are independent of the parameter. Based on the uniform estimates, the tightness of distributions of the solutions can be obtained.

The existence and properties of the solution of a class of nonlinear differential equations with switching at variable times
Huanting Li, Yunfei Peng and Kuilin Wu
2022, 27(10): 5631-5652 doi: 10.3934/dcdsb.2021289 +[Abstract](556) +[HTML](230) +[PDF](438.91KB)
Abstract:

In this paper, we deal with the qualitative theory for a class of nonlinear differential equations with switching at variable times (SSVT), such as the existence and uniqueness of the solution, the continuous dependence and differentiability of the solution with respect to parameters and the stability. Firstly, we obtain the existence and uniqueness of a global solution by defining a reasonable solution (see Definition 2.1). Secondly, the continuous dependence and differentiability of the solution with respect to the initial state and the switching line are investigated. Finally, the global exponential stability of the system is discussed. Moreover, we give the necessary and sufficient conditions of SSVT just switching \begin{document}$ k\in \mathbb{N} $\end{document} times on bounded time intervals.

Asymptotic $ H^2$ regularity of a stochastic reaction-diffusion equation
Hongyong Cui and Yangrong Li
2022, 27(10): 5653-5671 doi: 10.3934/dcdsb.2021290 +[Abstract](503) +[HTML](214) +[PDF](523.1KB)
Abstract:

In this paper we study the asymptotic dynamics for the weak solutions of the following stochastic reaction-diffusion equation defined on a bounded smooth domain \begin{document}$ {\mathcal{O}} \subset {\mathbb{R}}^N $\end{document}, \begin{document}$ N \leqslant 3 $\end{document}, with Dirichlet boundary condition:

where \begin{document}$ \beta>0 $\end{document}, \begin{document}$ g\in H $\end{document}, and \begin{document}$ W $\end{document} a scalar and two-sided Wiener process with a regular perturbation intensity \begin{document}$ h $\end{document}. We first construct an \begin{document}$ H^2 $\end{document} tempered random absorbing set of the system, and then prove an \begin{document}$ (H,H^2) $\end{document}-smoothing property and conclude that the random attractor of the system is in fact a finite-dimensional tempered random set in \begin{document}$ H^2 $\end{document} and pullback attracts tempered random sets in \begin{document}$ H $\end{document} under the topology of \begin{document}$ H^2 $\end{document}. The main technique we shall employ is comparing the regularity of the stochastic equation to that of the corresponding deterministic equation for which the asymptotic \begin{document}$ H^2 $\end{document} regularity is already known.

Boundedness of the complex Chen system
Xu Zhang and Guanrong Chen
2022, 27(10): 5673-5700 doi: 10.3934/dcdsb.2021291 +[Abstract](540) +[HTML](211) +[PDF](444.56KB)
Abstract:

Some ultimate bounds are derived for the complex Chen system.

Bifurcation and control of a predator-prey system with unfixed functional responses
Lizhi Fei and Xingwu Chen
2022, 27(10): 5701-5721 doi: 10.3934/dcdsb.2021292 +[Abstract](877) +[HTML](327) +[PDF](9642.14KB)
Abstract:

In this paper we investigate a discrete-time predator-prey system with not only some constant parameters but also unfixed functional responses including growth rate function of prey, conversion factor function and predation probability function. We prove that the maximal number of fixed points is \begin{document}$ 3 $\end{document} and give necessary and sufficient conditions of exactly \begin{document}$ j $\end{document}(\begin{document}$ j = 1,2,3 $\end{document}) fixed points, respectively. For transcritical bifurcation and Neimark-Sacker bifurcation, we provide bifurcation conditions depending on these unfixed functional responses. In order to regulate the stability of this biological system, a hybrid control strategy is used to control the Neimark-Sacker bifurcation. Finally, we apply our main results to some examples and carry out numerical simulations for each example to verify the correctness of our theoretical analysis.

Dynamics of stochastic retarded Benjamin-Bona-Mahony equations on unbounded channels
Qiangheng Zhang
2022, 27(10): 5723-5755 doi: 10.3934/dcdsb.2021293 +[Abstract](497) +[HTML](207) +[PDF](510.72KB)
Abstract:

This article is devoted to the asymptotic behaviour of solutions for stochastic Benjamin-Bona-Mahony (BBM) equations with distributed delay defined on unbounded channels. We first prove the existence, uniqueness and forward compactness of pullback random attractors (PRAs). We then establish the forward asymptotic autonomy of this PRA. Finally, we study the non-delay stability of this PRA. Due to the loss of usual compact Sobolev embeddings on unbounded domains, the forward uniform tail-estimates and forward flattening of solutions are used to prove the forward asymptotic compactness of solutions.

