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Discrete and Continuous Dynamical Systems - B

October 2019 , Volume 24 , Issue 10

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Alzheimer's disease and prion: An in vitro mathematical model
Ionel S. Ciuperca, Matthieu Dumont, Abdelkader Lakmeche, Pauline Mazzocco, Laurent Pujo-Menjouet, Human Rezaei and Léon M. Tine
2019, 24(10): 5225-5260 doi: 10.3934/dcdsb.2019057 +[Abstract](3236) +[HTML](426) +[PDF](17384.45KB)

Alzheimer's disease (AD) is a fatal incurable disease leading to progressive neuron destruction. AD is caused in part by the accumulation in the brain of Aβ monomers aggregating into oligomers and fibrils. Oligomers are amongst the most toxic structures as they can interact with neurons via membrane receptors, including PrPc proteins. This interaction leads to the misconformation of PrPc into pathogenic oligomeric prions, PrPol.

We develop here a model describing in vitro Aβ polymerization process. We include interactions between oligomers and PrPc, causing the misconformation of PrPc into PrPol. The model consists of nine equations, including size structured transport equations, ordinary differential equations and delayed differential equations. We analyse the well-posedness of the model and prove the existence and uniqueness of the solution of our model using Schauder fixed point and Cauchy-Lipschitz theorems. Numerical simulations are also provided to some specific profiles.

Evolution of a spiral-shaped polygonal curve by the crystalline curvature flow with a pinned tip
Tetsuya Ishiwata and Takeshi Ohtsuka
2019, 24(10): 5261-5295 doi: 10.3934/dcdsb.2019058 +[Abstract](2808) +[HTML](353) +[PDF](979.72KB)

We present a new ODE approach for an evolving polygonal spiral by the crystalline eikonal-curvature flow with a fixed center. In this approach, we introduce a mechanism of new facet generation at the center of the growing spiral, which is based on the theory of two-dimensional nucleation. We prove the existence, uniqueness and intersection free of solution to our formulation globally-in-time. In the proof of the existence we also prove that new facets are generated repeatedly in time. The comparison result of the normal velocity between inner and outer facets with the same normal direction leads intersection-free result. The normal velocities are positive after the next new facet is generated, so that the center is always behind of the moving facets.

$ L^\sigma $-measure criteria for boundedness in a quasilinear parabolic-parabolic Keller-Segel system with supercritical sensitivity
Mengyao Ding and Xiangdong Zhao
2019, 24(10): 5297-5315 doi: 10.3934/dcdsb.2019059 +[Abstract](2809) +[HTML](435) +[PDF](451.59KB)

This paper studies the parabolic-parabolic Keller-Segel system with supercritical sensitivity: \begin{document}$u_{t}=\nabla\cdot(\phi (u) \nabla u)-\nabla \cdot(\varphi(u)\nabla v)$\end{document}, \begin{document}$v_{t}=\Delta v -v+u$\end{document}, subject to homogeneous Neumann boundary conditions in a bounded and smooth domain \begin{document}$\Omega\subset\mathbb{R}^n$\end{document} \begin{document}$(n\ge2)$\end{document}, the diffusivity fulfills \begin{document}$\phi(u)\ge a_0(u+1)^{\gamma}$\end{document} with \begin{document}$\gamma\ge0$\end{document} and \begin{document}$a_0>0$\end{document}, while the chemotactic sensitivity satisfies \begin{document}$0\le \varphi(u)\le b_0u(u+1)^{\alpha+\gamma-1}$\end{document} with \begin{document}$\alpha>\frac{2}{n}$\end{document} and \begin{document}$b_0>0$\end{document}. It is proved that the problem possesses a globally bounded solution for \begin{document}$\frac{4}{n+2}<\alpha<2$\end{document}, whenever \begin{document}$\|u_0\|_{L^{\frac{n\alpha}{2}}(\Omega)}$\end{document} and \begin{document}$\|\nabla v_0\|_{L^{\frac{n\alpha+2\gamma}{2-\alpha}}(\Omega)}$\end{document} is sufficiently small. Similarly, the above conclusion still holds for \begin{document}$\alpha>2$\end{document} provided that \begin{document}$\|u_{0}\|_{L^{n\alpha-n}(\Omega)}$\end{document} and \begin{document}$\|\nabla v_0\|_{L^{\infty}(\Omega)}$\end{document} are small enough.

