On the absence of global solutions for quantum versions of Schrödinger equations and systems

We, first, consider the quantum version of the nonlinear Schrödinger equation

$i^q{D}_{q|t}u(t,x)+\Delta u(qt,x)=\lambda |u(qt,x){|}^p,t>0,x\in {\mathbb{R}}^N,$

where $0<q<1$, $i^q$ is the principal value of $i^q$, $D_{q|t}$ is the $q$-derivative with respect to $t$, $\Delta$ is the Laplacian operator in ${\mathbb{R}}^N$, $\lambda \in ??\left\{0\right\}$, $p>1$, and $u(t,x)$ is a complex-valued function. Sufficient conditions for the nonexistence of global weak solution to the considered equation are obtained under suitable initial data. Next, we study the system of nonlinear coupled equations

$i^q{D}_{q|t}u(t,x)+\Delta u(qt,x)=\lambda |v(qt,x){|}^p,t>0,x\in {\mathbb{R}}^N,$

$i^q{D}_{q|t}v(t,x)+\Delta v(qt,x)=\lambda |u(qt,x){|}^m,t>0,x\in {\mathbb{R}}^N,$     

Dynamic behavior of droplet through a confining orifice:A lattice Boltzmann study

The motion of gravity-driven deformable droplets passing through a confining orifice in two-dimensional ($2D$) space is numerically studied by the phase-field-based multiple-relaxation-time (MRT) lattice Boltzmann (LB) model, and the ratio of orifice-to-droplet diameter is less than 1. Droplets are placed just above a sink with an orifice in the middle, accelerate under gravity and encounter the orifice plate. In this work, we mainly consider the effects of the Bond number ($Bo$), orifice-to-droplet diameter ratio ($r=d∕D$), plate thickness ($H_t$), wettability (or contact angle) and the diameter ratio of two droplets ($r_d=D_1∕D_2$) on the dynamic behavior of droplet through the orifice. The results show that these issues have great influences on the typical flow patterns (i.e., release and capture). With the decrease of contact angle, the droplet is more easily captured, and there exists a critical equilibrium contact angle ${\theta }^{eq}$ when the Bond number and the orifice-to-droplet diameter ratio as well as the thickness of the plate are specified. For the case with $\theta >{\theta }^{eq}$, the droplet can finally pass through the orifice, otherwise, the droplet cannot pass through the orifice. In addition, the droplet is more likely to pass through the orifice as the thickness of the obstacle increases. Actually, when the obstacle thickness is large enough, droplet breaks into three segments and a liquid slug is formed in a hydrophilic orifice. Finally, for the evolution of two droplets with a larger diameter ratio ($r_d=1.0$), the combined droplet finally passes through the orifice due to greater inertia than the cases with $r_d=0$ and $r_d=0.43$. Besides, we also establish the relation $r=0.57^{\frac{2}{3}}B{o}^{?\frac{1}{3}}$ which can be used to separate droplet release from capture at $H_t=1.2mm$.     

Note on nonparametric spectral analysis of wideband spectrum with missing data via sample-and-hold interpolation and deconvolution

In [1] a procedure for bias-free estimation of the autocorrelation function is introduced for equidistantly sampled data with randomly occurring samples being invalid. The method incorporates sample-and-hold interpolation of the missing data points. The occurring dynamic error of the primary estimate of the correlation function is treated by a deconvolution procedure with two parameters $c_0$ and $c_1$ with $c_0+2c_1=1$, which are the on-diagonal and the aside-diagonal parameters of a specific correction matrix (at all lag times except zero). The parameters $c_0$ and $c_1$ were obtained as a function of the probability α of a sample to be valid by numerical simulation. However, explicit expressions for the parameters $c_0(\alpha )=1-\frac{2}{\alpha }+\frac{2}{{\alpha }^2}$ and $c_1(\alpha )=\frac{1}{\alpha }-\frac{1}{{\alpha }^2}$ can be derived, which might improve the usability of the deconvolution procedure in [1].     