Modeling the second outbreak of COVID-19 with isolation and contact tracing
Haitao Song, Fang Liu, Feng Li, Xiaochun Cao, Hao Wang, Zhongwei Jia, Huaiping Zhu, Michael Y. Li, Wei Lin, Hong Yang, Jianghong Hu and Zhen Jin
2022, 27(10): 5757-5777 doi: 10.3934/dcdsb.2021294 +[Abstract](891) +[HTML](475) +[PDF](1217.24KB)
Abstract:

The first case of Corona Virus Disease 2019 (COVID-19) was reported in Wuhan, China in December 2019. Since then, COVID-19 has quickly spread out to all provinces in China and over 150 countries or territories in the world. With the first level response to public health emergencies (FLRPHE) launched over the country, the outbreak of COVID-19 in China is achieving under control in China. We develop a mathematical model based on the epidemiology of COVID-19, incorporating the isolation of healthy people, confirmed cases and contact tracing measures. We calculate the basic reproduction numbers 2.5 in China (excluding Hubei province) and 2.9 in Hubei province with the initial time on January 30 which shows the severe infectivity of COVID-19, and verify that the current isolation method effectively contains the transmission of COVID-19. Under the isolation of healthy people, confirmed cases and contact tracing measures, we find a noteworthy phenomenon that is the second epidemic of COVID-19 and estimate the peak time and value and the cumulative number of cases. Simulations show that the contact tracing measures can efficiently contain the transmission of the second epidemic of COVID-19. With the isolation of all susceptible people or all infectious people or both, there is no second epidemic of COVID-19. Furthermore, resumption of work and study can increase the transmission risk of the second epidemic of COVID-19.

Propagation of stochastic travelling waves of cooperative systems with noise
Hao Wen, Jianhua Huang and Yuhong Li
2022, 27(10): 5779-5803 doi: 10.3934/dcdsb.2021295 +[Abstract](658) +[HTML](211) +[PDF](372.25KB)
Abstract:

We consider the cooperative system driven by a multiplicative It\^o type white noise. The existence and their approximations of the travelling wave solutions are proven. With a moderately strong noise, the travelling wave solutions are constricted by choosing a suitable marker of wavefront. Moreover, the stochastic Feynman-Kac formula, sup-solution, sub-solution and equilibrium points of the dynamical system corresponding to the stochastic cooperative system are utilized to estimate the asymptotic wave speed, which is closely related to the white noise.

Global strong solution to the nonhomogeneous micropolar fluid equations with large initial data and vacuum
Xin Zhong
2022, 27(10): 5805-5820 doi: 10.3934/dcdsb.2021296 +[Abstract](540) +[HTML](181) +[PDF](371.22KB)
Abstract:

We study the Cauchy problem of nonhomogeneous micropolar fluid equations with zero density at infinity in the whole plane \begin{document}$ \mathbb{R}^2 $\end{document}. We derive the global existence and uniqueness of strong solutions if the initial density decays not too slowly at infinity. Note that the initial data can be arbitrarily large and the initial density can contain vacuum states and even have compact support. Our method relies upon the delicate weighted energy estimates and the structural characteristics of the system under consideration.

Orbital dynamics on invariant sets of contact Hamiltonian systems
Qihuai Liu and Pedro J. Torres
2022, 27(10): 5821-5844 doi: 10.3934/dcdsb.2021297 +[Abstract](636) +[HTML](223) +[PDF](758.28KB)
Abstract:

In this paper, we shall give new insights on dynamics of contact Hamiltonian flows, which are gaining importance in several branches of physics as they model a dissipative behaviour. We divide the contact phase space into three parts, which are corresponding to three differential invariant sets \begin{document}$ \Omega_\pm, \Omega_0 $\end{document}. On the invariant sets \begin{document}$ \Omega_\pm $\end{document}, under some geometric conditions, the contact Hamiltonian system is equivalent to a Hamiltonian system via the Hölder transformation. The invariant set \begin{document}$ \Omega_0 $\end{document} may be composed of several equilibrium points and heteroclinic orbits connecting them, on which contact Hamiltonian system is conservative. Moreover, we have shown that, under general conditions, the zero energy level domain is a domain of attraction. In some cases, such a domain of attraction does not have nontrivial periodic orbits. Some interesting examples are presented.