Blowup rate of solutions of a degenerate nonlinear parabolic equation
Chi-Cheung Poon
2019, 24(10): 5317-5336 doi: 10.3934/dcdsb.2019060 +[Abstract](2872) +[HTML](389) +[PDF](438.21KB)

We study a nonlinear parabolic equation arising from heat combustion and plane curve evolution problems. Suppose that a solution satisfies a symmetry condition and blows up of type Ⅱ. We give an upper bound and a lower bound for the blowup rate of the solution. The lower bound obtained here is probably optimal.

Numerical solution of partial differential equations with stochastic Neumann boundary conditions
Minoo Kamrani
2019, 24(10): 5337-5354 doi: 10.3934/dcdsb.2019061 +[Abstract](3409) +[HTML](514) +[PDF](450.08KB)

The aim of this paper is to study the numerical solution of partial differential equations with boundary forcing. For spatial discretization we apply the Galerkin method and for time discretization we will use a method based on the accelerated exponential Euler method. Our purpose is to investigate the convergence of the proposed method, but the main difficulty in carrying out this construction is that at the forced boundary the solution is expected to be unbounded. Therefore the error estimates are performed in the \begin{document}$ L_p $\end{document} spaces.

On the asymptotic behavior of highly nonlinear hybrid stochastic delay differential equations
Tian Zhang, Huabin Chen, Chenggui Yuan and Tomás Caraballo
2019, 24(10): 5355-5375 doi: 10.3934/dcdsb.2019062 +[Abstract](3689) +[HTML](385) +[PDF](549.27KB)

In this paper, the existence and uniqueness, the stability analysis for the global solution of highly nonlinear stochastic differential equations with time-varying delay and Markovian switching are analyzed under a locally Lipschitz condition and a monotonicity condition. In order to overcome a difficulty stemming from the existence of the time-varying delay, one integral lemma is established. It should be mentioned that the time-varying delay is a bounded measurable function. By utilizing the integral inequality, the Lyapunov function and some stochastic analysis techniques, some sufficient conditions are proposed to guarantee the stability in both moment and almost sure senses for such equations, which can be also used to yield the almost surely asymptotic behavior. As a by-product, the exponential stability in \begin{document}$ p $\end{document}th\begin{document}$ (p\geq 1) $\end{document}-moment and the almost sure exponential stability are analyzed. Finally, two examples are given to show the usefulness of the results obtained.

A backscattering model based on corrector theory of homogenization for the random Helmholtz equation
Wenjia Jing and Olivier Pinaud
2019, 24(10): 5377-5407 doi: 10.3934/dcdsb.2019063 +[Abstract](2364) +[HTML](398) +[PDF](519.3KB)

This work concerns the analysis of wave propagation in random media. Our medium of interest is sea ice, which is a composite of a pure ice background and randomly located inclusions of brine and air. From a pulse emitted by a source above the sea ice layer, the main objective of this work is to derive a model for the backscattered signal measured at the source/detector location. The problem is difficult in that, in the practical configuration we consider, the wave impinges on the layer with a non-normal incidence. Since the sea ice is seen by the pulse as an effective (homogenized) medium, the energy is specularly reflected and the backscattered signal vanishes in a first order approximation. What is measured at the detector consists therefore of corrections to leading order terms, and we focus in this work on the homogenization corrector. We describe the propagation by a random Helmholtz equation, and derive an expression of the corrector in this layered framework. We moreover obtain a transport model for quadratic quantities in the random wavefield in a high frequency limit.

