Finite iterative Hermitian R-conjugate solutions of the generalized coupled Sylvester-conjugate matrix equations

In this paper, an iterative algorithm for solving a generalized coupled Sylvester-conjugate matrix equations over Hermitian $R$-conjugate matrices given by $A_{1}V{B}_1+C_{1}W{D}_1=E_1\overline{V}{F}_1+G_1$ and $A_{2}V{B}_2+C_{2}W{D}_2=E_2\overline{V}{F}_2+G_2$ is presented. When these two matrix equations are consistent, the convergence theorem shows that a solution can be obtained within finite iterative steps in the absence of round-off error for any initial arbitrary Hermitian $R$-conjugate solution matrices $V_1$, $W_1$. Some lemmas and theorems are stated and proved where the iterative solutions are obtained. A numerical example is given to demonstrate the behavior of the proposed method and to support the theoretical results.     

Blow-up analysis in quasilinear reaction–diffusion problems with weighted nonlocal source

In this paper, we consider the blow-up of solutions to a class of quasilinear reaction–diffusion problems

where $\Omega$ is a bounded convex domain in ${\mathbb{R}}^n(n\ge 2)$, weighted nonlocal source satisfies $a\left(x\right)f\left(u(x,t)\right)\le a_1+a_2{\left(u(x,t)\right)}^p{\left({\int }_{\Omega }{\left(u(x,t)\right)}^l\mathrm{d}x\right)}^m,$ and $a_1,a_2,p,l,$ and $m$ are positive constants. By utilizing a differential inequality technique and maximum principles, we establish conditions to guarantee that the solution remains global or blows up in a finite time. Moreover, an upper and a lower bound for blow-up time are derived. Furthermore, two examples are given to illustrate the applications of obtained results.     

Multiplicity of positive solutions for the Kirchhoff-type equations with critical exponent in RN

In this work we study the existence and multiplicity of solutions to the following Kirchhoff-type problem with critical nonlinearity in ${\mathbb{R}}^N$

where $N\ge 2p$, $\mu ,\lambda ,a,b>0$ and the nonlinearity $f(x,u)$ satisfies certain subcritical growth conditions. By using topological and variational methods, infinitely many positive solutions are obtained.     

Global asymptotic stability of steady states in a chemotaxis-growth system with singular sensitivity

This paper deals with a fully parabolic chemotaxis-growth system with singular sensitivity

under homogeneous Neumann boundary conditions in a smooth bounded domain $\Omega ?{\mathbb{R}}^2$, where the parameters $\chi ,\mu >0$ and $r\in \mathbb{R}$. Global existence and boundedness of solutions to the above system were established under some suitable conditions by Zhao and Zheng (2017). The main aim of this paper is further to show the large time behavior of global solutions which cannot be derived in the previous work.     

Existence and multiplicity of solutions for a nonlocal problem with critical Sobolev exponent

In this work, we are interested in considering the following nonlocal problem

where $\Omega ?{\mathbb{R}}^N(N\ge 4)$ is a smooth bounded domain, $a\ge 0,b>0,1<q<2,\mu ,\lambda >0$ and $2^1=\frac{2N}{N?2}$ is the critical Sobolev exponent. By using the variational method and the critical point theorem, some existence and multiplicity results are obtained.     

A variable exponent nonlocal p(x)-Laplacian equation for image restoration

In this paper, we focus on the mathematical and numerical study of a variable exponent nonlocal $p\left(X\right)$-Laplacian equation for image denoising. Based on the Semigroup Theory, we prove the existence and uniqueness of solution for the proposed model. To illustrate the efficiency and effectiveness of our model, we provide the denoising experimental results as well we compare it with some existing models in the literature.     

Global and blow-up solutions for nonlinear parabolic equations with Robin boundary conditions

In this paper we discuss the blow-up for classical solutions to the following class of parabolic equations with Robin boundary condition: $\{\begin{array}{cc}{\left(b\left(u\right)\right)}_t=??\left(g\left(u\right)?u\right)+f\left(u\right)\hfill & \text{in }\Omega \times (0,T)\text{,}\hfill \\ \frac{?u}{?n}+\gamma u=0\hfill & \text{on }?\Omega \times (0,T)\text{,}\hfill \\ u(x,0)=h\left(x\right)\ge 0\hfill & \text{in }\overline{\Omega }\text{,}\hfill \end{array}$ where $\Omega$ is a bounded domain of ${\mathbb{R}}^N(N\ge 2)$ with smooth boundary $?\Omega$. By constructing some appropriate auxiliary functions and using a first-order differential inequality technique, we derive conditions on the data which guarantee the blow-up or the global existence of the solution. For the blow-up solution, a lower bound on blow-up time is also obtained. Moreover, some examples are presented to illustrate the applications.     

