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Discrete & Continuous Dynamical Systems - B

2017 , Volume 22 , Issue 2

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Convergence rate of synchronization of systems with additive noise
Shahad Al-azzawi , Jicheng Liu and  Xianming Liu
2017, 22(2): 227-245 doi: 10.3934/dcdsb.2017012 +[Abstract](44) +[HTML](0) +[PDF](429.1KB)

The synchronization of stochastic differential equations (SDEs) with additive noise is investigated in pathwise sense, moreover convergence rate of synchronization is obtained. The optimality of the convergence rate is illustrated through examples.

Dynamics of a nonlocal SIS epidemic model with free boundary
Jia-Feng Cao , Wan-Tong Li and  Fei-Ying Yang
2017, 22(2): 247-266 doi: 10.3934/dcdsb.2017013 +[Abstract](129) +[HTML](2) +[PDF](500.2KB)

This paper is concerned with the spreading or vanishing of a epidemic disease which is characterized by a diffusion SIS model with nonlocal incidence rate and double free boundaries. We get the full information about the sufficient conditions that ensure the disease spreading or vanishing, which exhibits a detailed description of the communicable mechanism of the disease. Our results imply that the nonlocal interaction may enhance the spread of the disease.

Optimality conditions for a controlled sweeping process with applications to the crowd motion model
Tan H. Cao and  Boris S. Mordukhovich
2017, 22(2): 267-306 doi: 10.3934/dcdsb.2017014 +[Abstract](57) +[HTML](4) +[PDF](700.8KB)

The paper concerns the study and applications of a new class of optimal control problems governed by a perturbed sweeping process of the hysteresis type with control functions acting in both play-and-stop operator and additive perturbations. Such control problems can be reduced to optimization of discontinuous and unbounded differential inclusions with pointwise state constraints, which are immensely challenging in control theory and prevent employing conventional variation techniques to derive necessary optimality conditions. We develop the method of discrete approximations married with appropriate generalized differential tools of modern variational analysis to overcome principal difficulties in passing to the limit from optimality conditions for finite-difference systems. This approach leads us to nondegenerate necessary conditions for local minimizers of the controlled sweeping process expressed entirely via the problem data. Besides illustrative examples, we apply the obtained results to an optimal control problem associated with of the crowd motion model of traffic flow in a corridor, which is formulated in this paper. The derived optimality conditions allow us to develop an effective procedure to solve this problem in a general setting and completely calculate optimal solutions in particular situations.

Uniform $L^{∞}$ boundedness for a degenerate parabolic-parabolic Keller-Segel model
Wenting Cong and  Jian-Guo Liu
2017, 22(2): 307-338 doi: 10.3934/dcdsb.2017015 +[Abstract](45) +[HTML](4) +[PDF](539.8KB)

This paper investigates the existence of a uniform in time \begin{document} $L^{∞}$ \end{document} bounded weak entropy solution for the quasilinear parabolic-parabolic Keller-Segel model with the supercritical diffusion exponent \begin{document} $0<m<2-\frac{2}{d}$ \end{document} in the multi-dimensional space \begin{document} ${\mathbb{R}}^d$ \end{document} under the condition that the \begin{document} $L^{\frac{d(2-m)}{2}}$ \end{document} norm of initial data is smaller than a universal constant. Moreover, the weak entropy solution \begin{document} $u(x,t)$ \end{document} satisfies mass conservation when \begin{document} $m>1-\frac{2}{d}$ \end{document}. We also prove the local existence of weak entropy solutions and a blow-up criterion for general \begin{document} $L^1\cap L^{∞}$ \end{document} initial data.

Analysis of a nonlocal-in-time parabolic equation
Qiang Du , Jiang Yang and  Zhi Zhou
2017, 22(2): 339-368 doi: 10.3934/dcdsb.2017016 +[Abstract](154) +[HTML](1) +[PDF](592.9KB)

In this paper, we consider an initial boundary value problem for nonlocal-in-time parabolic equations involving a nonlocal in time derivative. We first show the uniqueness and existence of the weak solution of the nonlocal-in-time parabolic equation, and also the spatial smoothing properties. Moreover, we develop a new framework to study the local limit of the nonlocal model as the horizon parameter δ approaches 0. Exploiting the spatial smoothing properties, we develop a semi-discrete scheme using standard Galerkin finite element method for the spatial discretization, and derive error estimates dependent on data smoothness. Finally, extensive numerical results are presented to illustrate our theoretical findings.

