Boundedness in a quasilinear 2D parabolic-parabolic attraction-repulsion chemotaxis system
Yilong Wang Zhaoyin Xiang
In this paper, we consider the following quasilinear attraction-repulsion chemotaxis system of parabolic-parabolic type \begin{equation*} \left\{ \begin{split} &u_t=\nabla\cdot(D(u)\nabla u)-\nabla\cdot(\chi u\nabla v)+\nabla\cdot(\xi u\nabla w),\qquad & x\in\Omega,\,\, t>0,\\ &v_t=\Delta v+\alpha u-\beta v,\qquad &x\in\Omega, \,\,t>0,\\ &w_t=\Delta w+\gamma u-\delta w,\qquad &x\in\Omega,\,\, t>0 \end{split} \right. \end{equation*} under homogeneous Neumann boundary conditions, where $D(u)\geq c_D (u+\varepsilon)^{m-1}$ and $\Omega\subset\mathbb{R}^2$ is a bounded domain with smooth boundary. It is shown that whenever $m>1$, for any sufficiently smooth nonnegative initial data, the system admits a global bounded classical solution for the case of non-degenerate diffusion (i.e., $\varepsilon>0$), while the system possesses a global bounded weak solution for the case of degenerate diffusion (i.e., $\varepsilon=0$).
keywords: attraction-repulsion boundedness. Parabolic-parabolic chemotaxis global existence
Global existence and blow-up to a reaction-diffusion system with nonlinear memory
Lili Du Chunlai Mu Zhaoyin Xiang
In this paper, we consider a reaction-diffusion system coupled by nonlinear memory. Under appropriate hypotheses, we prove that the solution either exists globally or blows up in finite time. Furthermore, the blow-up rate estimates are obtained.
keywords: blow-up reaction-diffusion system nonlinear memory Global existence
Blowup behaviors for degenerate parabolic equations coupled via nonlinear boundary flux
Chunlai Mu Zhaoyin Xiang
This paper deals with the blow-up properties of solutions to a degenerate parabolic system coupled via nonlinear boundary flux. Firstly, we construct the self-similar supersolution and subsolution to obtain the critical global existence curve. Secondly, we establish the precise blow-up rate estimates for solutions which blow up in a finite time. Finally, we investigate the localization of blow-up points. The critical curve of Fujita type is conjectured with the aid of some new results.
keywords: Degenerate parabolic system critical global existence curve critical Fujita curve blow-up rate estimates blow-up sets.
Visualization analysis of traffic congestion based on floating car data
Jingmei Zhou Xiangmo Zhao Xin Cheng Zhigang Xu
Traffic congestion visualization is an important part in traffic information service. However, the real-time data is difficult to obtain and its analysis method is not accurate, so the reliability of congestion state visualization is low. This paper proposes a visualization analysis algorithm of traffic congestion based on Floating Car Data (FCD), which utilizes the FCD to estimate and display dynamic traffic state on the electronic map. Firstly, an improved map matching method is put forward to match rapidly the FCD with road sections, which includes two steps of coarse and precise matching. Then, the traffic speed is estimated and classified to display different traffic states. Eventually, multi-group experiments have been conducted based on more than 8000 taxies in Xi’an. The experimental results show that FCD can be matched accurately with the selected road sections which accuracy can reach up to \({\rm{96\% }}\), and the estimated traffic real-time state can achieve \({\rm{94\% }}\) in terms of reliability. So this visualization analysis algorithm can display accurately road traffic state in real time.
keywords: FCD map matching speed estimation. Visualization analysis traffic congestion
Global controllability and stabilizability of Kawahara equation on a periodic domain
Xiangqing Zhao Bing-Yu Zhang
In this paper we study controllability and stabilizability of a class of distributed parameter control system described by the Kawahara equation posed on a periodic domain $\mathbb{T}$ with internal control acting on a sub-domain $\omega $ of $\mathbb{T}$. Earlier in [42], aided by Bourgain smoothing property of the system, we showed that the system is locally exactly controllable and exponentially stabilizable. In this paper, helped further by certain properties of propagation of compactness and regularity in Bourgain spaces for the solutions of the associated linear system, we show that the system is globally exactly controllable and globally exponentially stabilizable.
keywords: propagation of compactness Kawahara equation global stabilizability propagation of regularity global exact controllability Bourgain space and Bourgain smoothing.
