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Discrete & Continuous Dynamical Systems - B

December 2018 , Volume 23 , Issue 10

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Spreading speed and traveling waves for a non-local delayed reaction-diffusion system without quasi-monotonicity
Zhenguo Bai and Tingting Zhao
2018, 23(10): 4063-4085 doi: 10.3934/dcdsb.2018126 +[Abstract](670) +[HTML](443) +[PDF](519.85KB)

A non-local delayed reaction-diffusion model with a quiescent stage is investigated. It is shown that the spreading speed of this model without quasi-monotonicity is linearly determinate and coincides with the minimal wave speed of traveling waves.

Time asymptotics of structured populations with diffusion and dynamic boundary conditions
Mustapha Mokhtar-Kharroubi and Quentin Richard
2018, 23(10): 4087-4116 doi: 10.3934/dcdsb.2018127 +[Abstract](582) +[HTML](367) +[PDF](467.24KB)

This work revisits and extends in various directions a work by J.Z. Farkas and P. Hinow (Math. Biosc and Eng, 8 (2011) 503-513) on structured populations models (with bounded sizes) with diffusion and generalized Wentzell boundary conditions. In particular, we provide first a self-contained \begin{document}$L^{1}$\end{document} generation theory making explicit the domain of the generator. By using Hopf maximum principle, we show that the semigroup is always irreducible regardless of the reproduction function. By using weak compactness arguments, we show first a stability result of the essential type and then deduce that the semigroup has a spectral gap and consequently the asynchronous exponential growth property. Finally, we show how to extend this theory to models with arbitrary sizes and point out an open problem pertaining to this extension.

On a free boundary problem for a nonlocal reaction-diffusion model
Jia-Feng Cao, Wan-Tong Li and Meng Zhao
2018, 23(10): 4117-4139 doi: 10.3934/dcdsb.2018128 +[Abstract](583) +[HTML](380) +[PDF](474.87KB)

This paper is concerned with the spreading or vanishing dichotomy of a species which is characterized by a reaction-diffusion Volterra model with nonlocal spatial convolution and double free boundaries. Compared with classical reaction-diffusion equations, the main difficulty here is the lack of a comparison principle in nonlocal reaction-diffusion equations. By establishing some suitable comparison principles over some different parabolic regions, we get the sufficient conditions that ensure the species spreading or vanishing, as well as the estimates of the spreading speed if species spreading happens. Particularly, we establish the global attractivity of the unique positive equilibrium by a method of successive improvement of lower and upper solutions.

Global phase portraits of a degenerate Bogdanov-Takens system with symmetry (Ⅱ)
Hebai Chen and Xingwu Chen
2018, 23(10): 4141-4170 doi: 10.3934/dcdsb.2018130 +[Abstract](627) +[HTML](460) +[PDF](1375.03KB)

The degenerate Bogdanov-Takens system \begin{document}$\dot x = y-(a_1x+a_2x^3),~\dot y = a_3x+a_4x^3$\end{document} has two normal forms, one of which is investigated in [Disc. Cont. Dyn. Syst. B (22)2017,1273-1293] and global behavior is analyzed for general parameters. To continue this work, in this paper we study the other normal form and perform all global phase portraits on the Poincaré disc. Since the parameters are not restricted to be sufficiently small, some classic bifurcation methods for small parameters, such as the Melnikov method, are no longer valid. We find necessary and sufficient conditions for existences of limit cycles and homoclinic loops respectively by constructing a distance function among orbits on the vertical isocline curve and further give the number of limit cycles for parameters in different regions. Finally we not only give the global bifurcation diagram, where global existences and monotonicities of the homoclinic bifurcation curve and the double limit cycle bifurcation curve are proved, but also classify all global phase portraits.

On the Cauchy problem for the XFEL Schrödinger equation
Binhua Feng and Dun Zhao
2018, 23(10): 4171-4186 doi: 10.3934/dcdsb.2018131 +[Abstract](511) +[HTML](368) +[PDF](412.45KB)

In this paper, we consider the Cauchy problem for the nonlinear Schrödinger equation with a time-dependent electromagnetic field and a Coulomb potential, which arises as an effective single particle model in X-ray free electron lasers(XFEL). We firstly show the local and global well-posedness for the Cauchy problem under the assumption that the magnetic potential is unbounded and time-dependent, and then obtain the regularity by a fixed point argument.