Global existence of weak solution in a chemotaxis-fluid system with nonlinear diffusion and rotational flux
Feng Li and Yuxiang Li
2019, 24(10): 5409-5436 doi: 10.3934/dcdsb.2019064 +[Abstract](3058) +[HTML](418) +[PDF](540.54KB)

In this paper, we consider the chemotaxis-Navier-Stokes system with nonlinear diffusion and rotational flux given by

in a bounded domain \begin{document}$\Omega\subset\mathbb{R}^3$\end{document}, where \begin{document}$k\in\mathbb{R}$\end{document}, \begin{document}$\phi\in W^{2,\infty}(\Omega)$\end{document} and the given tensor-valued function \begin{document}$S$\end{document}: \begin{document}$\overline\Omega\times[0,\infty)^2\rightarrow\mathbb{R}^{3\times 3}$\end{document} satisfies

Imposing no restriction on the size of the initial data, we establish the global existence of a very weak solution while assuming \begin{document}$m+\alpha>\frac{4}{3}$\end{document} and \begin{document}$m>\frac{1}{3}$\end{document}.

Effects of the noise level on nonlinear stochastic fractional heat equations
Kexue Li
2019, 24(10): 5437-5460 doi: 10.3934/dcdsb.2019065 +[Abstract](2616) +[HTML](370) +[PDF](462.21KB)

We consider the stochastic fractional heat equation \begin{document}$\partial_{t}u=\triangle^{\alpha/2}u+\lambda\sigma(u)\dot{w}$\end{document} on \begin{document}$[0,L]$\end{document} with Dirichlet boundary conditions, where \begin{document}$\dot{w}$\end{document} denotes the space-time white noise. For any \begin{document}$\lambda>0$\end{document}, we prove that the \begin{document}$p$\end{document}th moment of \begin{document}$\sup_{x\in [0,L]}|u(t,x)|$\end{document} grows at most exponentially. If \begin{document}$\lambda$\end{document} is small, we prove that the \begin{document}$p$\end{document}th moment of \begin{document}$\sup_{x\in [0,L]}|u(t,x)|$\end{document} is exponentially stable. At last, we obtain the noise excitation index of \begin{document}$p$\end{document}th energy of \begin{document}$u(t,x)$\end{document} is \begin{document}$\frac{2\alpha}{\alpha-1}$\end{document}.

Emergence of large densities and simultaneous blow-up in a two-species chemotaxis system with competitive kinetics
Yan Li
2019, 24(10): 5461-5480 doi: 10.3934/dcdsb.2019066 +[Abstract](2773) +[HTML](336) +[PDF](466.86KB)

In this paper, a fully parabolic chemotaxis system for two species

is considered associated with homogeneous Neumann boundary conditions in a smooth bounded domain \begin{document}$ \Omega\subset\mathbb{R}^n $\end{document}, \begin{document}$ n\geq3 $\end{document}, with parameters \begin{document}$ \chi_i, \mu_i, a_i>0 $\end{document}, \begin{document}$ i = 1, 2 $\end{document}. It is shown that for some low energy initial data, the influence of chemotactic cross-diffusion coupled with proliferation may force some solutions to exceed any given threshold. Further, it is proved that if blow-up happens in a two-species chemotaxis(-growth) system, it is simultaneous for both of the chemotactic species.

Simulation of a simple particle system interacting through hitting times
Vadim Kaushansky and Christoph Reisinger
2019, 24(10): 5481-5502 doi: 10.3934/dcdsb.2019067 +[Abstract](2504) +[HTML](388) +[PDF](1282.59KB)

We develop an Euler-type particle method for the simulation of a McKean–Vlasov equation arising from a mean-field model with positive feedback from hitting a boundary. Under assumptions on the parameters which ensure differentiable solutions, we establish convergence of order \begin{document}$ 1/2 $\end{document} in the time step. Moreover, we give a modification of the scheme using Brownian bridges and local mesh refinement, which improves the order to \begin{document}$ 1 $\end{document}. We confirm our theoretical results with numerical tests and empirically investigate cases with blow-up.

