ISSN:

1531-3492

eISSN:

1553-524X

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## Discrete & Continuous Dynamical Systems - B

December 2018 , Volume 23 , Issue 10

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*+*[Abstract](1222)

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**Abstract:**

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.

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**Abstract:**

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

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**Abstract:**

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.

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**Abstract:**

The degenerate Bogdanov-Takens system

*+*[Abstract](1025)

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**Abstract:**

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.

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**Abstract:**

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.

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**Abstract:**

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.

*+*[Abstract](1114)

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**Abstract:**

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

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**Abstract:**

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.

*+*[Abstract](913)

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**Abstract:**

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.

*+*[Abstract](962)

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**Abstract:**

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 *H* and provide the upper bound of its fractal dimension by the methods of

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**Abstract:**

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.

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**Abstract:**

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

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**Abstract:**

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.

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**Abstract:**

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.

*+*[Abstract](817)

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**Abstract:**

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.

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**Abstract:**

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,

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**Abstract:**

We consider a random map

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**Abstract:**

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.

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**Abstract:**

In this paper, we deal with a diffusive SIR epidemic model with nonlinear incidence of the form

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**Abstract:**

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.

*+*[Abstract](875)

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**Abstract:**

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

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.

*+*[Abstract](861)

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**Abstract:**

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

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**Abstract:**

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.

*+*[Abstract](755)

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**Abstract:**

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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