Stability of equilibria of randomly perturbed maps
PaweŁ Hitczenko and  Georgi S. Medvedev
2017, 22(2): 369-381 doi: 10.3934/dcdsb.2017017 +[Abstract](34) +[HTML](0) +[PDF](427.8KB)

We derive a sufficient condition for stability in probability of an equilibrium of a randomly perturbed map in \begin{document}$\mathbb{R}^d$\end{document}. This condition can be used to stabilize unstable equilibria by random forcing. Analytical results on stabilization are illustrated with numerical examples of randomly perturbed nonlinear maps in one-and two-dimensional spaces.

A self-organizing criticality mathematical model for contamination and epidemic spreading
Stelian Ion and  Gabriela Marinoschi
2017, 22(2): 383-405 doi: 10.3934/dcdsb.2017018 +[Abstract](32) +[HTML](0) +[PDF](494.9KB)

We introduce a new model to predict the spread of an epidemic, focusing on the contamination process and simulating the disease propagation by the means of a unique function viewed as a measure of the local infective energy. The model is intended to illustrate a map of the epidemic spread and not to compute the densities of various populations related to an epidemic, as in the classical models. First, the model is constructed as a cellular automaton exhibiting a self-organizing-type criticality process with two thresholds. This induces the consideration of an associate continuous model described by a nonlinear equation with two singularities, for whose solution we prove existence, uniqueness and certain properties. We provide numerical simulations to put into evidence the effect of some model parameters in various scenarios of the epidemic spread.

Asymptotic behavior of the solution of a diffusion equation with nonlocal boundary conditions
Bhargav Kumar Kakumani and  Suman Kumar Tumuluri
2017, 22(2): 407-419 doi: 10.3934/dcdsb.2017019 +[Abstract](42) +[HTML](2) +[PDF](267.6KB)

In this paper, we consider a particular type of nonlinear McKend-rick-von Foerster equation with a diffusion term and Robin boundary condition. We prove the existence of a global solution to this equation. The steady state solutions to the equations that we consider have a very important role to play in the study of long time behavior of the solution. Therefore we address the issues pertaining to the existence of solution to the corresponding state equation. Furthermore, we establish that the solution of McKendrick-von Foerster equation with diffusion converges pointwise to the solution of its steady state equations as time tends to infinity.

On tamed milstein schemes of SDEs driven by Lévy noise
Chaman Kumar and  Sotirios Sabanis
2017, 22(2): 421-463 doi: 10.3934/dcdsb.2017020 +[Abstract](40) +[HTML](4) +[PDF](683.4KB)

We extend the taming techniques developed in [3,19] to construct explicit Milstein schemes that numerically approximate Lévy driven stochastic differential equations with super-linearly growing drift coefficients. The classical rate of convergence is recovered when the first derivative of the drift coefficient satisfies a polynomial Lipschitz condition.

Asymptotic behavior in a chemotaxis-growth system with nonlinear production of signals
Yuanyuan Liu and  Youshan Tao
2017, 22(2): 465-475 doi: 10.3934/dcdsb.2017021 +[Abstract](36) +[HTML](4) +[PDF](384.3KB)

We consider the chemotaxis-growth system

under no-flux boundary conditions, in a convex bounded domain \begin{document} $Ω\subset\mathbb{R}^3$ \end{document} with smooth boundary, where \begin{document} $χ>0$ \end{document} and \begin{document} $μ>0$ \end{document} are given parameters, and \begin{document} $h(s)$ \end{document} is a prescribed function on \begin{document} $[0, ∞)$ \end{document}.

It is shown that under the assumption that

for any given nonnegative \begin{document} $u_0∈ C^0(\bar{Ω})$ \end{document} and \begin{document} $v_0∈ W^{1, ∞}(Ω)$ \end{document} the system possesses a global classical solution which is bounded in \begin{document} $Ω× (0, ∞)$ \end{document}. Moreover, whenever

any bounded classical solution constructed above stabilizes to the constant stationary solution \begin{document} $(1, h(1))$ \end{document} as the time goes to infinity.