Superconvergence analysis of nonconforming finite element method for time-fractional nonlinear parabolic equations on anisotropic meshes

In this paper, we prove a novel result of the consistency error estimate with order $O\left(h^2\right)$ for $E{Q}_1^{rot}$ element (see Lemma 2) on anisotropic meshes. Then, a linearized fully discrete Galerkin finite element method (FEM) is studied for the time-fractional nonlinear parabolic problems, and the superclose and superconvergent estimates of order $O(\tau +h^2)$ in broken $H^1$-norm on anisotropic meshes are derived by using the proved character of $E{Q}_1^{rot}$ element, which improve the results in the existing literature. Numerical results are provided to confirm the theoretical analysis.     

Global solution of a diffusive predator–prey model with prey-taxis

We consider the prey-taxis system:

$\left\{\begin{array}{cc}{u}_t=d_1\Delta u?\chi ??(u?v)+u(a?\mu u)+buf\left(v\right),\hfill & x\in \Omega ,t>0,\hfill \\ v_t=d_2\Delta v+v(c?\beta v)?uf\left(v\right),\hfill & x\in \Omega ,t>0\hfill \end{array}\right.$

in a smoothly bounded domain $\Omega ?{\mathbb{R}}^n$, with zero-flux boundary condition, where $a,d_1,d_2,\chi ,\mu ,b,c$ are positive constants and $\beta$ is a non-negative constant. We first investigate the global existence and local boundedness of solution for the case $\beta =0$. Moreover, when $\beta >0$, we show that the solution exists globally and is uniformly bounded provided $\mu$ is large enough.     

A fast singular boundary method for 3D Helmholtz equation

This paper presents a fast singular boundary method (SBM) for three-dimensional (3D) Helmholtz equation. The SBM is a boundary-type meshless method which incorporates the advantages of the boundary element method (BEM) and the method of fundamental solutions (MFS). It is easy-to-program, and attractive to the problems with complex geometries. However, the SBM is usually limited to small-scale problems, because of the operation count of $O\left(N^3\right)$ with direct solvers or $O\left(N^2\right)$ with iterative solvers, as well as the memory requirement of $O\left(N^2\right)$. To overcome this drawback, this study makes the first attempt to employ the precorrected-FFT (PFFT) to accelerate the SBM matrix–vector multiplication at each iteration step of the GMRES for 3D Helmholtz equation. Consequently, the computational complexity can be reduced from $O\left(N^2\right)$ to $O\left(NlogN\right)$ or $O\left(N\right)$. Three numerical examples are successfully tested on a desktop computer. The results clearly demonstrate the accuracy and efficiency of the developed fast PFFT-SBM strategy.     

Existence of multiple solutions for nonhomogeneous Schrödinger–Kirchhoff system involving the fractional p-Laplacian with sign-changing potential

In this paper, we prove the existence of multiple solutions for the following Schrödinger–Kirchhoff system involving the fractional $p$-Laplacian

$\begin{array}{c}\hfill \left\{\begin{array}{c}M\left({?}_{{\mathbb{R}}^{2N}}\frac{{|u\left(x\right)?u\left(y\right)|}^p}{{|x?y|}^{N+ps}}dxdy\right){(?\Delta )}_p^{s}u+V\left(x\right){\left|u\right|}^{p?2}u=F_u(x,u,v)+\lambda g\left(x\right),x\in {\mathbb{R}}^N,\hfill \\ M\left({?}_{{\mathbb{R}}^{2N}}\frac{{|v\left(x\right)?v\left(y\right)|}^p}{{|x?y|}^{N+ps}}dxdy\right){(?\Delta )}_p^{s}u+V\left(x\right){\left|v\right|}^{p?2}v=F_v(x,u,v)+\lambda h\left(x\right),x\in {\mathbb{R}}^N,\hfill \\ u\left(x\right)\to 0,v\left(x\right)\to 0,\text{as}\left|x\right|\to +\infty ,\hfill \end{array}\right.\end{array}$