A predator-prey model with cooperative hunting in the predator and group defense in the prey
Yanfei Du, Ben Niu and Junjie Wei
2022, 27(10): 5845-5881 doi: 10.3934/dcdsb.2021298 +[Abstract](756) +[HTML](212) +[PDF](1764.09KB)
Abstract:

In this paper we propose a predator-prey model with a non-differentiable functional response in which the prey exhibits group defense and the predator exhibits cooperative hunting. There is a separatrix curve dividing the phase portrait. The species with initial population above the separatrix result in extinction of prey in finite time, and the species with initial population below it can coexist, oscillate sustainably or leave the prey surviving only. Detailed bifurcation analysis is carried out to explore the effect of cooperative hunting in the predator and aggregation in the prey on the existence and stability of the coexistence state as well as the dynamics of system. The model undergoes transcritical bifurcation, Hopf bifurcation, homoclinic (heteroclinic) bifurcation, saddle-node bifurcation, and Bogdanov-Takens bifurcation, and through numerical simulations it is found that it possesses rich dynamics including bubble loop of limit cycles, and open ended branch of periodic orbits disappearing through a homoclinic cycle or a loop of heteroclinic orbits. Also, a continuous transition of different types of Hopf branches are investigated which forms a global picture of Hopf bifurcation in the model.

The cyclicity of a class of quadratic reversible centers defining elliptic curves
Guilin Ji and Changjian Liu
2022, 27(10): 5883-5903 doi: 10.3934/dcdsb.2021299 +[Abstract](753) +[HTML](197) +[PDF](770.14KB)
Abstract:

In this paper, the cyclicity of period annulus of an one-parameter family quadratic reversible system under quadratic perturbations is studied which is equivalent to the number of zeros of any nontrivial linear combination of three Abelian integrals. By the criteria established in [28] and the asymptotic expansions of Abelian integrals, we obtain that the cyclicity is two when the parameter in \begin{document}$ (-\infty,-2)\cup[-\frac{8}{5},+\infty) $\end{document}. Moreover, we develop new criteria which combined with the asymptotic expansions of Abelian integrals show that the cyclicity is three when the parameter belongs to \begin{document}$ (-2,-\frac{8}{5}) $\end{document}.

Numerical study of the Serre-Green-Naghdi equations and a fully dispersive counterpart
Vincent Duchêne and Christian Klein
2022, 27(10): 5905-5933 doi: 10.3934/dcdsb.2021300 +[Abstract](640) +[HTML](195) +[PDF](5092.14KB)
Abstract:

We perform numerical experiments on the Serre-Green-Naghdi (SGN) equations and a fully dispersive "Whitham-Green-Naghdi" (WGN) counterpart in dimension 1. In particular, solitary wave solutions of the WGN equations are constructed and their stability, along with the explicit ones of the SGN equations, is studied. Additionally, the emergence of modulated oscillations and the possibility of a blow-up of solutions in various situations is investigated.

We argue that a simple numerical scheme based on a Fourier spectral method combined with the Krylov subspace iterative technique GMRES to address the elliptic problem and a fourth order explicit Runge-Kutta scheme in time allows to address efficiently even computationally challenging problems.

Stability analysis of stochastic delay differential equations with Markovian switching driven by Lévy noise
Yanqiang Chang and Huabin Chen
2022, 27(10): 5935-5952 doi: 10.3934/dcdsb.2021301 +[Abstract](818) +[HTML](191) +[PDF](421.94KB)
Abstract:

In this paper, the existence and uniquenesss, stability analysis for stochastic delay differential equations with Markovian switching driven by L\begin{document}$ \acute{e} $\end{document}vy noise are studied. The existence and uniqueness of such equations is simply shown by using the Picard iterative methodology. By using the generalized integral, the Lyapunov-Krasovskii function and the theory of stochastic analysis, the exponential stability in \begin{document}$ p $\end{document}th(\begin{document}$ p\geq2 $\end{document}) for stochastic delay differential equations with Markovian switching driven by L\begin{document}$ \acute{e} $\end{document}vy noise is firstly investigated. The almost surely exponential stability is also applied. Finally, an example is provided to verify our results derived.

Global dynamics of a Huanglongbing model with a periodic latent period
Yan Hong, Xiuxiang Liu and Xiao Yu
2022, 27(10): 5953-5976 doi: 10.3934/dcdsb.2021302 +[Abstract](598) +[HTML](202) +[PDF](774.1KB)
Abstract:

Huanglongbing (HLB) is a disease of citrus that caused by phloem-restricted bacteria of the Candidatus Liberibacter group. In this paper, we present a HLB transmission model to investigate the effects of temperature-dependent latent periods and seasonality on the spread of HLB. We first establish disease free dynamics in terms of a threshold value \begin{document}$ R^p_0 $\end{document}, and then introduce the basic reproduction number \begin{document}$ \mathcal{R}_0 $\end{document} and show the threshold dynamics of HLB with respect to \begin{document}$ R^p $\end{document} and \begin{document}$ \mathcal{R}_0 $\end{document}. Numerical simulations are further provided to illustrate our analytic results.