A perturbed fourth order elliptic equation with negative exponent
Zongming Guo and Long Wei
2018, 23(10): 4187-4205 doi: 10.3934/dcdsb.2018132 +[Abstract](530) +[HTML](373) +[PDF](415.14KB)

By a new type of comparison principle for a fourth order elliptic problem in general domains, we investigate the structure of positive solutions to Navier boundary value problems of a perturbed fourth order elliptic equation with negative exponent, which arises in the study of the deflection of charged plates in electrostatic actuators in the modeling of electrostatic micro-electromechanical systems (MEMS). It is seen that the structure of solutions relies on the boundary values. The global branches of solutions to the Navier boundary value problems are established. We also show that the behaviors of these branches are relatively "stable" with respect to the Navier boundary values.

Carleman estimate for solutions to a degenerate convection-diffusion equation
Chunpeng Wang, Yanan Zhou, Runmei Du and Qiang Liu
2018, 23(10): 4207-4222 doi: 10.3934/dcdsb.2018133 +[Abstract](505) +[HTML](359) +[PDF](448.88KB)

This paper concerns a control system governed by a convection-diffusion equation, which is weakly degenerate at the boundary. In the governing equation, the convection is independent of the degeneracy of the equation and cannot be controlled by the diffusion. The Carleman estimate is established by means of a suitable transformation, by which the diffusion and the convection are transformed into a complex union, and complicated and detailed computations. Then the observability inequality is proved and the control system is shown to be null controllable.

Global stability of a diffusive and delayed HBV infection model with HBV DNA-containing capsids and general incidence rate
Ting Guo, Haihong Liu, Chenglin Xu and Fang Yan
2018, 23(10): 4223-4242 doi: 10.3934/dcdsb.2018134 +[Abstract](603) +[HTML](414) +[PDF](1076.74KB)

The aim of this paper is to study the dynamics of a new chronic HBV infection model that includes spatial diffusion, three time delays and a general incidence function. First, we analyze the well-posedness of the initial value problem of the model in the bounded domain. Then, we define a threshold parameter \begin{document}$R_{0}$\end{document} called the basic reproduction number and show that our model admits two possible equilibria, namely the infection-free equilibrium \begin{document}$E_{1}$\end{document} as well as the chronic infection equilibrium \begin{document}$E_{2}$\end{document}. Further, by constructing two appropriate Lyapunov functionals, we prove that \begin{document}$E_{1}$\end{document} is globally asymptotically stable when \begin{document}$R_{0}<1$\end{document}, corresponding to the viruses are cleared and the disease dies out; if \begin{document}$R_{0}>1$\end{document}, then \begin{document}$E_{1}$\end{document} becomes unstable and the equilibrium point \begin{document}$E_{2}$\end{document} appears and is globally asymptotically stable, which means that the viruses persist in the host and the infection becomes chronic. An application is provided to confirm the main theoretical results. Additionally, it is worth saying that, our results suggest theoretically useful method to control HBV infection and these results can be applied to a variety of possible incidence functions presented in a series of other papers.

Upper and lower bounds for the blow-up time in quasilinear reaction diffusion problems
Juntang Ding and Xuhui Shen
2018, 23(10): 4243-4254 doi: 10.3934/dcdsb.2018135 +[Abstract](507) +[HTML](381) +[PDF](397.38KB)

In this paper, we consider a quasilinear reaction diffusion equation with Neumann boundary conditions in a bounded domain. Basing on Sobolev inequality and differential inequality technique, we obtain upper and lower bounds for the blow-up time of the solution. An example is also given to illustrate the abstract results obtained of this paper.

Invasion and coexistence of competition-diffusion-advection system with heterogeneous vs homogeneous resources
Benlong Xu and Hongyan Jiang
2018, 23(10): 4255-4266 doi: 10.3934/dcdsb.2018136 +[Abstract](488) +[HTML](339) +[PDF](378.83KB)

This paper mainly study the dynamics of a Lotka-Volterra reaction-diffusion-advection model for two competing species which disperse by both random diffusion and advection along environmental gradient. In this model, the species are assumed to be identical except spatial resource distribution: heterogeneity vs homogeneity. It is shown that the species with heterogeneous resources distribution is always in a better position, that is, it can always invade when rare. The ratio of advection strength and diffusion rate of the species with heterogeneous distribution plays a crucial role in the dynamics behavior of the system. Some conditions of invasion, driving extinction, and coexistence are given in term of this ratio and the diffusion rate of its competitor.

