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Discrete & Continuous Dynamical Systems - B

June 2018 , Volume 23 , Issue 4

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On optimal controls in coefficients for ill-posed non-Linear elliptic Dirichlet boundary value problems
Olha P. Kupenko and Rosanna Manzo
2018, 23(4): 1363-1393 doi: 10.3934/dcdsb.2018155 +[Abstract](625) +[HTML](140) +[PDF](599.55KB)
Abstract:

We consider an optimal control problem associated to Dirichlet boundary value problem for non-linear elliptic equation on a bounded domain \begin{document}$Ω$ \end{document}. We take the coefficient \begin{document}$u(x)∈ L^∞(Ω)\cap BV(Ω)$ \end{document} in the main part of the non-linear differential operator as a control and in the linear part of differential operator we consider coefficients to be unbounded skew-symmetric matrix \begin{document}$A_{skew}∈ L^q(Ω;\mathbb{S}^N_{skew})$ \end{document}. We show that, in spite of unboundedness of the non-linear differential operator, the considered Dirichlet problem admits at least one weak solution and the corresponding OCP is well-possed and solvable. At the same time, optimal solutions to such problem can inherit a singular character of the matrices \begin{document}$A^{skew}$ \end{document}. We indicate two types of optimal solutions to the above problem and show that one of them can be attained by optimal solutions of regularized problems for coercive elliptic equations with bounded coefficients, using the two-parametric regularization of the initial OCP.

Well-posedness in critical spaces for a multi-dimensional compressible viscous liquid-gas two-phase flow model
Haibo Cui, Qunyi Bie and Zheng-An Yao
2018, 23(4): 1395-1410 doi: 10.3934/dcdsb.2018156 +[Abstract](654) +[HTML](151) +[PDF](459.69KB)
Abstract:

This paper is dedicated to the study of the Cauchy problem for a compressible viscous liquid-gas two-phase flow model in \begin{document}$\mathbb{R}^N\,(N≥2)$ \end{document}. We concentrate on the critical Besov spaces based on the \begin{document}$L^p$ \end{document} setting. We improve the range of Lebesgue exponent \begin{document}$p$ \end{document}, for which the system is locally well-posed, compared to [22]. Applying Lagrangian coordinates is the key to our statements, as it enables us to obtain the result by means of Banach fixed point theorem.

Global analysis of strong solutions for the viscous liquid-gas two-phase flow model in a bounded domain
Guochun Wu and Yinghui Zhang
2018, 23(4): 1411-1429 doi: 10.3934/dcdsb.2018157 +[Abstract](523) +[HTML](143) +[PDF](439.63KB)
Abstract:

In this paper, we investigate global existence and asymptotic behavior of strong solutions for the viscous liquid-gas two-phase flow model in a bounded domain with no-slip boundary. The global existence and uniqueness of strong solutions are obtained when the initial data is near its equilibrium in \begin{document}$H^2(Ω)$ \end{document}. Furthermore, the exponential convergence rates of the pressure and velocity are also proved by delicate energy methods.

A time-delay in the activator kinetics enhances the stability of a spike solution to the gierer-meinhardt model
Nabil T. Fadai, Michael J. Ward and Juncheng Wei
2018, 23(4): 1431-1458 doi: 10.3934/dcdsb.2018158 +[Abstract](713) +[HTML](162) +[PDF](752.47KB)
Abstract:

We study the spectrum of a new class of nonlocal eigenvalue problems (NLEPs) that characterize the linear stability properties of localized spike solutions to the singularly perturbed two-component Gierer-Meinhardt (GM) reaction-diffusion (RD) system with a fixed time-delay \begin{document}$T$\end{document} in only the nonlinear autocatalytic activator kinetics. Our analysis of this model is motivated by the computational study of Seirin Lee et al. [Bull. Math. Bio., 72(8), (2010)] on the effect of gene expression time delays on spatial patterning for both the GM and some related RD models. For various limiting forms of the GM model, we show from a numerical study of the associated NLEP, together with an analytical scaling law analysis valid for large delay \begin{document}$T$\end{document}, that a time-delay in only the activator kinetics is stabilizing in the sense that there is a wider region of parameter space where the spike solution is linearly stable than when there is no time delay. This enhanced stability behavior with a delayed activator kinetics is in marked contrast to the de-stabilizing effect on spike solutions of having a time-delay in both the activator and inhibitor kinetics. Numerical results computed from the RD system with delayed activator kinetics are used to validate the theory for the 1-D case.