Continuity of random attractors on a topological space and fractional delayed FitzHugh-Nagumo equations with WZ-noise
Yangrong Li, Shuang Yang and Guangqing Long
2022, 27(10): 5977-6008 doi: 10.3934/dcdsb.2021303 +[Abstract](492) +[HTML](204) +[PDF](451.48KB)
Abstract:

We study the continuity of a family of random attractors parameterized in a topological space (perhaps non-metrizable). Under suitable conditions, we prove that there is a residual dense subset \begin{document}$ \Lambda^* $\end{document} of the parameterized space such that the binary map \begin{document}$ (\lambda, s)\mapsto A_\lambda(\theta_s \omega) $\end{document} is continuous at all points of \begin{document}$ \Lambda^*\times \mathbb{R} $\end{document} with respect to the Hausdorff metric. The proofs are based on the generalizations of Baire residual Theorem (by Hoang et al. PAMS, 2015), Baire density Theorem and a convergence theorem of random dynamical systems from a complete metric space to the general topological space, and thus the abstract result, even restricted in the deterministic case, is stronger than those in literature. Finally, we establish the residual dense continuity and full upper semi-continuity of random attractors for the random fractional delayed FitzHugh-Nagumo equation driven by nonlinear Wong-Zakai noise, where the size of noise belongs to the parameterized space \begin{document}$ (0, \infty] $\end{document} and the infinity of noise means that the equation is deterministic.

Blowup results for the fractional Schrödinger equation without gauge invariance
Qihong Shi, Congming Peng and Qingxuan Wang
2022, 27(10): 6009-6022 doi: 10.3934/dcdsb.2021304 +[Abstract](601) +[HTML](205) +[PDF](376.62KB)
Abstract:

This paper is concerned with the nonexistence of global solutions to the fractional Schrödinger equations with order \begin{document}$ \alpha $\end{document} and nongauge power type nonlinearity \begin{document}$ |u|^p $\end{document} for any space dimensions, where \begin{document}$ \alpha\in (0, 2] $\end{document} is assumed to be any fractional numbers. A modified test function is employed to overcome some difficulties caused by the fractional operator and to establish blowup results. Some restrictions with respect to \begin{document}$ \alpha, p $\end{document} and initial data in the previous literature are removed.

On the well-posedness of the anisotropically-reduced two-dimensional Kuramoto-Sivashinsky Equation
David Massatt
2022, 27(10): 6023-6036 doi: 10.3934/dcdsb.2021305 +[Abstract](492) +[HTML](188) +[PDF](315.31KB)
Abstract:

We address the global existence and uniqueness of solutions for the anisotropically reduced 2D Kuramoto-Sivashinsky equations in a periodic domain with initial data \begin{document}$ u_{01} \in L^2 $\end{document} and \begin{document}$ u_{02} \in H^{-1 + \eta} $\end{document} for \begin{document}$ \eta > 0 $\end{document}.

Boundedness in a two species attraction-repulsion chemotaxis system with two chemicals
Aichao Liu, Binxiang Dai and Yuming Chen
2022, 27(10): 6037-6062 doi: 10.3934/dcdsb.2021306 +[Abstract](543) +[HTML](347) +[PDF](424.67KB)
Abstract:

This paper deals with a class of attraction-repulsion chemotaxis systems in a smoothly bounded domain. When the system is parabolic-elliptic-parabolic-elliptic and the domain is \begin{document}$ n $\end{document}-dimensional, if the repulsion effect is strong enough then the solutions of the system are globally bounded. Meanwhile, when the system is fully parabolic and the domain is either one-dimensional or two-dimensional, the system also possesses a globally bounded classical solution.

On weak (measure-valued)-strong uniqueness for compressible MHD system with non-monotone pressure law
Yu Liu and Ting Zhang
2022, 27(10): 6063-6081 doi: 10.3934/dcdsb.2021307 +[Abstract](624) +[HTML](178) +[PDF](397.73KB)
Abstract:

In this paper, we define a renormalized dissipative measure-valued (rDMV) solution of the compressible magnetohydrodynamics (MHD) equations with non-monotone pressure law. We prove the existence of the rDMV solutions and establish a suitable relative energy inequality. And we obtain the weak (measure-valued)-strong uniqueness property of this rDMV solution with the help of the relative energy inequality.