Low Mach number limit of strong solutions for 3-D full compressible MHD equations with Dirichlet boundary condition
Lan Zeng, Guoxi Ni and Yingying Li
2019, 24(10): 5503-5522 doi: 10.3934/dcdsb.2019068 +[Abstract](3057) +[HTML](395) +[PDF](493.12KB)

In this paper, we consider the low Mach number limit of the full compressible MHD equations in a 3-D bounded domain with Dirichlet boundary condition for velocity field, Neumann boundary condition for temperature and perfectly conducting boundary condition for magnetic field. First, the uniform estimates in the Mach number for the strong solutions are obtained in a short time interval, provided that the initial density and temperature are close to the constant states. Then, we prove the solutions of the full compressible MHD equations converge to the isentropic incompressible MHD equations as the Mach number tends to zero.

Fully decoupled schemes for the coupled Schrödinger-KdV system
Jiaxiang Cai, Juan Chen and Bin Yang
2019, 24(10): 5523-5538 doi: 10.3934/dcdsb.2019069 +[Abstract](2852) +[HTML](380) +[PDF](628.3KB)

The coupled numerical schemes are inefficient for the time-dependent coupled Schrödinger-KdV system. In this study, some splitting schemes are proposed for the system based on the operator splitting method and coordinate increment discrete gradient method. The schemes are decoupled, so that each of the variables can be solved separately at each time level. Ample numerical experiments are carried out to demonstrate the efficiency and accuracy of our schemes.

Maintaining gene expression levels by positive feedback in burst size in the presence of infinitesimal delay
Pavol Bokes
2019, 24(10): 5539-5552 doi: 10.3934/dcdsb.2019070 +[Abstract](2432) +[HTML](350) +[PDF](739.33KB)

Synthesis of individual molecules in the expression of genes often occurs in bursts of multiple copies. Gene regulatory feedback can affect the frequency with which these bursts occur or their size. Whereas frequency regulation has traditionally received more attention, we focus specifically on the regulation of burst size. It turns out that there are (at least) two alternative formulations of feedback in burst size. In the first, newly produced molecules immediately partake in feedback, even within the same burst. In the second, there is no within-burst regulation due to what we call infinitesimal delay. We describe both alternatives using a minimalistic Markovian drift-jump framework combining discrete and continuous dynamics. We derive detailed analytic results and efficient simulation algorithms for positive non-cooperative autoregulation (whether infinitesimally delayed or not). We show that at steady state both alternatives lead to a gamma distribution of protein level. The steady-state distribution becomes available only after a transcritical bifurcation point is passed. Interestingly, the onset of the bifurcation is postponed by the inclusion of infinitesimal delay.

Existence and approximation of strong solutions of SDEs with fractional diffusion coefficients
Hao Yang, Fuke Wu and Peter E. Kloeden
2019, 24(10): 5553-5567 doi: 10.3934/dcdsb.2019071 +[Abstract](3065) +[HTML](375) +[PDF](421.98KB)

In stochastic financial and biological models, the diffusion coefficients often involve the terms \begin{document}$ \sqrt{|x|} $\end{document} and \begin{document}$ \sqrt{|x(1-x)|} $\end{document}, or more general \begin{document}$ |x|^{r} $\end{document} and \begin{document}$ |x(1-x)|^r $\end{document} for \begin{document}$ r $\end{document} \begin{document}$ \in $\end{document} \begin{document}$ (0, 1) $\end{document}. These coefficients do not satisfy the local Lipschitz condition, which implies that the existence and uniqueness of the solution cannot be obtained by the standard conditions. This paper establishes the existence and uniqueness of the strong solution and the strong convergence of the Euler-Maruyama approximations under certain conditions for systems of stochastic differential equations for which one component has such a diffusion coefficient with \begin{document}$ r $\end{document} \begin{document}$ \in $\end{document} \begin{document}$ [1/2, 1) $\end{document}.