Convergence of a finite volume scheme for a stochastic conservation law involving a $Q$-brownian motion
Tadahisa Funaki, Yueyuan Gao and Danielle Hilhorst
2018, 23(4): 1459-1502 doi: 10.3934/dcdsb.2018159 +[Abstract](678) +[HTML](151) +[PDF](665.68KB)
Abstract:

We study a time explicit finite volume method for a first order conservation law with a multiplicative source term involving a \begin{document}$Q$\end{document}-Wiener process. After having presented the definition of a measure-valued weak entropy solution of the stochastic conservation law, we apply a finite volume method together with Godunov scheme for the space discretization, and we denote by \begin{document}$\{u_{\mathcal{T}, k}\}$\end{document} its discrete solution. We present some a priori estimates including a weak BV estimate. After performing a time interpolation, we prove two entropy inequalities for the discrete solution. We show that the discrete solution \begin{document}$\{u_{\mathcal{T}, k}\}$\end{document} converges along a subsequence to a measure-valued entropy solution of the conservation law in the sense of Young measures as the maximum diameter of the volume elements and the time step tend to zero. Some numerical simulations are presented in the case of the stochastic Burgers equation. The empirical average turns out to be a regularization of the deterministic solution; moreover, the variance in the case of the \begin{document}$Q$\end{document}-Brownian motion converges to a constant while that in the Brownian motion case keeps increasing as time tends to infinity.

An N-barrier maximum principle for elliptic systems arising from the study of traveling waves in reaction-diffusion systems
Chiun-Chuan Chen and Li-Chang Hung
2018, 23(4): 1503-1521 doi: 10.3934/dcdsb.2018054 +[Abstract](792) +[HTML](287) +[PDF](720.55KB)
Abstract:

By employing the N-barrier method developed in C.-C. Chen and L.-C. Hung, 2016 ([6]), we establish a new N-barrier maximum principle for diffusive Lotka-Volterra systems of two competing species. To this end, this gives rise to the N-barrier maximum principle for a second-order elliptic equation involving two distinct unknown functions and a quadratic nonlinearity. An immediate consequence of the N-barrier maximum principle is an a priori estimate for the total populations of the two species. As an application of this maximum principle, we show under certain conditions the existence and nonexistence of traveling waves solutions for systems of three competing species. In addition, new \begin{document}$(1, 0, 0)$\end{document}-\begin{document}$(u^{*}, v^{*}, 0)$\end{document} waves are given in terms of the tanh function, provided that the system's parameters satisfy certain conditions.

Time fractional and space nonlocal stochastic boussinesq equations driven by gaussian white noise
Tianlong Shen, Jianhua Huang and Caibin Zeng
2018, 23(4): 1523-1533 doi: 10.3934/dcdsb.2018056 +[Abstract](894) +[HTML](281) +[PDF](397.0KB)
Abstract:

We present the time-spatial regularity of the nonlocal stochastic convolution for Caputo-type time fractional nonlocal Ornstein-Ulenbeck equations, and establish the existence and uniqueness of mild solutions for time fractional and space nonlocal stochastic Boussinesq equations driven by Gaussian white noise.

Longtime robustness and semi-uniform compactness of a pullback attractor via nonautonomous PDE
Yangrong Li, Lianbing She and Jinyan Yin
2018, 23(4): 1535-1557 doi: 10.3934/dcdsb.2018058 +[Abstract](812) +[HTML](276) +[PDF](489.11KB)
Abstract:

This paper is concerned with the robustness of a pullback attractor as the time tends to infinity. A pullback attractor is called forward (resp. backward) compact if the union over the future (resp. the past) is pre-compact. We prove that the forward (resp. backward) compactness is a necessary and sufficient condition such that a pullback attractor is upper semi-continuous to a compact set at positive (resp. negative) infinity, and also obtain the minimal limit-set. We further prove the lower semi-continuity of the pullback attractor and get the maximal limit-set at infinity. Some criteria for such robustness are established when the evolution process is forward or backward omega-limit compact. Those theoretical criteria are applied to prove semi-uniform compactness and robustness at infinity in pullback dynamics for a Ginzburg-Landau equation with variable coefficients and a forward or backward tempered nonlinearity.