Dynamics of weak solutions for the three dimensional Navier-Stokes equations with nonlinear damping
Fang Li, Bo You and Yao Xu
2018, 23(10): 4267-4284 doi: 10.3934/dcdsb.2018137 +[Abstract](509) +[HTML](378) +[PDF](488.66KB)

The main objective of this paper is to study the existence of a finite dimensional global attractor for the three dimensional Navier-Stokes equations with nonlinear damping for \begin{document}$r>4.$\end{document} Motivated by the idea of [1], even though we can obtain the existence of a global attractor for \begin{document}$r≥ 2$\end{document} by the multi-valued semi-flow, it is very difficult to provide any information about its fractal dimension. Therefore, we prove the existence of a global attractor in H and provide the upper bound of its fractal dimension by the methods of \begin{document}$\ell$\end{document}-trajectories in this paper.

Exponential stability of an incompressible non-Newtonian fluid with delay
Linfang Liu, Tomás Caraballo and Xianlong Fu
2018, 23(10): 4285-4303 doi: 10.3934/dcdsb.2018138 +[Abstract](537) +[HTML](406) +[PDF](591.07KB)

The existence and uniqueness of stationary solutions to an incompressible non-Newtonian fluid are first established. The exponential stability of steady-state solutions is then analyzed by means of four different approaches. The first is the classical Lyapunov function method, while the second one is based on a Razumikhin type argument. Then, a method relying on the construction of Lyapunov functionals and another one using a Gronwall-like lemma are also exploited to study the stability, respectively. Some comments concerning several open research directions about this model are also included.

Finite dimensionality of global attractor for the solutions to 3D viscous primitive equations of large-scale moist atmosphere
Boling Guo and Guoli Zhou
2018, 23(10): 4305-4327 doi: 10.3934/dcdsb.2018160 +[Abstract](306) +[HTML](250) +[PDF](457.5KB)

Under general boundary conditions we consider the finiteness of the Hausdorff and fractal dimensions of the global attractor for the strong solution of the 3D moist primitive equations with viscosity. Firstly, we obtain time-uniform estimates of the first-order time derivative of the strong solutions in \begin{document}$L^2(\mho)$\end{document}. Then, to prove the finiteness of the Hausdorff and fractal dimensions of the global attractor, the common method is to obtain the uniform boundedness of the strong solution in \begin{document}$H^2(\mho)$\end{document} to establish the squeezing property of the solution operator. But it is difficult to achieve due to the boundary conditions and complicated structure of the 3D moist primitive equations. To overcome the difficulties, we try to use the uniform boundedness of the derivative of the strong solutions with respect to time \begin{document}$t$\end{document} in \begin{document}$L^2(\mho)$\end{document} to prove the uniform continuity of the global attractor. Finally, using the uniform continuity of the global attractor we establish the squeezing property of the solution operator which implies the finiteness of the Hausdorff and fractal dimensions of the global attractor.

Stabilization of turning processes using spindle feedback with state-dependent delay
Qingwen Hu and Huan Zhang
2018, 23(10): 4329-4360 doi: 10.3934/dcdsb.2018167 +[Abstract](346) +[HTML](209) +[PDF](374.58KB)

We develop a stabilization strategy of turning processes by means of delayed spindle control. We show that turning processes which contain intrinsic state-dependent delays can be stabilized by a spindle control with state-dependent delay, and develop analytical description of the stability region in the parameter space. Numerical simulations stability region are also given to illustrate the general results.

Well-posedeness and energy decay of solutions to a bresse system with a boundary dissipation of fractional derivative type
Abbes Benaissa and Abderrahmane Kasmi
2018, 23(10): 4361-4395 doi: 10.3934/dcdsb.2018168 +[Abstract](402) +[HTML](238) +[PDF](532.0KB)

We consider the Bresse system with three control boundary conditions of fractional derivative type. We prove the polynomial decay result with an estimation of the decay rates. Our result is established using the semigroup theory of linear operators and a result obtained by Borichev and Tomilov.

Existence and uniqueness of global classical solutions to a two dimensional two species cancer invasion haptotaxis model
Jan Giesselmann, Niklas Kolbe, Mária Lukáčová-Medvi${\rm{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over d} }}$ová and Nikolaos Sfakianakis
2018, 23(10): 4397-4431 doi: 10.3934/dcdsb.2018169 +[Abstract](361) +[HTML](224) +[PDF](988.3KB)

We consider a haptotaxis cancer invasion model that includes two families of cancer cells. Both families migrate on the extracellular matrix and proliferate. Moreover the model describes an epithelial-to-mesenchymal-like transition between the two families, as well as a degradation and a self-reconstruction process of the extracellular matrix.