Emergent behaviors of discrete Lohe aggregation flows
Hyungjun Choi, Seung-Yeal Ha and Hansol Park
2022, 27(10): 6083-6123 doi: 10.3934/dcdsb.2021308 +[Abstract](601) +[HTML](181) +[PDF](523.4KB)
Abstract:

The Lohe sphere model and the Lohe matrix model are prototype continuous aggregation models on the unit sphere and the unitary group, respectively. These models have been extensively investigated in recent literature. In this paper, we propose several discrete counterparts for the continuous Lohe type aggregation models and study their emergent behaviors using the Lyapunov function method. For suitable discretization of the Lohe sphere model, we employ a scheme consisting of two steps. In the first step, we solve the first-order forward Euler scheme, and in the second step, we project the intermediate state onto the unit sphere. For this discrete model, we present a sufficient framework leading to the complete state aggregation in terms of system parameters and initial data. For the discretization of the Lohe matrix model, we use the Lie group integrator method, Lie-Trotter splitting method and Strang splitting method to propose three discrete models. For these models, we also provide several analytical frameworks leading to complete state aggregation and asymptotic state-locking.

Exponential ergodicity for regime-switching diffusion processes in total variation norm
Jun Li and Fubao Xi
2022, 27(10): 6125-6146 doi: 10.3934/dcdsb.2021309 +[Abstract](486) +[HTML](321) +[PDF](376.9KB)
Abstract:

We investigate the long time behavior for a class of regime-switching diffusion processes. Based on direct evaluation of moments and exponential functionals of hitting time of the underlying process, we adopt coupling method to obtain existence and uniqueness of the invariant probability measure and establish explicit exponential bounds for the rate of convergence to the invariant probability measure in total variation norm. In addition, we provide some concrete examples to illustrate our main results which reveal impact of random switching on stochastic stability and convergence rate of the system.

Multi-valued random dynamics of stochastic wave equations with infinite delays
Jingyu Wang, Yejuan Wang and Tomás Caraballo
2022, 27(10): 6147-6172 doi: 10.3934/dcdsb.2021310 +[Abstract](467) +[HTML](238) +[PDF](409.37KB)
Abstract:

This paper is devoted to the asymptotic behavior of solutions to a non-autonomous stochastic wave equation with infinite delays and additive white noise. The nonlinear terms of the equation are not expected to be Lipschitz continuous, but only satisfy continuity assumptions along with growth conditions, under which the uniqueness of the solutions may not hold. Using the theory of multi-valued non-autonomous random dynamical systems, we prove the existence and measurability of a compact global pullback attractor.

Statistical solution and Liouville type theorem for coupled Schrödinger-Boussinesq equations on infinite lattices
Congcong Li, Chunqiu Li and Jintao Wang
2022, 27(10): 6173-6196 doi: 10.3934/dcdsb.2021311 +[Abstract](473) +[HTML](183) +[PDF](357.48KB)
Abstract:

In this article, we are concerned with statistical solutions for the nonautonomous coupled Schrödinger-Boussinesq equations on infinite lattices. Firstly, we verify the existence of a pullback-\begin{document}$ {\mathcal{D}} $\end{document} attractor and establish the existence of a unique family of invariant Borel probability measures carried by the pullback-\begin{document}$ {\mathcal{D}} $\end{document} attractor for this lattice system. Then, it will be shown that the family of invariant Borel probability measures is a statistical solution and satisfies a Liouville type theorem. Finally, we illustrate that the invariant property of the statistical solution is indeed a particular case of the Liouville type theorem.

Boundary layer effects on ionic flows via Poisson-Nernst-Planck systems with nonuniform ion sizes
Jianing Chen and Mingji Zhang
2022, 27(10): 6197-6216 doi: 10.3934/dcdsb.2021312 +[Abstract](493) +[HTML](228) +[PDF](459.65KB)
Abstract:

We study a one-dimensional Poisson-Nernst-Planck model with two oppositely charged particles, zero permanent charges and nonuniform finite ion sizes through a local hard-sphere model. Of particular interest is to examine the boundary layer effects on ionic flows systematically in terms of individual fluxes, the total flow rate of charges (current-voltage relations) and the total flow rate of matter. This is particularly important because boundary layers of charge are particularly likely to create artifacts over long distances, and this could dramatically affect the behavior of ionic flows. Several critical potentials are identified, which play unique and critical roles in examining the dynamics of ionic flows. Some can be estimated experimentally. Numerical simulations are performed for a better understanding and further illustrating our analytical results. We believe the analysis can provide complementary information of the qualitative properties of ionic flows and help one better understand the mechanism of ionic flow through membrane channels.

2021 Impact Factor: 1.497
5 Year Impact Factor: 1.527
2021 CiteScore: 2.3

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