Interplay of time-delay and velocity alignment in the Cucker-Smale model on a general digraph
Jiu-Gang Dong, Seung-Yeal Ha and Doheon Kim
2019, 24(10): 5569-5596 doi: 10.3934/dcdsb.2019072 +[Abstract](3313) +[HTML](395) +[PDF](858.63KB)

We study dynamic interplay between time-delay and velocity alignment in the ensemble of Cucker-Smale (C-S) particles(or agents) on time-varying networks which are modeled by digraphs containing spanning trees. Time-delayed dynamical systems often appear in mathematical models from biology and control theory, and they have been extensively investigated in literature. In this paper, we provide sufficient frameworks for the mono-cluster flocking to the continuous and discrete C-S models, which are formulated in terms of system parameters and initial data. In our proposed frameworks, we show that the continuous and discrete C-S models exhibit exponential flocking estimates. For the explicit C-S communication weights which decay algebraically, our results exhibit threshold phenomena depending on the decay rate and depth of digraph. We also provide several numerical examples and compare them with our analytical results.

Minimal forward random point attractors need not exist
Michael Scheutzow
2019, 24(10): 5597-5600 doi: 10.3934/dcdsb.2019073 +[Abstract](2153) +[HTML](335) +[PDF](222.91KB)

It is well-known that random attractors of a random dynamical system are generally not unique. It was shown in [1] that if there exist more than one pullback or weak random attractor which attracts a given family of (possibly random) sets, then there exists a minimal (in the sense of smallest) one. This statement does not hold for forward random attractors. The same paper contains an example of a random dynamical system and a deterministic family of sets which has more than one forward attractor which attracts the given family but no minimal one. The question whether one can find an example which has multiple forward point attractors but no minimal one remained open. Here we provide such an example.

Cell-type switches induced by stochastic histone modification inheritance
Rongsheng Huang and Jinzhi Lei
2019, 24(10): 5601-5619 doi: 10.3934/dcdsb.2019074 +[Abstract](2783) +[HTML](438) +[PDF](1004.91KB)

Cell plasticity is important for tissue developments during which somatic cells may switch between distinct states. Genetic networks to yield multistability are usually required to yield multiple states, and either external stimuli or noise in gene expressions are trigger signals to induce cell-type switches. In many biological systems, cells show highly plasticity and can switch between different states spontaneously, but maintaining the dynamic equilibrium of the cell population. Here, we considered a mechanism of spontaneous cell-type switches through the combination between gene regulation network and stochastic epigenetic state transitions. We presented a mathematical model that consists of a standard positive feedback loop with changes of histone modifications during cell cycling. Based on the model, nucleosome state of an associated gene is a random process during cell cycling, and hence introduces an inherent noise to gene expression, which can automatically induce cell-type switches in cell cycling. Our model reveals a simple mechanism of spontaneous cell-type switches through a stochastic histone modification inheritance during cell cycle. This mechanism is inherent to the normal cell cycle process, and is independent to the external signals.

Exponential convergence for the 3D stochastic cubic Ginzburg-Landau equation with degenerate noise
Yan Zheng and Jianhua Huang
2019, 24(10): 5621-5632 doi: 10.3934/dcdsb.2019075 +[Abstract](2401) +[HTML](413) +[PDF](430.83KB)

The current paper is devoted to 3D stochastic Ginzburg-Landau equation with degenerate random forcing. We prove that the corresponding Markov semigroup possesses an exponentially attracting invariant measure. To accomplish this, firstly we establish a type of gradient inequality, which is also essential to proving asymptotic strong Feller property. Then we prove that the corresponding dynamical system possesses a strong type of Lyapunov structure and is of a relatively weak form of irreducibility.

Spatial dynamics of a Lotka-Volterra model with a shifting habitat
Yueding Yuan, Yang Wang and Xingfu Zou
2019, 24(10): 5633-5671 doi: 10.3934/dcdsb.2019076 +[Abstract](3344) +[HTML](476) +[PDF](746.9KB)

In this paper, we study a Lotka-Volterra competition-diffusion model that describes the growth, spread and competition of two species in a shifting habitat. Our results show that (Ⅰ) if the competition between the two species are either mutually strong or mutually weak against each other, the spatial dynamics mainly depend on environment worsening speed c and the spreading speed of each species in the absence of the other in the best possible environment; (Ⅱ) if one species is a strong competitor and the other is a weak competitor, then the interplay of the species' competing strengths and the spreading speeds also has an effect on the spatial dynamics. Particularly, we find that a strong but slower competitor can co-persist with a weak but faster competitor, provided that the environment worsening speed is not too fast.

