This paper is concerned with a class of reaction-diffusion system with density-suppressed motility
<disp-formula> <tex-math id="FE1"> \begin{document}$ \begin{equation*} \begin{cases} u_{t} = ...\Delta(\gamma(v) u)+\alpha u F(w), & x \in \Omega, \quad t>0, \\ v_{t} = D \Delta v+u-v, & x \in \Omega, \quad t>0, \\ w_{t} = \Delta w-u F(w), & x \in \Omega, \quad t>0, \end{cases} \end{equation*} $\end{document} </tex-math></disp-formula>
under homogeneous Neumann boundary conditions in a smooth bounded domain $ \Omega\subset \mathbb{R}^n\; (n\leq 2) $, where $ \alpha > 0 $ and $ D > 0 $ are constants. The random motility function $ \gamma $ satisfies
<disp-formula> <tex-math id="FE2"> \begin{document}$ \begin{equation*} \gamma\in C^3((0, +\infty)), \ \gamma>0, \ \gamma'<0\, \ \text{on}\, \ (0, +\infty) \ \ \text{and}\ \ \lim\limits_{v\rightarrow +\infty}\gamma(v) = 0. \end{equation*} $\end{document} </tex-math></disp-formula>
The intake rate function $ F $ satisfies $ F\in C^1(0, +\infty)), \, F(0) = 0\, \ \text{and}\ \, F > 0\, \ \text{on}\, \ (0, +\infty) $. We show that the above system admits a unique global classical solution for all non-negative initial data $ u_0\in W^{1, \infty}(\Omega), \, v_0\in W^{1, \infty}(\Omega), \, w_0\in W^{1, \infty}(\Omega) $. Moreover, if there exist $ k > 0 $ and $ \overline{v} > 0 $ such that
<disp-formula> <tex-math id="FE3"> \begin{document}$ \begin{equation*} \inf\limits_{v>\overline{v}}v^k\gamma(v)>0, \end{equation*} $\end{document} </tex-math></disp-formula>
then the global solution is bounded uniformly in time.
Revisiting a newly reported modified Chen system by both the definitions of $ \alpha $-limit and $ \omega $-limit set, Lyapunov function and Hamiltonian function, this paper seized a multitude of ...pairs of potential heteroclinic orbits to (1) $ E_{0} $ and $ E_{\pm} $, or (2) $ E_{+} $ or (3) $ E_{-} $, and homoclinic and heteroclinic orbits on its invariant algebraic surface $ Q = z - \frac{x^{2}}{2a} = 0 $ with cofactor $ -2a $, which is not available in the existing literature to the best of our knowledge. Particularly, the theoretical conclusions were verified via numerical examples.
In financial asset allocation, enterprises adjust their investment in R&D innovation according to their motives and the external environment. Based on a review of the literature related to enterprise ...financialization and R&D innovation, this paper proposes research hypotheses through theoretical analysis first; then, taking China's A-share non-financial listed companies from 2010 to 2019 as research objects, this paper explores the relationship between enterprise financialization and R&D innovation with a quantile panel data model; further, the heterogeneous relationship between the two under different business cycle phases is empirically analyzed. The following conclusions are drawn. First, there is a dynamic relationship between enterprise financialization and R&D innovation, varying with different financing constraints. Second, the dynamic relationship between enterprise financialization and R&D innovation stems from the motivation difference in enterprise asset allocation. Third, there are significant differences in the dynamic relationship at different business cycle phases.
The development of urban digital inclusive finance has a complex network connection, and the characteristics of this network affect urban innovation and entrepreneurship. By using 287 cities from ...2015 to 2019 in China, this paper constructs the network of Chinese digital inclusive finance, extracting two typical network characteristics. And it further studies the impact of the two network characteristics on innovation and entrepreneurship in various cities. The empirical results show that: first, the network centrality characteristics of digital inclusive finance have a significant promoting effect on urban innovation and entrepreneurship; second, the impact of the network centrality characteristics of digital inclusive finance is related to the speed of industrial structure transformation, but the impact mechanism is related to the node number instead of the node strength; third, the impact of the network centrality characteristics on innovation and entrepreneurship exists heterogeneity among cities and the moderately developed cities have the largest impact.
This work focuses on exploring pointwise error estimate of three-level conservative difference scheme for supergeneralized viscous Burgers' equation. The cut-off function method plays an important ...role in constructing difference scheme and presenting numerical analysis. We study the conservative invariant of proposed method, which is energy-preserving for all positive integers $ p $ and $ q $. Meanwhile, one could apply the discrete energy argument to the rigorous proof that the three-level scheme has unique solution combining the mathematical induction. In addition, we prove the $ L_2 $-norm and $ L_{\infty} $-norm convergence of proposed scheme in pointwise sense with separate and different ways, which is different from previous work in <xref ref-type="bibr" rid="b1">1 . Numerical results verify the theoretical conclusions.
