In this paper, we mainly discuss an important class of nonclassical diffusion equation which the additional damping terms
vary over time. The existence of global weak solution is obtained by using the method of Faedo-Galerkin and analytical techniques. Meanwhile, we also prove the uniqueness of the solution and the continuous dependence on initial value, where the nonlinearity f satisfies arbitrary polynomial growth.
This paper studies the following fourth order quasilinear elliptic equation
\begin{equation*}
\left\{\begin{aligned}
&\triangle^{2} u-\triangle u+V(x)u-\frac{1}{2}u\triangle (u^{2})=f(u),&x\in \mathbb{R}^{N},\\
&u\in H^{2}(\mathbb{R}^{N}),
\end{aligned}
\right.
\end{equation*}
where $\triangle^{2}:=\triangle(\triangle)$ is the biharmonic operator, $2<N\leq 6$. We prove that the equation admits a ground state solution of the Nehari-Poho\u{z}aev type.
Let $\gamma^{*}(D)$ denote the twin domination number of digraph $D$ and let $C_{m}\square C_{n}$ denote the Cartesian product of the directed cycle $C_{m}$ and $C_{n}$, for $m, n\geq 2$. In this paper, we give a lower bound for $\gamma^{*}(C_{m}\square C_{n})$ and we determine the exact values of $\gamma^{*}(C_{m}\square C_{n})$ when $m,~n\equiv 0~ ({\rm mod} ~3)$ and when $m\equiv 2~({\rm mod}~3)$.
Let $p$ be an odd prime. In this paper, we obtain the number of (connected)
Cayley graphs on the dicyclic groups $T_{4p}=\langle a,b~|~a^{p}=b^4=1,b^{-1}ab=a^{-1}\rangle$ up to isomorphism by using the P\'{o}lya enumeration theorem.
In this paper, a complete classification of 11-valent symmetric graphs of $4p$ order is given, where $p$ is a prime.
According to the results of the classification, there is only one the 11-valent graph of order $4p$, which is a complete graph $K_{12}$.
Since the classical ARMA residual control chart is susceptible to outlier values, this paper first establishes a robust ARMA model based on the Institute of Geodesy & Geophysics III IGGIII weight function to obtains an independent residual distribution sequence. Then a robust residual control chart is constructed based on the estimations of the mean and standard deviation by the weighted three mean and average absolute deviation, respectively. Simulations and empirical tests show that the robust control chart can resist the outlier interferences better, and the monitoring effect is better than that of the traditional control chart.
In this paper, we give a survey on the extension of (quasi-)plurisubharmonic functions from complex submanifolds.
We firstly review the extension of plurisubharmonic functions on Stein manifolds, and then review the extension of quasi-plurisubharmonic functions on compact complex manifolds, including some unpublished new results of the authors Wang and Zhou.
This paper investigates a time-delayed reaction-diffusion model with chemotaxis for phytoplankton. The stability of the positive steady states is obtained by analyzing the related characteristic equations. With the help of Crandall- Rabinowitz's local partial bifurcation theory, the chemotaxis sensitivity coefficient and delay are taken as bifurcation parameters respectively to investigate the existence of Turing bifurcation and Hopf bifurcation. Then, the direction and stability of Hopf bifurcation are studied by using the central manifold theorem and the normal form method. Finally, numerical simulations are presented to show the influence of chemotaxis and time delay on the bifurcation and pattern formation of the system. Our results show that: in the system without time delay, when the chemotaxis sensitivity coefficient exceeds a critical value, the positive steady-state solution of the system will change from stable to unstable (Turing instability); in the system with delay, when the chemotaxis sensitivity coefficient is less than the critical value, if the delay is below a certain value, the positive steady-state solution is locally asymptotically stable; if the time delay exceeds a certain value, then the system will undergo Hopf bifurcation at the positive steady-state solution, and a stable spatially homogeneous periodic solution will be bifurcated from the positive steady-state solution.
In this paper, we first study the exact peakon, periodic peakon, pseudo-peakon as well as the compacton solutions for the generalized Camassa-Holm equation and the Degasperis-Procesi equation. Based on the method of dynamical systems and the theory of singular traveling wave equations, the exact explicit parametric representations of the above mentioned solutions are derived. These solutions tell us that peakon is a limit solution of a family of periodic peakons or a limit solution of a family of pseudo-peakons under two classes of limit senses. The pseudo-peakon and pseudo-periodic peakon family are smooth classical solutions with two time scales.
Second, we use some nonlinear wave equation models to show that there exist various exact explicit peakon solutions, which are different from the peakon solutions given by the generalized Camassa-Holm equation and the Degasperis-Procesi equation.
Third, we point out that the so called ``peakon equations'' in some references have no peakons. Corresponding to these ``peakon equations'', their traveling systems are the singular traveling systems of the second kind, which can not have the peakon solution.
The symmetry problem for finite graphs has been extensively investigated over the past
few decades. This article is devoted to giving an introduction in one particular family of symmetric
graphs, namely (locally) $s$-geodesic-transitive graphs. Recently, substantial progress has been made on the study of this family of graphs, many
open problems have been solved, and many new research problems have arisen.
The methods used in this area range
from deep group theory, including the finite simple group classification, through to combinatorial
techniques.