where ${(?\Delta )}_p^s$ denotes the fractional $p$-Laplacian of order $s\in (0,1)$, $2\le p<\infty$, $ps<N$, $F_u=\frac{?F}{?u}$, $F_v=\frac{?F}{?v}$, $V\left(x\right)$ is allowed to be sign-changing, $\lambda >0$ and $g,h:{\mathbb{R}}^N\to \mathbb{R}$ is a perturbation. Under some certain assumptions on $f$, we obtain the existence of multiple solutions for this problem via Ekeland’s variational principle and mountain pass theorem.     

Existence and asymptotic behavior of positive ground state solutions for coupled nonlinear fractional Kirchhoff-type systems

This paper is concerned with the following linearly coupled fractional Kirchhoff-type system

$\left\{\begin{array}{c}\left(a+b{\int }_{{\mathbb{R}}^3}|{(?△)}^{\frac{\alpha }{2}}u{|}^2\mathrm{d}x\right){(?△)}^{\alpha }u+\lambda u=f\left(u\right)+\gamma v,\text{in}{\mathbb{R}}^3,\hfill \\ \left(c+d{\int }_{{\mathbb{R}}^3}|{(?△)}^{\frac{\alpha }{2}}v{|}^2\mathrm{d}x\right){(?△)}^{\alpha }v+\mu v=g\left(v\right)+\gamma u,\text{in}{\mathbb{R}}^3,\hfill \\ u,v\in H^{\alpha }\left({\mathbb{R}}^3\right),\hfill \end{array}\right.$

where $a,c,\lambda ,\mu >0$, $b,d\ge 0$ are constants, $\alpha \in [\frac{3}{4},1)$ and $\gamma >0$ is a coupling parameter. Under the general Berestycki–Lions conditions on the nonlinear terms $f$ and $g$, we prove the existence of positive vector ground state solutions of Poho?aev type for the above system via variational methods. Moreover, the asymptotic behavior of these solutions as $\gamma \to 0^{+}$ is explored as well. Recent results from the literature are generally improved and extended.     

Global well-posedness of the d-D Oldroyd-B type models with fractional Laplacian dissipation

This paper focuses on the Cauchy problem of the d-dimensional incompressible Oldroyd-B type models for viscoelastic flow with fractional Laplacian dissipation, namely, with ${(?\Delta )}^{{\eta }_1}u$ and ${(?\Delta )}^{{\eta }_2}\tau$. For ${\eta }_1\ge \frac{1}{2}+\frac{d}{4}$, ${\eta }_2>0$ and ${\eta }_1+{\eta }_2\ge 1+\frac{d}{2}$, we obtain the global regularity of strong solutions when the initial data $(u_0,{\tau }_0)$ are sufficiently smooth.     

An accelerated cyclic-reduction-based solvent method for solving quadratic eigenvalue problem of gyroscopic systems

The quadratic eigenvalue problem (QEP) $({\lambda }^{2}M+\lambda G+K)x=0$, with $M^T=M$ being positive definite, $K^T=K$ being negative definite and $G^T=?G$, is associated with gyroscopic systems. In Guo (2004), a cyclic-reduction-based solvent (CRS) method was proposed to compute all eigenvalues of the above mentioned QEP. Firstly, the problem is converted to find a suitable solvent of the quadratic matrix equation (QME) $M{X}^2+GX+K=0$. Then using a Cayley transformation and a proper substitution, the QME is transformed into the nonlinear matrix equation (NME) $Z+A^T{Z}^{?1}A=Q$ with $A=M+K+G$ and $Q=2(M?K)$. The problem finally can be solved by applying the CR method to obtain the maximal symmetric positive definite solution of the NME as long as the QEP has no eigenvalues on the imaginary axis or for some cases where the QEP has eigenvalues on the imaginary axis. However, when all eigenvalues of the QEP are far away from or near the origin, the Cayley transformation seems not to be the best one and the convergence rate of the CRS method proposed in Guo (2004) might be further improved. In this paper, inspired by using a doubling algorithm to solve the QME, we use a Möbius transformation instead of the Cayley transformation to present an accelerated CRS (ACRS) method for solving the QEP of gyroscopic systems. In addition, we discuss the selection strategies of optimal parameter for the ACRS method. Numerical results demonstrate the efficiency of our method.     