The Internet of Medical Things (IoMT) significantly impacts our healthcare system because it allows us to track and verify patient medical data before storing it in the cloud for future use. A ...rapidly expanding platform like IoMT requires high security to keep all data safe. The patient's prescription history and other sensitive information must be encrypted and managed with great care. Nevertheless, it is challenging to determine what data uses are acceptable while protecting patient privacy and security. Understanding the limits of current technologies and envisioning future research paths is crucial for establishing a safe and reliable data environment. An untrustworthy person can communicate with a trustworthy person via blockchain, a decentralized digital ledger that allows for end-to-end communication. Therefore, this research suggests that the healthcare industry with blockchain-integrated cyber-security based on artificial intelligence (BICS-AI) in medical care to preserve medical-related things. Blockchain applications have the potential to consistently identify the most severe, potentially life-threatening mistakes in the medical field. The use of blockchain for decentralized data protection helps to protect patient health records from compromise. With the help of an access control provider (ACP), here came up with a lightweight solution that addresses this issue by allowing the delegating of security operations. Medical data from IoMT and integrated devices can be collected and stored securely and distributed using a conventional in-depth approach combined with blockchain, making it suitable for healthcare professionals such as nursing homes, hospitals, and the healthcare industry where data exchange is required. The research findings indicate that the suggested system is viable and has a 94.84$ \% $ security rate, a security performance of 96.4$ \% $, a success rate of 89.9$ \% $, and a 5.1$ \% $ latency rate compared to traditional methods.
The cyclostationary spectrum (CS) method is one of the best at what it does because it effectively detects idle spectrum with low signal-to-noise ratios (SNR). In order to distinguish the signal in a ...noisy environment, gather more data that aids in a better analysis of signals, and use spectral correlation for dependable framework modelling, CS achieves optimal performance characteristics. High intricacy is seen as one of the CS's shortcomings. In this article, we suggest a novel CS algorithm for 5G waveforms. By restricting the computation of cyclostationary characteristics and the signal autocorrelation, the complexity of CS is reduced. To evaluate the performance of 5G waveforms, the Energy Detection (ED) and CS spectrum sensing algorithms based on cognitive radio (CR) are presented. The results of the study show that the suggested CS algorithm did a good job of detection and gained 2 dB compared to the conventional standards.
We study a continuous data assimilation (CDA) algorithm for a velocity-vorticity formulation of the 2D Navier-Stokes equations in two cases: nudging applied to the velocity and vorticity, and nudging ...applied to the velocity only. We prove that under a typical finite element spatial discretization and backward Euler temporal discretization, application of CDA preserves the unconditional long-time stability property of the velocity-vorticity method and provides optimal long-time accuracy. These properties hold if nudging is applied only to the velocity, and if nudging is also applied to the vorticity then the optimal long-time accuracy is achieved more rapidly in time. Numerical tests illustrate the theory, and show its effectiveness on an application problem of channel flow past a flat plate.
In this paper, we consider a plate equation with nonlinear damping and logarithmic source term. By the contraction mapping principle, we establish the local existence. The global existence and decay ...estimate of the solution at subcritical initial energy are obtained. We also prove that the solution with negative initial energy blows up in finite time under suitable conditions. Moreover, we also give the blow-up in finite time of solution at the arbitrarily high initial energy for linear damping (i.e. <inline-formula><tex-math id="M1">$ m = 2 $</tex-math></inline-formula>).
The development of the Internet of things (IoT) and online platforms enables companies and governments to collect data from a much broader spatial and temporal area in the logistics industry. The ...huge amount of data provides new opportunities to handle uncertainty in optimization problems within the logistics system. Accordingly, various prescriptive analytics frameworks have been developed to predict different parts of uncertain optimization problems, including the uncertain parameter, the combined coefficient consisting of the uncertain parameter, the objective function, and the optimal solution. This tutorial serves as the pioneer to introduce existing literature on state-of-the-art prescriptive analytics methods, such as the predict-then-optimize framework, the smart predict-then-optimize framework, the weighted sample average approximation framework, the empirical risk minimization framework, and the kernel optimization framework. Based on these frameworks, this tutorial further proposes possible improvements and practical tips to be considered when we use these methods. We hope that this tutorial will serve as a reference for future prescriptive analytics research on the logistics system in the era of big data.
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