Innovation based on Gaussian elimination to compute generalized inverse AT,S(2)

In this paper, we execute elementary row and column operations on the partitioned matrix $\left(\begin{array}{cc}\hfill GAG\hfill & \hfill G\hfill \\ \hfill G\hfill & \hfill 0\hfill \end{array}\right)$ into $\left(\begin{array}{cc}\hfill \left(\begin{array}{cc}\hfill I_s\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill \end{array}\right)\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill ?A_{T,S}^{\left(2\right)}\hfill \end{array}\right)$to compute generalized inverse $A_{T,S}^{\left(2\right)}$ of a given complex matrix $A$, where $G$ is a matrix such that $R\left(G\right)=T$ and $N\left(G\right)=S$. The total number of multiplications and divisions operations is $T(m,n,s)=2m{n}^2+\frac{4m?s?1}{2}ns+(m?s)ns+mns$ and the upper bound of $T(m,n,s)$ is less than $6m{n}^2?\frac{3}{2}{n}^3?\frac{1}{2}{n}^2$ when $n\le m$. A numerical example is shown to illustrate that this method is correct.     

An iteration method for solving the linear system Ax=b

In this paper, based on a convergence splitting of the matrix $A$, we present an inner–outer iteration method for solving the linear system $Ax=b$. We analyze the overall convergence of this method without any other restriction on its parameters. Moreover, we give the accelerated inner–outer iteration method, and discuss how to apply the inner–outer iterations as a preconditioner for the Krylov subspace methods. The inner–outer iteration method can also be used for the solution of $AXB=C$. Finally, several numerical examples are given to validate the performance of our proposed algorithms.     

Existence of positive ground state solutions for a critical Kirchhoff type problem with sign-changing potential

In this work, we are interested in studying the following Kirchhoff type problem

where $\Omega ?{\mathbb{R}}^N(N\ge 3)$ is a smooth bounded domain, $2^1=\frac{2N}{N?2}$ is the critical Sobolev exponent, $0<q<1,\lambda >0$, and $f\in L^{\infty }\left(\Omega \right)$ with the set $\{x\in \Omega :f\left(x\right)>0\}$ of positive measures, and $g\in L^{\infty }\left(\Omega \right)$ with $g\left(x\right)\ge 0,g?0.$ By the Nehari method and variational method, the existence of positive ground state solutions is obtained.     

Existence of ground state solutions of Nehari–Pohozaev type for fractional Schrödinger–Poisson systems with a general potential

In this paper, we consider the following fractional Schrödinger–Poissonproblem

where $0<s\le t<1$ and $2s+2t>3$, the potential $V\left(x\right)$ is weakly differentiable and $f\in C(\mathbb{R},\mathbb{R})$. By introducing some new tricks, we prove that the problem admits a ground state solution of Nehari–Pohozaev type under mild assumptions on $V$ and $f$. The results here extend the existing study.     

Existence of ground states for a Kirchhoff type problem without 4-superlinear condition

We consider the existence of ground state solutions for the Kirchhoff type problem

where $a,b>0$, $N=1,2,3$ and $2<p<2^1$. Here we are interested in the case that $2<p\le 4$ since the existence of ground state for $4<p\le 2^1$ is easily obtained by a standard variational argument. Our method is based on a Poho$\stackrel{?}{z}$aev type identity.     

Lattice Boltzmann method based on Dual-MRT model for three-dimensional natural convection and entropy generation in CuO–water nanofluid filled cuboid enclosure included with discrete active walls

In the present study, the three-dimensional natural convection and entropy generation in a cuboid enclosure included with various discrete active walls is analyzed using lattice Boltzmann method. The enclosure is filled with CuO–water nanofluid. To predict thermo-physical properties, dynamic viscosity and thermal conductivity, of CuO–water nanofluid, the KKL model is applied to consider the effect of Brownian motion on nanofluid properties. In lattice Boltzmann simulation, two different MRT models are used to solve the problem. The D3Q7-MRT model is used to solve the temperature filed, and the D3Q19 is employed to solve the fluid flow of natural convection within the enclosure. The influences of different Rayleigh numbers $\left(10^3<Ra<10^6\right)$ and solid volume fractions $\left(0<\phi <0.04\right)$ and four different arrangements of discrete active walls on the fluid flow, heat transfer, total entropy generation, local heat transfer irreversibility and local fluid friction irreversibility are presented comprehensively.     