We prove in two dimensional space positivity and conditional global existence and uniqueness of the classical solutions of the problem for large initial data.

A vicinal surface model for epitaxial growth with logarithmic free energy
Yuan Gao, Hangjie Ji, Jian-Guo Liu and Thomas P. Witelski
2018, 23(10): 4433-4453 doi: 10.3934/dcdsb.2018170 +[Abstract](311) +[HTML](227) +[PDF](1168.3KB)

We study a continuum model for solid films that arises from the modeling of one-dimensional step flows on a vicinal surface in the attachment-detachment-limited regime. The resulting nonlinear partial differential equation, \begin{document}$u_t = -u^2(u^3+α u)_{hhhh}$\end{document}, gives the evolution for the surface slope \begin{document}$u$\end{document} as a function of the local height \begin{document}$h$\end{document} in a monotone step train. Subject to periodic boundary conditions and positive initial conditions, we prove the existence, uniqueness and positivity of global strong solutions to this PDE using two Lyapunov energy functions. The long time behavior of \begin{document}$u$\end{document} converging to a constant that only depends on the initial data is also investigated both analytically and numerically.

Convergence rate of strong approximations of compound random maps, application to SPDEs
Emmanuel Gobet and Mohamed Mrad
2018, 23(10): 4455-4476 doi: 10.3934/dcdsb.2018171 +[Abstract](297) +[HTML](256) +[PDF](639.96KB)

We consider a random map \begin{document}$x\mapsto F(ω,x)$\end{document} and a random variable \begin{document}$Θ(ω)$\end{document}, and we denote by \begin{document}${{F}^{N}}(ω,x) $\end{document} and \begin{document}$ {{\Theta }^{N}}(ω) $\end{document} their approximations: We establish a strong convergence result, in \begin{document}${\bf{L}}_p$\end{document}-norms, of the compound approximation \begin{document}${{F}^{N}}(ω,{{\Theta }^{N}}(ω) )$\end{document} to the compound variable \begin{document}$F(ω,Θ(ω)) $\end{document}, in terms of the approximations of \begin{document}$F$\end{document} and \begin{document}$Θ$\end{document}. We then apply this result to the composition of two Stochastic Differential Equations (SDEs) through their initial conditions, which can give a way to solve some Stochastic Partial Differential Equations (SPDEs), in particular those from stochastic utilities.

Convergence of solutions to inverse problems for a class of variational-hemivariational inequalities
Stanisław Migórski and Biao Zeng
2018, 23(10): 4477-4498 doi: 10.3934/dcdsb.2018172 +[Abstract](383) +[HTML](210) +[PDF](475.26KB)

The paper investigates an inverse problem for a stationary variational-hemivariational inequality. The solution of the variational-hemivariational inequality is approximated by its penalized version. We prove existence of solutions to inverse problems for both the initial inequality problem and the penalized problem. We show that optimal solutions to the inverse problem for the penalized problem converge, up to a subsequence, when the penalty parameter tends to zero, to an optimal solution of the inverse problem for the initial variational-hemivariational inequality. The results are illustrated by a mathematical model of a nonsmooth contact problem from elasticity.

Theoretical analysis on a diffusive SIR epidemic model with nonlinear incidence in a heterogeneous environment
Chengxia Lei, Fujun Li and Jiang Liu
2018, 23(10): 4499-4517 doi: 10.3934/dcdsb.2018173 +[Abstract](362) +[HTML](272) +[PDF](455.28KB)

In this paper, we deal with a diffusive SIR epidemic model with nonlinear incidence of the form \begin{document}$I^pS^q$\end{document} for \begin{document}$0<p≤1$\end{document} in a heterogeneous environment. We establish the boundedness and uniform persistence of solutions to the system, and the global stability of the constant endemic equilibrium in the case of homogeneous environment. When the spatial environment is heterogeneous, we determine the asymptotic profile of endemic equilibrium if the diffusion rate of the susceptible or infected population is small. Our theoretical analysis shows that, different from the studies of [1,28,38,44] for the SIS models, restricting the movement of the susceptible or infected population can not lead to the extinction of infectious disease for such an SIR system.