Dynamics of a Lotka-Volterra competition-diffusion model with stage structure and spatial heterogeneity
Shuling Yan and Shangjiang Guo
2018, 23(4): 1559-1579 doi: 10.3934/dcdsb.2018059 +[Abstract](910) +[HTML](292) +[PDF](587.67KB)
Abstract:

This paper is concerned with a Lotka-Volterra competition-diffusion model with stage structure and spatial heterogeneity. By analyzing the sign of the principal eigenvalue corresponding to each semi-trivial solution, we obtain the linear stability and global attractivity of the semi-trivial solution. In addition, an attracting region was obtained by means of the method of upper and lower solutions.

Two codimension-two bifurcations of a second-order difference equation from macroeconomics
Jiyu Zhong and Shengfu Deng
2018, 23(4): 1581-1600 doi: 10.3934/dcdsb.2018062 +[Abstract](940) +[HTML](271) +[PDF](723.32KB)
Abstract:

In this paper we mainly investigate two codimension-two bifurcations of a second-order difference equation from macroeconomics. Applying the center manifold theorem and the normal form analysis, we firstly give the parameter conditions for the generalized flip bifurcation, and prove that the system does not produce a strong resonance. Then, we compute the normal forms to obtain the parameter conditions for the Neimark-Sacker bifurcation, from which we present the conditions for the Chenciner bifurcation. In order to verify the correctness of our results, we also numerically simulate a half stable invariant circle and two invariant circles, one stable and one unstable, arising from the Chenciner bifurcation.

Palindromic control and mirror symmetries in finite difference discretizations of 1-D Schrödinger equations
Katherine A. Kime
2018, 23(4): 1601-1621 doi: 10.3934/dcdsb.2018063 +[Abstract](844) +[HTML](321) +[PDF](432.54KB)
Abstract:

We consider discrete potentials as controls in systems of finite difference equations which are discretizations of a 1-D Schrödinger equation. We give examples of palindromic potentials which have corresponding steerable initial-terminal pairs which are not mirror-symmetric. For a set of palindromic potentials, we show that the corresponding steerable pairs that satisfy a localization property are mirror-symmetric. We express the initial and terminal states in these pairs explicitly as scalar multiples of vector-valued functions of a parameter in the control.

Ion size effects on individual fluxes via Poisson-Nernst-Planck systems with Bikerman's local hard-sphere potential: Analysis without electroneutrality boundary conditions
Hong Lu, Ji Li, Joseph Shackelford, Jeremy Vorenberg and Mingji Zhang
2018, 23(4): 1623-1643 doi: 10.3934/dcdsb.2018064 +[Abstract](849) +[HTML](307) +[PDF](709.89KB)
Abstract:

A quasi-one-dimensional steady-state Poisson-Nernst-Planck model with Bikerman's local hard-sphere potential for ionic flows of two oppositely charged ion species through a membrane channel is analyzed. Of particular interest is the qualitative properties of ionic flows in terms of individual fluxes without the assumption of electroneutrality conditions, which is more realistic to study ionic flow properties of interest. This is the novelty of this work. Our result shows that ⅰ) boundary concentrations and relative size of ion species play critical roles in characterizing ion size effects on individual fluxes; ⅱ) the first order approximation \begin{document} $\mathcal{J}_{k1} = D_kJ_{k1}$ \end{document} in ion volume of individual fluxes \begin{document} $\mathcal{ J}_k = D_kJ_k$ \end{document} is linear in boundary potential, furthermore, the signs of \begin{document} $\partial_V \mathcal{ J}_{k1}$ \end{document} and \begin{document} $\partial^2_{Vλ} \mathcal{J}_{k1}$ \end{document}, which play key roles in characterizing ion size effects on ionic flows can be both negative depending further on boundary concentrations while they are always positive and independent of boundary concentrations under electroneutrality conditions (see Corollaries 3.2-3.3, Theorems 3.4-3.5 and Proposition 3.7). Numerical simulations are performed to identify some critical potentials defined in (2). We believe our results will provide useful insights for numerical and even experimental studies of ionic flows through membrane channels.