Periodic solutions of some classes of continuous second-order differential equations
Jaume Llibre and  Amar Makhlouf
2017, 22(2): 477-482 doi: 10.3934/dcdsb.2017022 +[Abstract](248) +[HTML](2) +[PDF](309.9KB)

We study the periodic solutions of the second-order differential equations of the form \begin{document} $ \ddot x ± x^{n} = μ f(t), $ \end{document} or \begin{document} $ \ddot x ± |x|^{n} = μ f(t), $ \end{document} where \begin{document} $n=4,5,...$ \end{document}, \begin{document} $f(t)$ \end{document} is a continuous \begin{document} $T$ \end{document}-periodic function such that \begin{document} $\int_0^T {f\left( t \right)} dt\ne 0$ \end{document}, and \begin{document} $μ$ \end{document} is a positive small parameter. Note that the differential equations \begin{document} $ \ddot x ± x^{n} = μ f(t)$ \end{document} are only continuous in \begin{document} $t$ \end{document} and smooth in \begin{document} $x$ \end{document}, and that the differential equations \begin{document} $ \ddot x ± |x|^{n} = μ f(t)$ \end{document} are only continuous in \begin{document} $t$ \end{document} and locally-Lipschitz in \begin{document} $x$ \end{document}.

Expanding speed of the habitat for a species in an advective environment
Junfan Lu , Hong Gu and  Bendong Lou
2017, 22(2): 483-490 doi: 10.3934/dcdsb.2017023 +[Abstract](37) +[HTML](1) +[PDF](475.6KB)

Recently, Gu et al. [7,8] studied a reaction-diffusion-advection equation \begin{document}$u_t =u_{xx} -β u_x + f(u)$\end{document} in \begin{document}$(g(t), h(t))$\end{document}, where \begin{document}$g(t)$\end{document} and \begin{document}$h(t)$\end{document} are two free boundaries satisfying Stefan conditions, \begin{document}$f(u)$\end{document} is a Fisher-KPP type of nonlinearity. When \begin{document}$β ∈ [0,c_0)$\end{document}, where \begin{document}$c_0 := 2\sqrt{f'(0)}$\end{document}, they found that for a spreading solution \begin{document}$(u,g,h)$\end{document}, \begin{document}$h(t)/t \to c^*_r (β)$\end{document} and \begin{document}$g(t)/t \to c^*_l (β)$\end{document} as \begin{document}$t \to ∞$\end{document}, and \begin{document}$c^*_r (β) > c^*_r(0) = - c^*_l (0) > - c^*_l (β) >0$\end{document}. In this paper we study the expanding speed \begin{document}$C^*(β) :=c^*_r(β) - c^*_l (β)$\end{document} of the habitat \begin{document}$(g(t), h(t))$\end{document}, and show that \begin{document}$C^*(β)$\end{document} is strictly increasing in \begin{document}$β ∈ [0,c_0)$\end{document}. When \begin{document}$β ∈ [c_0, β^*)$\end{document} for some \begin{document}$β^*>c_0$\end{document}, [8] also found a virtual spreading phenomena: \begin{document}$h(t)/t \to c^*_r(β)$\end{document} as \begin{document}$t\to∞$\end{document}, and a back forms in the solution which moves rightward with a speed \begin{document}$β - c_0$\end{document}. Hence the expanding speed of the main habitat for such a solution is \begin{document}$C^*(β) := c^*_r(β) -[β -c_0]$\end{document}. In this paper we show that \begin{document}$C^*(β)$\end{document} is strictly decreasing in \begin{document}$β∈ [c_0, β^*)$\end{document} with \begin{document}$C^*(β^* -0)=0$\end{document}, and so there exists a unique \begin{document}$β_0∈ (c_0, β^*)$\end{document} such that the advection is favorable to the expanding speed of the habitat if and only if \begin{document}$β∈ (0,β_0)$\end{document}.