Numerical study of phase transition in van der Waals fluid
Qiaolin He, Chang Liu and Xiaoding Shi
2018, 23(10): 4519-4540 doi: 10.3934/dcdsb.2018174 +[Abstract](325) +[HTML](241) +[PDF](9009.6KB)

In this article, we use a relaxation scheme for conservation laws to study liquid-vapor phase transition modeled by the van der Waals equation, which introduces a small parameter $ε$ and a new variable. We solve the relaxation system in Lagrangian coordinates for one dimension and solve the system in Eulerian coordinates for two dimension. A second order TVD Runge-Kutta splitting scheme is used in time discretization and upwind or MUSCL scheme is used in space discretization. The long time behavior of the fluid is numerically investigated. If the initial data belongs to elliptic region, the solution converges to two Maxwell states. When the initial data lies in metastable region, the solution either remains in the same phase or converges to the Maxwell states depending to the initial perturbation. If the initial state is in the stable region, the solution remains in that region for all time.

Identification of generic stable dynamical systems taking a nonlinear differential approach
Mahdi Khajeh Salehani
2018, 23(10): 4541-4555 doi: 10.3934/dcdsb.2018175 +[Abstract](526) +[HTML](300) +[PDF](467.84KB)

Identifying new stable dynamical systems, such as generic stable mechanical or electrical control systems, requires questing for the desired systems parameters that introduce such systems. In this paper, a systematic approach to construct generic stable dynamical systems is proposed. In fact, our approach is based on a simple identification method in which we intervene directly with the dynamics of our system by considering a continuous \begin{document}$1$\end{document}-parameter family of system parameters, being parametrized by a positive real variable \begin{document}$\ell$\end{document}, and then identify the desired parameters that introduce a generic stable dynamical system by analyzing the solutions of a special system of nonlinear functional-differential equations associated with the \begin{document}$\ell$\end{document}-varying parameters. We have also investigated the reliability and capability of our proposed approach.

To illustrate the utility of our result and as some applications of the nonlinear differential approach proposed in this paper, we conclude with considering a class of coupled spring-mass-dashpot systems, as well as the compartmental systems - the latter of which provide a mathematical model for many complex biological and physical processes having several distinct but interacting phases.

A reaction-diffusion-advection SIS epidemic model in a spatially-temporally heterogeneous environment
Danhua Jiang, Zhi-Cheng Wang and Liang Zhang
2018, 23(10): 4557-4578 doi: 10.3934/dcdsb.2018176 +[Abstract](455) +[HTML](212) +[PDF](477.44KB)

In this paper, we study the effects of diffusion and advection for an SIS epidemic reaction-diffusion-advection model in a spatially and temporally heterogeneous environment. We introduce the basic reproduction number \begin{document} $\mathcal{R}_{0}$ \end{document} and establish the threshold-type results on the global dynamics in terms of \begin{document} $\mathcal{R}_{0}$ \end{document}. Some general qualitative properties of \begin{document} $\mathcal{R}_{0}$ \end{document} are presented, then the paper is devoted to studying how the advection and diffusion of the infected individuals affect the reproduction number \begin{document} $\mathcal{R}_{0}$ \end{document} for the special case that \begin{document} $γ(x,t)-β(x,t) = V(x,t)$ \end{document} is monotone with respect to spatial variable \begin{document} $x$ \end{document}. Our results suggest that if \begin{document} $V_{x}(x,t)≥0,\not\equiv0$ \end{document} and \begin{document} $V(x, t)$ \end{document} changes sign about \begin{document} $x$ \end{document}, the advection is beneficial to eliminate the disease, whereas if \begin{document} $V_{x}(x,t)≤0,\not\equiv0$ \end{document} and \begin{document} $V(x, t)$ \end{document} changes sign about \begin{document} $x$ \end{document}, the advection is bad for the elimination of disease.

Prevalence of stable periodic solutions in the forced relativistic pendulum equation
Feng Wang, Jifeng Chu and Zaitao Liang
2018, 23(10): 4579-4594 doi: 10.3934/dcdsb.2018177 +[Abstract](369) +[HTML](217) +[PDF](400.41KB)

We study the prevalence of stable periodic solutions of the forced relativistic pendulum equation for external force which guarantees the existence of periodic solutions. We prove the results for a general planar system.

Time-dependent asymptotic behavior of the solution for plate equations with linear memory
Tingting Liu and Qiaozhen Ma
2018, 23(10): 4595-4616 doi: 10.3934/dcdsb.2018178 +[Abstract](405) +[HTML](229) +[PDF](509.83KB)

In this article, we consider the long-time behavior of solutions for the plate equation with linear memory. Within the theory of process on time-dependent spaces, we investigate the existence of the time-dependent attractor by using the operator decomposition technique and compactness of translation theorem and more detailed estimates. Furthermore, the asymptotic structure of time-dependent attractor, which converges to the attractor of fourth order parabolic equation with memory, is proved. Besides, we obtain a further regular result.

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