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.     

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.     

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).     

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.     

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

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

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

Unsteady micropolar fluid flow in a thin domain with Tresca fluid–solid interface law

We consider a micropolar fluid flow in a two-dimensional domain. We assume that the velocity field satisfies a non-linear slip boundary condition of friction type on a part of the boundary while the micro-rotation field satisfies non-homogeneous Dirichlet boundary conditions. We prove the existence and uniqueness of a solution. Then motivated by lubrication problems we assume that the thickness and the roughness of the domain are of order $0<\epsilon <<1$ and we study the asymptotic behaviour of the flow as $\epsilon$ tends to zero. By using the two-scale convergence technique we derive the limit problem which is totally decoupled for the limit velocity and pressure $(v^0,p^0)$ on one hand and the limit micro-rotation $Z^0$ on the other hand. Moreover we prove that $v^0$, $p^0$ and $Z^0$ are uniquely determined via auxiliary well-posed problems.     

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.     

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.     

Quantification of thermodynamic properties for vaporisation reactions above solid Ga2O3 and In2O3 by Knudsen Effusion Mass Spectrometry

Even though Ga₂O₃ and In₂O₃ are broadly used as semi-conductors, thermodynamic data for their vaporisation reactions exhibit a large spread. Therefore, the vaporisation behaviour of solid Ga₂O₃ and In₂O₃ was determined by means of Knudsen Effusion Mass Spectrometry (KEMS). Ga₂O₃ and In₂O₃ were studied in an iridium Knudsen cell and heated over a temperature range of 1200–1750 K in order to identify the species present in the vapour phase, and determine their partial pressures. We find that M₂O (where M = Ga or In) is the most abundant gas species above the solid oxide, followed by M and MO, in accord with tabulated data. Following the calculation of partial pressures and equilibrium constants, we propose ${\Delta }_f{H}_{\mathrm{298,}3\mathrm{rd}}^{\mathrm{o}}\left(\mathrm{G}{\mathrm{a}}_2\mathrm{O}\left(\mathrm{g}\right)\right)$ = −68966 ± 7442 Jmol^-1 and ${\Delta }_f{H}_{\mathrm{298,}3\mathrm{rd}}^{\mathrm{o}}\left(\mathrm{I}{\mathrm{n}}_2\mathrm{O}\left(\mathrm{g}\right)\right)$ = −22245 ± 964 Jmol^-1 from the 3^rd law method. Deviations in ${\Delta }_f{H}_{\mathrm{298,}3\mathrm{rd}}^{\mathrm{o}}\left(i\right)$ relative to literature KEMS measurements are generally within ∼2% relative, and can be ascribed to the use of different ionisation cross sections, Knudsen cell material, temperature calibrations, as well as tabulated Gibbs energy functions. However, comparison with ab initio studies suggests the data reported in this work is more accurate than in previous studies, given that the ${\Delta }_f{H}_{\mathrm{298,}3\mathrm{rd}}^{\mathrm{o}}\left(\mathrm{I}\mathrm{n}\mathrm{O}\left(\mathrm{g}\right)\right)$ = 157744 ± 3681 Jmol^-1 deviates by only ∼0.2% from the theoretical value.     

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.     

Divide and capture: An improved cryptanalysis of the encryption standard algorithm RSA

RSA is a well known standard algorithm used by modern computers to encrypt and decrypt messages. In some applications, to save the decryption time, it is desirable to have a short secret key d compared to the modulus N. The first significant attack that breaks RSA with short secret key given by Wiener in 1990 is based on the continued fraction technique and it works with $d<\frac{1}{\sqrt[4]{18}}{N}^{.25}$. A decade later, in 2000, Boneh and Durfee presented an improved attack based on lattice technique which works with d < N^.292. Until this day, Boneh–Durfee attack remain as the best attack on RSA with short secret key. In this paper, we revisit the continued fraction technique and propose a new attack on RSA. Our main result shows that when $d<\sqrt{t(2\sqrt{2}+8/3)}{N}^{.75}/\sqrt{e},$ where e is the public exponent and t is a chosen parameter, our attack can break the RSA with the running time of O(tlog (N)). Our attack is especially well suited for the case where e is much smaller than N. When e ≈ N, the Boneh–Durfee attack outperforms ours. As a result, we could simultaneously run both attacks, our new attack and the classical Boneh–Durfee attack as a backup.