Energy decay of solutions for the wave equation with a time-varying delay term in the weakly nonlinear internal feedbacks
Ferhat Mohamed and  Hakem Ali
2017, 22(2): 491-506 doi: 10.3934/dcdsb.2017024 +[Abstract](43) +[HTML](20) +[PDF](400.6KB)

In this paper, we investigate the nonlinear wave equation in a bounded domain with a time-varying delay term in the weakly nonlinear internal feedback

The asymptotic behavior of solutions is studied by using an appropriate Lyapunov functional. Moreover, we extend and improve the previous results in the literature.

Lyapunov functionals for multistrain models with infinite delay
Yoji Otani , Tsuyoshi Kajiwara and  Toru Sasaki
2017, 22(2): 507-536 doi: 10.3934/dcdsb.2017025 +[Abstract](27) +[HTML](0) +[PDF](546.4KB)

We construct Lyapunov functionals for delay differential equation models of infectious diseases in vivo to analyze the stability of the equilibria. The Lyapunov functionals contain the terms that integrate over all previous states. An appropriate evaluation of the logarithm functions in those terms guarantees the existence of the integrals. We apply the rigorous analysis for the one-strain models to multistrain models by using mathematical induction.

Global classical solutions to the free boundary problem of planar full magnetohydrodynamic equations with large initial data
Yaobin Ou and  Pan Shi
2017, 22(2): 537-567 doi: 10.3934/dcdsb.2017026 +[Abstract](28) +[HTML](0) +[PDF](517.3KB)

The free boundary problem of planar full compressible magnetohydrodynamic equations with large initial data is studied in this paper, when the initial density connects to vacuum smoothly. The global existence and uniqueness of classical solutions are established, and the expanding rate of the free interface is shown. Using the method of Lagrangian particle path, we derive some L estimates and weighted energy estimates, which lead to the global existence of classical solutions. The main difficulty of this problem is the degeneracy of the system near the free boundary, while previous results (cf. [4,30]) require that the density is bounded from below by a positive constant.

Randomly perturbed switching dynamics of a dc/dc converter
Chetan D. Pahlajani
2017, 22(2): 569-584 doi: 10.3934/dcdsb.2017027 +[Abstract](34) +[HTML](0) +[PDF](591.5KB)

In this paper, we study the effect of small Brownian noise on a switching dynamical system which models a first-order DC/DC buck converter. The state vector of this system comprises a continuous component whose dynamics switch, based on the ON/OFF configuration of the circuit, between two ordinary differential equations (ODE), and a discrete component which keeps track of the ON/OFF configurations. Assuming that the parameters and initial conditions of the unperturbed system have been tuned to yield a stable periodic orbit, we study the stochastic dynamics of this system when the forcing input in the ON state is subject to small white noise fluctuations of size \begin{document}$\varepsilon $\end{document}, \begin{document}$0<\varepsilon \ll 1$\end{document}. For the ensuing stochastic system whose dynamics switch at random times between a small noise stochastic differential equation (SDE) and an ODE, we prove a functional law of large numbers which states that in the limit of vanishing noise, the stochastic system converges to the underlying deterministic one on time horizons of order \begin{document}$\mathscr{O}(1/\varepsilon ^ν)$\end{document}, \begin{document}$0 ≤ ν < 2/3$\end{document}.

Global stability in the 2D Ricker equation revisited
Brian Ryals and  Robert J. Sacker
2017, 22(2): 585-604 doi: 10.3934/dcdsb.2017028 +[Abstract](40) +[HTML](4) +[PDF](2496.9KB)

We offer two improvements to prior results concerning global stability of the 2D Ricker Equation. We also offer some new methods of approach for the more pathological cases and prove some miscellaneous results including a special nontrivial case in which the mapping is conjugate to the 1D Ricker map along an invariant line and a proof of the non-existence of period-2 points.

Ergodicity of the stochastic coupled fractional Ginzburg-Landau equations driven by α-stable noise
Tianlong Shen and  Jianhua Huang
2017, 22(2): 605-625 doi: 10.3934/dcdsb.2017029 +[Abstract](33) +[HTML](1) +[PDF](456.0KB)

The current paper is devoted to the ergodicity of stochastic coupled fractional Ginzburg-Landau equations driven by \begin{document}$α$\end{document}-stable noise on the Torus \begin{document}$\mathbb{T}$\end{document}. By the maximal inequality for stochastic \begin{document}$α$\end{document}-stable convolution and commutator estimates, the well-posedness of the mild solution for stochastic coupled fractional Ginzburg-Landau equations is established. Due to the discontinuous trajectories and non-Lipschitz nonlinear term, the existence and uniqueness of the invariant measures are obtained by the strong Feller property and the accessibility to zero.

