Long-time dynamics of an epidemic model with nonlocal diffusion and free boundaries

<abstract><p>In this paper, we consider a reaction-diffusion epidemic model with nonlocal diffusion and free boundaries, which generalises the free-boundary epidemic model by Zhao et al. ^{[<xref ref-type="bibr" rid="b1">1</xref>]} by including spatial mobility of the infective host population. We obtain a rather complete description of the long-time dynamics of the model. For the reproduction number $ R_0 $ arising from the corresponding ODE model, we establish its relationship to the spreading-vanishing dichotomy via an associated eigenvalue problem. If $ R_0 \le 1 $, we prove that the epidemic vanishes eventually. On the other hand, if $ R_0 &gt; 1 $, we show that either spreading or vanishing may occur depending on its initial size. In the case of spreading, we make use of recent general results by Du and Ni ^{[<xref ref-type="bibr" rid="b2">2</xref>]} to show that finite speed or accelerated spreading occurs depending on whether a threshold condition is satisfied by the kernel functions in the nonlocal diffusion operators. In particular, the rate of accelerated spreading is determined for a general class of kernel functions. Our results indicate that, with all other factors fixed, the chance of successful spreading of the disease is increased when the mobility of the infective host is decreased, reaching a maximum when such mobility is 0 (which is the situation considered by Zhao et al. ^{[<xref ref-type="bibr" rid="b1">1</xref>]}).</p></abstract>

An analysis of two degenerate double-Hopf bifurcations

<abstract><p>The generic double-Hopf bifurcation is presented in detail in literature in textbooks like references. In this paper we complete the study of the double-Hopf bifurcation with two degenerate (or nongeneric) cases. In each case one of the generic conditions is not satisfied. The normal form and the corresponding bifurcation diagrams in each case are obtained. New possibilities of behavior which do not appear in the generic case were found.</p></abstract>

Pullback $ \mathcal{D} $-attractors of the three-dimensional non-autonomous micropolar equations with damping

<abstract><p>In this paper, we consider the three-dimensional non-autonomous micropolar equations with damping term in periodic domain $ \mathbb{T}^{3} $. By assuming external forces satisfy certain condtions, the existence of pullback $ \mathcal{D} $-attractors for the three-dimensional non-autonomous micropolar equations with damping term is proved in $ V_{1}\times V_{2} $ and $ H^{2}\times H^{2} $ with $ 3 &lt; \beta &lt; 5 $.</p></abstract>

A comparison of analytical solutions of nonlinear complex generalized Zakharov dynamical system for various definitions of the differential operator

<abstract><p>This paper considers deriving new exact solutions of a nonlinear complex generalized Zakharov dynamical system for two different definitions of derivative operators called conformable and $ M- $ truncated. The system models the spread of the Langmuir waves in ionized plasma. The extended rational $ sine-cosine $ and $ sinh-cosh $ methods are used to solve the considered system. The paper also includes a comparison between the solutions of the models containing separately conformable and $ M- $ truncated derivatives. The solutions are compared in the $ 2D $ and $ 3D $ graphics. All computations and representations of the solutions are fulfilled with the help of Mathematica 12. The methods are efficient and easily computable, so they can be applied to get exact solutions of non-linear PDEs (or PDE systems) with the different types of derivatives.</p></abstract>

The algebraic classification of nilpotent commutative algebras

<p style='text-indent:20px;'>This paper is devoted to the complete algebraic classification of complex <inline-formula><tex-math id="M1">\begin{document}$ 5 $\end{document}</tex-math></inline-formula>-dimensional nilpotent commutative algebras. Our method of classification is based on the standard method of classification of central extensions of smaller nilpotent commutative algebras and the recently obtained classification of complex <inline-formula><tex-math id="M2">\begin{document}$ 5 $\end{document}</tex-math></inline-formula>-dimensional nilpotent commutative <inline-formula><tex-math id="M3">\begin{document}$ \mathfrak{CD} $\end{document}</tex-math></inline-formula>-algebras.</p>

On the construction of $ \mathbb Z^n_2- $grassmannians as homogeneous $ \mathbb Z^n_2- $spaces

<abstract><p>In this paper, we construct the $ \mathbb Z^n_2- $grassmannians by gluing of the $ \mathbb Z^n_2- $domains and give an explicit description of the action of the $ \mathbb Z^n_2- $Lie group $ GL(\overrightarrow{\textbf{m}}) $ on the $ \mathbb Z^n_2- $grassmannian $ G_{ \overrightarrow{\textbf{k}}}(\overrightarrow{\textbf{m}}) $ in the functor of points language. In particular, we give a concrete proof of the transitively of this action, and the gluing of the local charts of the $ \mathbb Z^n_2- $grassmannian.</p></abstract>

Optimal time-decay rates of the compressible Navier–Stokes–Poisson system in $ \mathbb R^3 $

<p style='text-indent:20px;'>We are concerned with the Cauchy problem of the 3D compressible Navier–Stokes–Poisson system. Compared to the previous related works, the main purpose of this paper is two–fold: First, we prove the optimal decay rates of the higher spatial derivatives of the solution. Second, we investigate the influences of the electric field on the qualitative behaviors of solution. More precisely, we show that the density and high frequency part of the momentum of the compressible Navier–Stokes–Poisson system have the same <inline-formula><tex-math id="M2">\begin{document}$ L^2 $\end{document}</tex-math></inline-formula> decay rates as the compressible Navier–Stokes equation and heat equation, but the <inline-formula><tex-math id="M3">\begin{document}$ L^2 $\end{document}</tex-math></inline-formula> decay rate of the momentum is slower due to the effect of the electric field.</p>