Ground state solutions of Pohoz̆aev type for fractional Choquard equations with general nonlinearities

In this paper, we study the fractional Choquard equation

${(?\Delta )}^{s}u+u=\left(\right|x{|}^{?\mu }1F\left(u\right))f\left(u\right),\text{in}{\mathbb{R}}^N,$

where $N\ge 3$, $0<s<1$, $0<\mu <min\{N,4s\}$, and $f\in C(\mathbb{R},\mathbb{R})$ satisfies the general Berestycki–Lions conditions. Combining constrained variational method with deformation lemma, we obtain a ground state solution of Pohoz?aev type for the above equation. The result improves some ones in Shen et al. (2016).     

Global well-posedness of the incompressible fractional Navier–Stokes equations in Fourier–Besov spaces with variable exponents

We study the Cauchy problem of the fractional Navier–Stokes equations in critical variable exponent Fourier–Besov spaces $F{\stackrel{?}{B}}_{p\left(?\right),q}^{4?2\alpha ?\frac{3}{p\left(?\right)}}$. We discuss some properties of variable exponent Fourier–Besov spaces and prove a general global well-posedness result which covers some recent works about classical Navier–Stokes equations.     

Effect of Porosity on free and forced vibration characteristics of the GPL reinforcement composite nanostructures

This article investigates the influence of porosity on free and forced vibration characteristics of a nanoshell reinforced by graphene platelets (GPL). The material properties of piece-wise graphene-reinforced composites (GPLRCs) are assumed to be graded in the thickness direction of a cylindrical nanoshell and estimated using a nanomechanical model. In addition, because of imperfection of the current structure, three kinds of porosity distributions are considered. The nanostructure is modeled using modified strain gradient theory (MSGT) which is a size-dependent theory with three length scale parameters. The novelty of the current study is to consider the effects of porosity, GPLRC and MSGT on dynamic and static behaviors of the nanostructure. Considering three length scale parameters ( $l_0=5h$, $l_1=3h$, $l_2=5h$ ) in MSGT leads to a better agreement with MD simulation in comparison by other theories. Finally, effects of different factors on static and dynamic behaviors of the porous nanostructure are examined in detail.     

Blow-up phenomena in porous medium equation systems with nonlinear boundary conditions

This paper deals with the blow-up phenomena for the following porous medium equation systems with nonlinear boundary conditions $\left\{\begin{array}{cc}{u}_t=\Delta u^m+k_1\left(t\right)f_1\left(v\right),v_t=\Delta v^n+k_2\left(t\right)f_2\left(u\right)\hfill & in\Omega \times (0,t^1),\hfill \\ \hfill \\ \hfill \\ \frac{?u}{?\nu }=g_1\left(u\right),\frac{?v}{?\nu }=g_2\left(v\right)\hfill & on?\Omega \times (0,t^1),\hfill \\ \hfill \\ \hfill \\ u(x,0)=u_0\left(x\right)\ge 0,v(x,0)=v_0\left(x\right)\ge 0\hfill & in\overline{\Omega },\hfill \end{array}\right.$ where $m,n>1$, $\Omega ?{\mathbb{R}}^N(N\ge 2)$ is bounded convex domain with smooth boundary. Using a differential inequality technique and a Sobolev inequality, we prove that under certain conditions on data, the solution blows up in finite time. We also derive an upper and a lower bound for blow-up time. In addition, as applications of the abstract results obtained in this paper, an example is given.