Existence and nonexistence of local/global solutions for a nonhomogeneous heat equation
Xie Li Zhaoyin Xiang
In this paper, we study the existence of local/global solutions to the Cauchy problem \begin{eqnarray} \rho(x)u_t=\Delta u+q(x)u^p, (x,t)\in R^N \times (0,T),\\ u(x,0)=u_{0}(x)\ge 0, x \in R^N \end{eqnarray} with $p > 0$ and $N\ge 3$. We describe the sharp decay conditions on $\rho, q$ and the data $u_0$ at infinity that guarantee the local/global existence of nonnegative solutions.
keywords: global existence Fujita exponent Cauchy problem inhomogeneous heat equation. Local existence
Boundedness in quasilinear Keller-Segel equations with nonlinear sensitivity and logistic source
Xie Li Zhaoyin Xiang
In this paper, we investigate the quasilinear Keller-Segel equations (q-K-S): \[ \left\{ \begin{split} &n_t=\nabla\cdot\big(D(n)\nabla n\big)-\nabla\cdot\big(\chi(n)\nabla c\big)+\mathcal{R}(n), \qquad x\in\Omega,\,t>0,\\ &\varrho c_t=\Delta c-c+n, \qquad x\in\Omega,\,t>0, \end{split} \right. \] under homogeneous Neumann boundary conditions in a bounded domain $\Omega\subset\mathbb{R}^N$. For both $\varrho=0$ (parabolic-elliptic case) and $\varrho>0$ (parabolic-parabolic case), we will show the global-in-time existence and uniform-in-time boundedness of solutions to equations (q-K-S) with both non-degenerate and degenerate diffusions on the non-convex domain $\Omega$, which provide a supplement to the dichotomy boundedness vs. blow-up in parabolic-elliptic/parabolic-parabolic chemotaxis equations with degenerate diffusion, nonlinear sensitivity and logistic source. In particular, we improve the recent results obtained by Wang-Li-Mu (2014, Disc. Cont. Dyn. Syst.) and Wang-Mu-Zheng (2014, J. Differential Equations).
keywords: parabolic-parabolic Keller-Segel systems. Global existence parabolic-elliptic Keller-Segel system boundedness
Blow-up and asymptotic behavior of solutions to a semilinear integrodifferential system
Qiong Chen Chunlai Mu Zhaoyin Xiang
This paper deals with the blow-up properties and asymptotic behavior of solutions to a semilinear integrodifferential system with nonlocal reaction terms in space and time. The blow-up conditions are given by a variant of the eigenfunction method combined with new properties on systems of differential inequalities. At the same time, the blow-up set is obtained. For some special cases, the asymptotic behavior of the blow-up solution is precisely characterized.
keywords: blow-up set Integrodifferential system asymptotic behavior. blow-up
Time series based urban air quality predication
Ruiqi Li Yifan Chen Xiang Zhao Yanli Hu Weidong Xiao
Urban air pollution post a great threat to human health, and has been a major concern of many metropolises in developing countries. Lately, a few air quality monitoring stations have been established to inform public the real-time air quality indices based on fine particle matters, e.g. $PM_{2.5}$, in countries suffering from air pollutions. Air quality, unfortunately, is fairly difficult to manage due to multiple complex human activities from driving to smelting. We observe that human activities' hidden regular pattern offers possibility in predication, and this motivates us to infer urban air condition from the perspective of time series. In this paper, we focus on $PM_{2.5}$ based urban air quality, and introduce two kinds of time-series methods for real-time and fine-grained air quality prediction, harnessing historical air quality data reported by existing monitoring stations. The methods are evaluated based in the real-life $PM_{2.5}$ concentration data in the year of 2013 (January - December) in Wuhan, China.
keywords: $PM_{2.5}$ multiplicative model time series Urban air quality ARIMA.

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