Entropy generation analysis and heatline visualization of free convection in nanofluid (KKL model-based)-filled cavity including internal active fins using lattice Boltzmann method

Two-dimensional natural convection and entropy generation in a square cavity filled with CuO–water nanofluid is performed. The lattice Boltzmann method is employed to solve the problem numerically. The influences of different Rayleigh numbers $\left(10^3<Ra<10^6\right)$ and solid volume fractions $\left(0<\phi <0.05\right)$ on the fluid flow, heat transfer and total/local entropy generation are presented comprehensively. Also, the heatline visualization is employed to identify the heat energy flow. To predict the thermo-physical properties, dynamic viscosity and thermal conductivity, of CuO–water nanofluid, the KKL model is applied to consider the effect of Brownian motion on nanofluid properties. It is concluded that the configurations of active fins have pronounced effect on the fluid flow, heat transfer and entropy generation. Furthermore, the Nusselt number has direct relationship with Rayleigh number and solid volume fraction, and the entropy generation has direct and reverse relationships with Rayleigh number and solid volume fraction, respectively.     

C1 continuous h-adaptive least-squares spectral element method for phase-field models

To describe the interfacial dynamics between two phases using the phase-field method, the interfacial region needs to be close enough to a sharp interface so as to reproduce the correct physics. Due to the high gradients of the solution within the interfacial region and consequent high computational cost, the use of the phase-field method has been limited to the small-scale problems whose characteristic length is similar to the interfacial thickness. By using finer mesh at the interface and coarser mesh in the rest of computational domain, the phase-field methods can handle larger scale of problems with realistic interface thicknesses. In this work, a $C^1$ continuous $h$-adaptive mesh refinement technique with the least-squares spectral element method is presented. It is applied to the Navier–Stokes-Cahn–Hilliard (NSCH) system and the isothermal Navier–Stokes–Korteweg (NSK) system. Hermite polynomials are used to give global differentiability in the approximated solution, and a space–time coupled formulation and the element-by-element technique are implemented. Two refinement strategies based on the solution gradient and the local error estimators are suggested, and they are compared in two numerical examples.     

On the blow-up criterion for the quasi-geostrophic equations in homogeneous Besov spaces

In this paper, we consider the blow-up criterion for the quasi-geostrophic equations with dissipation ${\Lambda }^{\gamma }$ ($0<\gamma <1$). By establishing a new trilinear estimate, we show that if

$\theta \in L^{\frac{\gamma }{\gamma +s?1}}(0,T;{\stackrel{?}{B}}_{\infty ,\infty }^s\left({\mathbb{R}}^2\right))$

for some $s\in \left(1?\frac{\gamma }{2},1\right)$, then the solution can be extended smoothly past $T$. This improves and extends the corresponding results in Dong and Pavlovi? (2009) ([32]) and Yuan (2010).     

Even-order linear dynamic equations with mixed derivatives

We consider the time scale $k$th-order differential operators $D_k^{\Delta }y≔\{\begin{array}{cc}{y}^{\Delta \nabla \dots \Delta \nabla }& k\text{ even ,}\\ y^{\Delta \nabla \dots \nabla \Delta }& k\text{ odd ,}\end{array}{\tilde{D}}_k^{\Delta }y≔\{\begin{array}{cc}{y}^{\nabla \Delta \dots \nabla \Delta }& k\text{ even ,}\\ y^{\Delta \nabla \dots \nabla \Delta }& k\text{ odd ,}\end{array}$$D_k^{\nabla }y≔\{\begin{array}{cc}{y}^{\nabla \Delta \dots \nabla \Delta }& k\text{ even ,}\\ y^{\nabla \Delta \dots \Delta \nabla }& k\text{ odd ,}\end{array}{\tilde{D}}_k^{\nabla }y≔\{\begin{array}{cc}{y}^{\Delta \nabla \dots \Delta \nabla }& k\text{ even ,}\\ y^{\nabla \Delta \dots \Delta \nabla }& k\text{ odd ,}\end{array}$ and the higher-order dynamic equations $L\left(y\right)≔\sum _{\nu =0}^n{(-1)}^{\nu }{\tilde{D}}_{\nu }^{\nabla }\left(r_{\nu }\left(t\right)D_{\nu }^{\Delta }y\right)=0\text{,}$$M\left(y\right)≔\sum _{\nu =0}^n{(-1)}^{\nu }{\tilde{D}}_{\nu }^{\Delta }\left(r_{\nu }\left(t\right)D_{\nu }^{\nabla }y\right)=0\text{.}$ We will show that these equations can be investigated as special cases of the so-called (delta or nabla) symplectic dynamic systems $z^{\Delta }=S\left(t\right)z\text{,}{z}^{\nabla }=S\left(t\right)z\text{,}$ whose qualitative theory is well developed. We also suggest further perspectives of the investigation of the qualitative properties of higher-order equations with mixed derivatives.