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2688-1594

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## Electronic Research Archive

### Special issue on infinite dimensional dynamical systems and applications

Xiaoying Han, Auburn University, Auburn, AL, USA xzh0003@auburn.edu

Wan-Tong Li, Lanzhou University, Chinawtli@lzu.edu.cn

Zhi-Cheng Wang, School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, Chinawangzhch@lzu.edu.cn

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The asymptotic behaviour of an autonomous neural field lattice system with delays is investigated. It is based on the Amari model, but with the Heaviside function in the interaction term replaced by a sigmoidal function. First, the lattice system is reformulated as an infinite dimensional ordinary delay differential equation on weighted sequence state space

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In this paper, two properties of the pullback attractor for a 2D non-autonomous micropolar fluid flows with delay on unbounded domains are investigated. First, we establish the

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In this paper we study the continuity in initial data of a classical reaction-diffusion equation with arbitrary

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We study the random dynamics for the stochastic non-autonomous Kuramoto-Sivashinsky equation in the possibly non-dissipative case. We first prove the existence of a pullback attractor in the Lebesgue space of odd functions, then show that the fiber of the odd pullback attractor semi-converges to a nonempty compact set as the time-parameter goes to minus infinity and finally prove the measurability of the attractor. In a word, we obtain a longtime stable random attractor formed from odd functions. A key tool is the existence of a bridge function between Lebesgue and Sobolev spaces of odd functions.

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This paper deals with the limiting dynamical behavior of non-autonomous stochastic reaction-diffusion equations on thin domains. Firstly, we prove the existence and uniqueness of the regular random attractor. Then we prove the upper semicontinuity of the regular random attractors for the equations on a family of

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In this paper, we investigate a class of stochastic recurrent neural networks with discrete and distributed delays for both biological and mathematical interests. We do not assume any Lipschitz condition on the nonlinear term, just a continuity assumption together with growth conditions so that the uniqueness of the Cauchy problem fails to be true. Moreover, the existence of pullback attractors with or without periodicity is presented for the multi-valued noncompact random dynamical system. In particular, a new method for checking the asymptotical compactness of solutions to the class of nonautonomous stochastic lattice systems with infinite delay is used.

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**Abstract:**

Crime in urban environment is a major social problem nowadays. As such, many efforts have been made to develop mathematical models for this type of crime. The pioneering work [M. B. Short, M. R. D'Orsogna, V. B. Pasour, G. E. Tita, P. J. Brantingham, A. L. Bertozzi and L. B. Chayes, Math. Models Methods Appl. Sci., 18, (2008), pp. 1249-1267] establishes an agent-based human-environment interaction model of criminal behavior for residential burglary, where aggregate pattern formation of "hotspots" is quantitatively studied for the first time. Potential offenders are assumed to interact with environment according to well-known criminology and sociology notions. However long-term simulations for the coupled dynamics are computationally costly due to all components evolving on slow time scales. In this paper, we introduce a new-generation criminal behavior model with separated spatio-temporal scales for the agent actions and the environment parameter reactions. The computational cost is reduced significantly, while the essential stochastic features of the pioneering model are preserved. Moreover, the separation of scales brings the model into the theoretical framework of piecewise deterministic Markov processes (PDMP). A martingale approach is applicable which will be useful to analyze both stochastic and statistical features of the model in subsequent studies.

2020
Impact Factor: 1.833

5 Year Impact Factor: 1.833

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