Dynamics for the damped wave equations on time-dependent domains
Feng Zhou, Chunyou Sun and Xin Li
2018, 23(4): 1645-1674 doi: 10.3934/dcdsb.2018068 +[Abstract](1180) +[HTML](394) +[PDF](619.97KB)
Abstract:

We consider the asymptotic dynamics of a damped wave equations on a time-dependent domains with homogeneous Dirichlet boundary condition, the nonlinearity is allowed to have a cubic growth rate which is referred to as the critical exponent. To this end, we establish the existence and uniqueness of strong and weak solutions satisfying energy inequality under the assumption that the spatial domains $\mathcal{O}_{t}$ in $\mathbb{R}^{3}$ are obtained from a bounded base domain $\mathcal{O}$ by a $C^{3}$-diffeomorphism $r(·, t)$. Furthermore, we establish the pullback attractor under a slightly weaker assumption that the measure of the spatial domains are uniformly bounded above.

Boundedness and global solvability to a chemotaxis-haptotaxis model with slow and fast diffusion
Chunhua Jin
2018, 23(4): 1675-1688 doi: 10.3934/dcdsb.2018069 +[Abstract](1212) +[HTML](454) +[PDF](434.52KB)
Abstract:

In this paper, we deal with the following coupled chemotaxis-haptotaxis system modeling cancer invasionwith nonlinear diffusion, where \begin{document} $Ω\subset\mathbb R^N$ \end{document} (\begin{document} $N≥ 3$ \end{document}) is a bounded domain. Under zero-flux boundary conditions, we showed that for any \begin{document} $m>0$ \end{document}, the problem admits a global bounded weak solution for any large initial datum if \begin{document} $\frac{χ}{μ}$ \end{document} is appropriately small. The slow diffusion case (\begin{document} $m>1$ \end{document}) of this problem have been studied by many authors [14,7,19,23], in which, the boundedness and the global in time solution are established for \begin{document} $m>\frac{2N}{N+2}$ \end{document}, but the cases \begin{document} $m≤ \frac{2N}{N+2}$ \end{document} remain open.

Asymptotic behavior of random fitzhugh-nagumo systems driven by colored noise
Anhui Gu and Bixiang Wang
2018, 23(4): 1689-1720 doi: 10.3934/dcdsb.2018072 +[Abstract](1205) +[HTML](917) +[PDF](550.74KB)
Abstract:

In this paper, we prove the existence and uniqueness of random attractors for the FitzHugh-Nagumo system driven by colored noise with a nonlinear diffusion term. We demonstrate that the colored noise is much easier to deal with than the white noise for studying the pathwise dynamics of stochastic systems. In addition, we show the attractors of the random FitzHugh-Nagumo system driven by a linear multiplicative colored noise converge to that of the corresponding stochastic system driven by a linear multiplicative white noise.

Dynamics of a diffusive prey-predator system with strong Allee effect growth rate and a protection zone for the prey
Na Min and Mingxin Wang
2018, 23(4): 1721-1737 doi: 10.3934/dcdsb.2018073 +[Abstract](1427) +[HTML](500) +[PDF](391.76KB)
Abstract:

In this paper, a diffusive prey-predator model with strong Allee effect growth rate and a protection zone $\Omega _0$ for the prey is investigated. We analyze the global existence, long time behaviors of positive solutions and the local stabilities of semi-trivial solutions. Moreover, the conditions of the occurrence and avoidance of overexploitation phenomenon are obtained. Furthermore, we demonstrate that the existence and stability of non-constant steady state solutions branching from constant semi-trivial solutions by using bifurcation theory. Our results show that the protection zone is effective when Allee threshold is small and the protection zone is large.

Laplace-Beltrami operator for the heat conduction in polymer coating of electronic devices
Micol Amar and Roberto Gianni
2018, 23(4): 1739-1756 doi: 10.3934/dcdsb.2018078 +[Abstract](688) +[HTML](216) +[PDF](584.91KB)
Abstract:

In this paper we study a model for the heat conduction in a composite having a microscopic structure arranged in a periodic array. We obtain the macroscopic behaviour of the material and specifically the overall conductivity via an homogenization procedure, providing the equation satisfied by the effective temperature.

A regularity criterion for the 3D full compressible magnetohydrodynamic equations with zero heat conductivity
Jishan Fan, Fucai Li and Gen Nakamura
2018, 23(4): 1757-1766 doi: 10.3934/dcdsb.2018079 +[Abstract](481) +[HTML](190) +[PDF](346.78KB)
Abstract:

We establish a regularity criterion for the 3D full compressible magnetohydrodynamic equations with zero heat conductivity and vacuum in a bounded domain.

