# American Institute of Mathematical Sciences

ISSN:
1531-3492

eISSN:
1553-524X

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## Discrete & Continuous Dynamical Systems - B

May 2014 , Volume 19 , Issue 3

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2014, 19(3): 629-650 doi: 10.3934/dcdsb.2014.19.629 +[Abstract](1225) +[PDF](494.7KB)
Abstract:
In the present article, we study the numerical approximation of a system of Hamilton-Jacobi and transport equations arising in geometrical optics. We consider a semi-Lagrangian scheme. We prove the well posedness of the discrete problem and the convergence of the approximated solution toward the viscosity-measure valued solution of the exact problem.
2014, 19(3): 651-677 doi: 10.3934/dcdsb.2014.19.651 +[Abstract](979) +[PDF](544.4KB)
Abstract:
We study the long-time behavior of the solutions to a nonlinear damped driven Schrödinger type equation on a strip. We prove that this behavior is described by a regular compact global attractor.
2014, 19(3): 679-695 doi: 10.3934/dcdsb.2014.19.679 +[Abstract](1041) +[PDF](411.9KB)
Abstract:
This paper is devoted to the study of the existence, uniqueness, continuous dependence and spatial behaviour of the solutions for the backward in time problem determined by the Type III with two temperatures thermoelastodynamic theory. We first show the existence, uniqueness and continuous dependence of the solutions. Instability of the solutions for the Type II with two temperatures theory is proved later. For the one-dimensional Type III with two temperatures theory, the exponential instability is also pointed-out. We also analyze the spatial behaviour of the solutions. By means of the exponentially weighted Poincaré inequality, we are able to obtain a function that defines a measure on the solutions and, therefore, we obtain the usual exponential type alternative for the solutions of the problem defined in a semi-infinite cylinder.
2014, 19(3): 697-714 doi: 10.3934/dcdsb.2014.19.697 +[Abstract](1161) +[PDF](679.0KB)
Abstract:
In this study, we consider the traveling spots that were observed in the photosensitive Belousov-Zhabotinsky reaction experiment conducted by Mihailuk et al. in 2001. First, we introduce the interface equation by the singular limit analysis of a FitzHugh--Nagumo-type reaction-diffusion system. Then, we obtain the profile of the support of the solution. Next, we prove the uniqueness of the traveling spot by studying ordinary differential equations that describe its front and back. In addition, we provide an upper bound for the width of the spot. Furthermore, we compare the singular limit problem with the wave front interaction model proposed by Zykov and Showalter in 2005 and obtain traveling fingers.
2014, 19(3): 715-733 doi: 10.3934/dcdsb.2014.19.715 +[Abstract](1271) +[PDF](345.5KB)
Abstract:
In this paper, we consider global stability for a heroin model with two distributed delays. The basic reproduction number of the heroin spread is obtained, which completely determines the stability of the equilibria. Using the direct Lyapunov method with Volterra type Lyapunov function, we show that the drug use-free equilibrium is globally asymptotically stable if the basic reproduction number is less than one, and the unique drug spread equilibrium is globally asymptotically stable if the basic reproduction number is greater than one.
2014, 19(3): 735-745 doi: 10.3934/dcdsb.2014.19.735 +[Abstract](1040) +[PDF](458.7KB)
Abstract:
This paper formulates and analyzes an HIV-1 infection model with saturated infection rate. We first discuss the boundedness of the solution and the existence of the equilibrium. The local stability of the virus-free equilibrium and infected equilibrium is established by analyzing the roots of the characteristic equations. Furthermore, we study the global stability of the virus-free equilibrium and infected equilibrium by using suitable Lyapunov function and LaSalle's invariance principle, and obtain sufficient conditions for the global stability of the infected equilibrium. Finally, numerical simulations are presented to illustrate the main results.
2014, 19(3): 747-760 doi: 10.3934/dcdsb.2014.19.747 +[Abstract](1251) +[PDF](608.3KB)
Abstract:
In recent studies, global Hopf branches were investigated for delayed model of HTLV-I infection with delay-independent parameters. It is shown in [8,9] that when stability switches occur, global Hopf branches tend to be bounded, and different branches can overlap to produce coexistence of stable periodic solutions. In this paper, we investigate global Hopf branches for delayed systems with delay-dependent parameters. Using a delayed predator-prey model as an example, we demonstrate that stability switches caused by varying the time delay are accompanied by bounded global Hopf branches. When multiple Hopf branches exist, they are nested and the overlap produces coexistence of two or possibly more stable limit cycles.
2014, 19(3): 761-773 doi: 10.3934/dcdsb.2014.19.761 +[Abstract](1011) +[PDF](355.8KB)
Abstract:
By using Lyapunov functions and some recent estimates of Halanay type, new criteria are introduced for the global exponential stability of a class of cellular neural networks, with delay and periodic coefficients and inputs. The novelty of those criteria lies in the fact that they are very efficient in presence of oscillating coefficients, because they are given in average form.
2014, 19(3): 775-787 doi: 10.3934/dcdsb.2014.19.775 +[Abstract](1097) +[PDF](531.3KB)
Abstract:
Robust morphogen gradient formation is important for embryo development. Patterns of developmental tissue are encoded by the morphogen gradient that drives the process of cell differentiation in response to different morphogen levels. Experiments have shown that tissue patterning is robust with respect to morphogen overexpression. However, the mechanisms for this robust patterning remain unclear. The expansion-repression mechanism, which was proposed for achieving scaling of patterning with organ size, is a type of self-enhanced clearance through a non-local feedback regulation and may contribute to the robustness with respect to morphogen overexpression. In this paper, we study the role of the expansion-repression mechanism in morphogen gradient robustness through a two-equation model with general forms of feedback functions. We prove the existence of steady-state solutions, and, through model reduction and simplification, show that the expansion-repression mechanism is able to improve the robustness against changes in the morphogen production rate. However, this improvement is restricted by the biological requirement of multi-fate long-range morphogen gradient.
2014, 19(3): 789-799 doi: 10.3934/dcdsb.2014.19.789 +[Abstract](1113) +[PDF](390.7KB)
Abstract:
In this paper we consider a free boundary problem describing cell motility, which is a simple model of Umeda (see [11]). This model includes a non-local term and the interface equation with curvature. We prove that there exist at least two traveling waves of the model. First, we rewrite the problem into a fixed-point problem for a continuous map $T$ and then show that there exist at least two fixed points for the map $T$.
2014, 19(3): 801-815 doi: 10.3934/dcdsb.2014.19.801 +[Abstract](1160) +[PDF](440.1KB)
Abstract:
This paper is concerned with the stability of traveling front solutions for a viscous Fisher-KPP equation. By applying geometric singular perturbation method, special Evans function estimates, detailed spectral analysis and $C_0$ semigroup theories, each traveling front solution with wave speed $c<-2\sqrt{f^\prime(0)}$ is proved to be locally exponentially stable in some appropriate exponentially weighted spaces.
2014, 19(3): 817-826 doi: 10.3934/dcdsb.2014.19.817 +[Abstract](1101) +[PDF](393.9KB)
Abstract:
We study two systems of reaction diffusion equations with monostable or bistable type of nonlinearities and with free boundaries. These systems are used as multi-species competitive model. For two-species models, we prove the existence of traveling wave solutions, each of which consists of two semi-waves intersecting at the free boundary. For three-species models, we also obtain some traveling wave solutions. In this case, however, every traveling wave solution consists of two semi-waves and one compactly supported wave in between, each intersecting with its neighbors at the free boundaries.
2014, 19(3): 827-848 doi: 10.3934/dcdsb.2014.19.827 +[Abstract](1279) +[PDF](7765.3KB)
Abstract:
In this paper, a kind of second-order two-scale (SOTS) computation is developed for integrated heat transfer problem with conduction, convection and radiation in periodic porous materials, where the convection part is composed of long thin parallel pipes with periodic distribution, the conduction part occupied by solid materials and the radiation part is on the pipe's walls and the surfaces of cavities. First of all, by asymptotic expansion of the temperature field, the homogenization problem, first-order correctors and second-order correctors are obtained successively. Then, the error estimation of the second-order two-scale approximate solution is derived on some regularity hypothesis. Finally, the corresponding finite element algorithms are proposed and some numerical results are presented. The numerical tests indicate that the developed method can be successfully used for solving the integrated heat transfer problem, which can reduce the computational efforts greatly.
2014, 19(3): 849-865 doi: 10.3934/dcdsb.2014.19.849 +[Abstract](986) +[PDF](438.5KB)
Abstract:
In this work, two novel decoupling algorithms for the steady Stokes-Darcy model based on two-grid discretizations are proposed and analyzed. Optimal error estimates for these variables are presented. Two grid decoupled scheme proposed by Mu and Xu (2007) is used to develop the two novel decoupling algorithms. For Algorithm 3.2, the optimal error estimates are obtained for both ${\bf{u}}_f,\ p_f$ and $\phi$ with mesh sizes satisfying $H=\sqrt{h}$. For Algorithm 3.3, the convergence of $\phi$ in $H^1$-norm is improved form $H^2$ to $H^\frac{5}{2}$. Furthermore, the existing results in [17] are improved and supplemented. Finally, some numerical experiments are provided to show the efficiency and effectiveness of the developed algorithms.

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