Traveling wave solutions in a diffusive producer-scrounger model
Junhao Wen and  Peixuan Weng
2017, 22(2): 627-645 doi: 10.3934/dcdsb.2017030 +[Abstract](73) +[HTML](2) +[PDF](467.0KB)

This paper looks into the stability of equilibria, existence and non-existence of traveling wave solutions in a diffusive producer-scrounger model. We find that the existence and non-existence of traveling wave solutions are determined by a minimum wave speed \begin{document}$c_{m}$\end{document} and a threshold value \begin{document}$R_{0}$\end{document}. By constructing a suitable invariant convex set $Γ$ and applying Schauder fixed point theorem, the existence for \begin{document}$c>c_{m}, R_{0}>1$\end{document} was established. Besides, a Lyapunov function is constructed subtly to explore the asymptotic behaviors of traveling wave solutions. The non-existences of traveling wave solutions for both \begin{document}$c < c_{m}, R_{0}> 1$\end{document} and \begin{document}$R_{0}≤1, c > 0$\end{document} were obtained by two-sides Laplace transform and reduction method to absurdity.

The bifurcation analysis of turing pattern formation induced by delay and diffusion in the Schnakenberg system
Fengqi Yi , Eamonn A. Gaffney and  Sungrim Seirin-Lee
2017, 22(2): 647-668 doi: 10.3934/dcdsb.2017031 +[Abstract](247) +[HTML](6) +[PDF](1046.6KB)

A delayed reaction-diffusion Schnakenberg system with Neumann boundary conditions is considered in the context of long range biological self-organisation dynamics incorporating gene expression delays. We perform a detailed stability and Hopf bifurcation analysis and derive conditions for determining the direction of bifurcation and the stability of the bifurcating periodic solution. The delay-diffusion driven instability of the unique spatially homogeneous steady state solution and the diffusion-driven instability of the spatially homogeneous periodic solution are investigated, with limited simulations to support our theoretical analysis. These studies analytically demonstrate that the modelling of gene expression time delays in Turing systems can eliminate or disrupt the formation of a stationary heterogeneous pattern in the Schnakenberg system.

A note on global existence to a higher-dimensional quasilinear chemotaxis system with consumption of chemoattractant
Jiashan Zheng and  Yifu Wang
2017, 22(2): 669-686 doi: 10.3934/dcdsb.2017032 +[Abstract](40) +[HTML](22) +[PDF](527.8KB)

The Neumann boundary value problem for the chemotaxis system generalizing the prototype

\begin{document}$\left\{ \begin{array}{l}{u_t} = \nabla \cdot (D(u)\nabla u) - \nabla \cdot (u\nabla v),\;\;\;\;x \in \Omega ,t < 0,\\{v_t} = \Delta v - uv,\;\;\;\;\;x \in \Omega ,t < 0,\end{array} \right. \tag{KS}\label{KS} $ \end{document}

is considered in a smooth bounded convex domain \begin{document}$Ω\subset \mathbb{R}^N(N≥2)$\end{document}, where

\begin{document}$D(u)≥ C_D(u+1)^{m-1}~~ \mbox{for all}~~ u≥0~~\mbox{with some}~~ m > 1~~\mbox{and}~~ C_D>0.$ \end{document}

If \begin{document}$m >\frac{3N}{2N+2}$\end{document} and suitable regularity assumptions on the initial data are given, the corresponding initial-boundary problem possesses a global classical solution. Our paper extends the results of Wang et al. ([24]), who showed the global existence of solutions in the cases \begin{document}$m>2-\frac{6}{N+4}$\end{document} (\begin{document}$N≥3$\end{document}). If the flow of fluid is ignored, our result is consistent with and improves the result of Tao, Winkler ([15]) and Tao, Winkler ([17]), who proved the possibility of global boundedness, in the case that \begin{document}$N=2,m>1$\end{document} and \begin{document}$N= 3$\end{document}, \begin{document}$m > \frac{8}{7}$\end{document}, respectively.

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