Limiting behavior of trajectory attractors of perturbed reaction-diffusion equations
Gaocheng Yue
2019, 24(10): 5673-5694 doi: 10.3934/dcdsb.2019101 +[Abstract](1974) +[HTML](245) +[PDF](485.47KB)

In this paper, we study the relations between the long-time dynamical behavior of the perturbed reaction-diffusion equations and the exact reaction-diffusion equations with concave and convex nonlinear terms and prove that bounded sets of solutions of the perturbed reaction-diffusion equations converges to the trajectory attractor \begin{document}$ \mathscr{U}_0 $\end{document} of the exact reaction-diffusion equations when \begin{document}$ t\rightarrow\infty $\end{document} and \begin{document}$ \varepsilon\rightarrow 0^+. $\end{document} In particular, we show that the trajectory attractor \begin{document}$ \mathscr{U}_ \varepsilon $\end{document} of the perturbed reaction-diffusion equations converges to the trajectory attractor \begin{document}$ \mathscr{U}_0 $\end{document} of the exact reaction-diffusion equations when \begin{document}$ \varepsilon\rightarrow 0^+. $\end{document} Moreover, we derive the upper and lower bounds of the fractal dimension for the global attractor of the perturbed reaction-diffusion equations.

Existence and uniqueness of solutions to a family of semi-linear parabolic systems using coupled upper-lower solutions
Anne Mund, Christina Kuttler and Judith Pérez-Velázquez
2019, 24(10): 5695-5707 doi: 10.3934/dcdsb.2019102 +[Abstract](2495) +[HTML](257) +[PDF](476.3KB)

We prove existence and uniqueness of weak and classical solutions to certain semi-linear parabolic systems with Robin boundary conditions using the coupled upper-lower solution approach. Our interest lies in cross-dependencies on the gradient parts of the reaction term, which prevents the straight-forward application of standard theorems. Such cross-dependencies emerge e.g. in a model describing evolution of bacterial quorum sensing, but are interesting also in a more general context. We show the existence and uniqueness of solutions for this example.

Synchronization for stochastic differential equations with nonlinear multiplicative noise in the mean square sense
Zhen Li and Jicheng Liu
2019, 24(10): 5709-5736 doi: 10.3934/dcdsb.2019103 +[Abstract](2503) +[HTML](325) +[PDF](497.33KB)

We provide a more clear technique to deal with general synchronization problems for SDEs, where the multiplicative noise appears nonlinearly. Moreover, convergence rate of synchronization is obtained. A new method employed here is the techniques of moment estimates for general solutions based on the transformation of multi-scales equations. As a by-product, the relationship between general solutions and stationary solutions is constructed.

Asymptotic behavior of random lattice dynamical systems and their Wong-Zakai approximations
Anhui Gu
2019, 24(10): 5737-5767 doi: 10.3934/dcdsb.2019104 +[Abstract](2253) +[HTML](244) +[PDF](616.83KB)

In this paper, we study the Wong-Zakai approximations given by a smoothed approximation of the white noise and their associated long term pathwise behavior for the stochastic lattice dynamical systems. To be exactly, we first establish the existence of the random attractor for the random lattice dynamical system driven by the smoothed noise and then show the convergence of solutions and random attractors to these of stochastic lattice dynamical systems driven by a multiplicative noise and an additive white noise, respectively, when the perturbation parameters tend to zero.

Impacts of noise on heat equations
Guangying Lv and Hongjun Gao
2019, 24(10): 5769-5784 doi: 10.3934/dcdsb.2019105 +[Abstract](1766) +[HTML](279) +[PDF](417.77KB)

In this paper, we consider the impacts of noise on heat equations. Our results show that the noise can induce singularities (finite time blow up of solutions) and that the nonlinearity can prevent the singularities. Moreover, suitable noise can prevent the solution vanishing. Besides that, we obtain the solutions of some reaction-diffusion equations keep positive, included stochastic Burgers' equation.

2021 Impact Factor: 1.497
5 Year Impact Factor: 1.527
2021 CiteScore: 2.3




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