Nonlocal elliptic system arising from the growth of cancer stem cells
Manuel Delgado, Ítalo Bruno Mendes Duarte and Antonio Suárez Fernández
2018, 23(4): 1767-1795 doi: 10.3934/dcdsb.2018083 +[Abstract](555) +[HTML](208) +[PDF](641.08KB)
Abstract:

In this work we show the existence of coexistence states for a nonlocal elliptic system arising from the growth of cancer stem cells. For this, we use the bifurcation method and the theory of the fixed point index in cones. Moreover, in some cases we study the behaviour of the coexistence region, depending on the parameters of the problem.

Parking 3-sphere swimmer I. Energy minimizing strokes
François Alouges and Giovanni Di Fratta
2018, 23(4): 1797-1817 doi: 10.3934/dcdsb.2018085 +[Abstract](510) +[HTML](191) +[PDF](1059.04KB)
Abstract:

The paper is about the parking 3-sphere swimmer (sPr3), a low-Reynolds number model swimmer composed of three balls of equal radii. The three balls can move along three horizontal axes (supported in the same plane) that mutually meet at the center of sPr3 with angles of 120°. The governing dynamical system is introduced and the implications of its geometric symmetries revealed. It is then shown that, in the first order range of small strokes, optimal periodic strokes are ellipses embedded in 3d space, i.e., closed curves of the form \begin{document} $t ∈ [0, 2 π] \mapsto (\cos t) u + (\sin t) v$ \end{document} for suitable vectors u and v of \begin{document} $\mathbb{R}^3$ \end{document}. A simple analytic expression for the vectors u and v is derived.

Singular perturbed renormalization group theory and its application to highly oscillatory problems
Wenlei Li and Shaoyun Shi
2018, 23(4): 1819-1833 doi: 10.3934/dcdsb.2018089 +[Abstract](574) +[HTML](210) +[PDF](404.83KB)
Abstract:

Renormalization group method in the singular perturbation theory, originally introduced by Chen et al, has been proven to be very practicable in a large number of singular perturbed problems. In this paper, we will firstly reconsider the Renormalization group method under some general conditions to get several newly rigorous approximate results. Then we will apply the obtained results to investigate a class of second order differential equations with the highly oscillatory phenomenon of highly oscillatory properties, which occurs in many multiscale models from applied mathematics, physics and material science, etc. Our strategy, in fact, can be also used to analyze the same problem for related evolution equations with multiple scales, such as nonlinear Klein-Gordon equations in the nonrelativistic limit regime.

Determination of the area of exponential attraction in one-dimensional finite-time systems using meshless collocation
Peter Giesl and James McMichen
2018, 23(4): 1835-1850 doi: 10.3934/dcdsb.2018094 +[Abstract](563) +[HTML](204) +[PDF](473.43KB)
Abstract:

We consider a non-autonomous ordinary differential equation over a finite time interval \begin{document}$[T_1,T_2]$\end{document}. The area of exponential attraction consists of solutions such that the distance to adjacent solutions exponentially contracts from \begin{document}$T_1$\end{document} to \begin{document}$T_2$\end{document}. One can use a contraction metric to determine an area of exponential attraction and to provide a bound on the rate of attraction.

In this paper, we will give the first method to algorithmically construct a contraction metric for finite-time systems in one spatial dimension. We will show the existence of a contraction metric, given by a function which satisfies a second-order partial differential equation with boundary conditions. We then use meshless collocation to approximately solve this equation, and show that the resulting approximation itself defines a contraction metric, if the collocation points are sufficiently dense. We give error estimates and apply the method to an example.

Stability and robustness analysis for a multispecies chemostat model with delays in the growth rates and uncertainties
Frederic Mazenc, Gonzalo Robledo and Michael Malisoff
2018, 23(4): 1851-1872 doi: 10.3934/dcdsb.2018098 +[Abstract](746) +[HTML](245) +[PDF](550.22KB)
Abstract:

We study a chemostat model with an arbitrary number of competing species, one substrate, and constant dilution rates. We allow delays in the growth rates and additive uncertainties. Using constant inputs of certain species, we derive bounds on the sizes of the delays that ensure asymptotic stability of an equilibrium when the uncertainties are zero, which can allow persistence of multiple species. Under delays and uncertainties, we provide bounds on the delays and on the uncertainties that ensure input-to-state stability with respect to uncertainties.

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