Solutions in Bessel‐potential spaces for wave equations with nonlinear damping

We consider a semilinear wave equation with nonlinear damping in the whole space . Local‐in‐time existence and uniqueness results are obtained in the class of Bessel‐potential spaces . Copyright © 2017 John Wiley & Sons, Ltd.     

Asymptotics for the third‐order nonlinear Schrödinger equation in the critical case

We consider the Cauchy problem for the third‐order nonlinear Schrödinger equation where and is the Fourier transform. Our purpose in this paper is to prove the large time asymptoitic behavior of solutions for the defocusing case λ > 0 with a logarithmic correction under the non zero mass condition Copyright © 2016 John Wiley & Sons, Ltd.     

Global smooth solution to the 2D Boussinesq equations with fractional dissipation

In this paper, we consider the 2D incompressible Boussinesq system with fractional Laplacian dissipation and thermal diffusion. On the basis of the previous works and some new observations, we show that the condition with suffices in order for the solution pair of velocity and temperature to remain smooth for all time. Copyright © 2017 John Wiley & Sons, Ltd.     

The periodic solution bifurcated from homoclinic orbit for coupled ordinary differential equations

We consider the problem of the periodic solutions bifurcated from a homoclinic orbit for a pair of coupled ordinary differential equations in . Assume that the autonomous system has a degenerate homoclinic solution γ in . A functional analytic approach is used to consider the existence of periodic solution for the autonomous system with periodic perturbations. By exponential dichotomies and the method of Lyapunov–Schmidt, the bifurcation function defined between two finite dimensional subspaces is obtained, where the zeros correspond to the existence of periodic solutions for the coupled ordinary differential equations near . Copyright © 2016 John Wiley & Sons, Ltd.     

Regular wavelets and Triebel‐Lizorkin type oscillation spaces

In this paper, we apply wavelets to study the Triebel‐Lizorkin type oscillation spaces and identify them with the well‐known Triebel‐Lizorkin‐Morrey spaces. Further, we prove that Calderón‐Zygmund operators are bounded on .     

Global well‐posedness and blow‐up criterion for the periodic quasi‐geostrophic equations in Lei‐Lin‐Gevrey spaces

In this paper we consider a periodic 2‐dimensional quasi‐geostrophic equations with subcritical dissipation. We show the global existence and uniqueness of the solution for small initial data in the Lei‐Lin‐Gevrey spaces . Moreover, we establish an exponential type explosion in finite time of this solution.     

Matrix measure on time scales and almost periodic analysis of the impulsive Lasota–Wazewska model with patch structure and forced perturbations

In this paper, a new impulsive Lasota–Wazewska model with patch structure and forced perturbed terms is proposed and analyzed on almost periodic time scales. For this, we introduce the concept of matrix measure on time scales and obtain some of its properties. Then, sufficient conditions are established which ensure the existence and exponential stability of positive almost periodic solutions of the proposed biological model. Our results are new even when the time scale is almost periodic, in particular, for periodic time scales on or . An example is given to illustrate the theory. Finally, we present some phenomena which are triggered by almost periodic time scales. Copyright © 2016 John Wiley & Sons, Ltd.     

Global stability of a discrete multigroup SIR model with nonlinear incidence rate

In this paper, by applying nonstandard finite difference scheme, we propose a discrete multigroup Susceptible‐Infective‐Removed (SIR) model with nonlinear incidence rate. Using Lyapunov functions, it is shown that the global dynamics of this model are completely determined by the basic reproduction number . If , then the disease‐free equilibrium is globally asymptotically stable; if , then there exists a unique endemic equilibrium and it is globally asymptotically stable. Example and numerical simulations are presented to illustrate the results. Copyright © 2017 John Wiley & Sons, Ltd.     

Galerkin finite element method for generalized Forchheimer equation of slightly compressible fluids in porous media

We consider the generalized Forchheimer flows for slightly compressible fluids. Using Muskat's and Ward's general form of Forchheimer equations, we describe the fluid dynamics by a nonlinear degenerate parabolic equation for the density. We study Galerkin finite elements method for the initial boundary value problem. The existence and uniqueness of the approximation are proved. A prior estimates for the solutions in , time derivative in and gradient in , with a∈(0,1) are established. Error estimates for the density variable are derived in several norms for both continuous and discrete time procedures. Numerical experiments using backward Euler scheme confirm the theoretical analysis regarding convergence rates. Copyright © 2017 John Wiley & Sons, Ltd.     

Block‐Transitive and Point‐Primitive 2‐ Designs with Sporadic Socle

The purpose of this paper is to classify all pairs , where is a nontrivial 2‐ design, and acts transitively on the set of blocks of and primitively on the set of points of with sporadic socle. We prove that there exists only one such pair : is the unique 2‐(176,8,2) design and , the Higman–Sims simple group.     

Exponential dichotomies of impulsive dynamic systems with applications on time scales

In this paper, it is the first time that some important inequalities are obtained to estimate the exponential type of upper and lower bounds of solutions for the three representative classes of homogeneous impulsive dynamic systems on time scales. Based on these, some new criteria are established for admitting an exponential dichotomy of the impulsive dynamic systems. The obtained results are essentially new, even the time scale or . In addition, in applications, we apply the obtained results to discuss the almost periodic problems of a class of integro‐differential systems, and the numerical simulations are given to illustrate that our timescale methods are feasible and effective. Finally, we present the conclusion and further discussion related to this topic. Copyright © 2014 John Wiley & Sons, Ltd.     

Existence of positive solutions for integral boundary value problems of fractional differential equations on infinite interval

In this paper, we consider a class of nonlinear fractional differential equations on the infinite interval with the integral boundary conditions By using Krasnoselskii fixed point theorem, the existence results of positive solutions for the boundary value problem in three cases and , are obtained, respectively. We also give out two corollaries as applications of the existence theorems and some examples to illustrate our results. Copyright © 2016 John Wiley & Sons, Ltd.     

Global stability of a multi‐group model with distributed delay and vaccination

A delayed multi‐group SVEIR epidemic model with vaccination and a general incidence function has been formulated and studied in this paper. Mathematical analysis shows that the basic reproduction number plays a key role in the dynamics of the model: the disease‐free equilibrium is globally asymptotically stable when , while the endemic equilibrium exists uniquely and is globally asymptotically stable when . For the proofs, we exploit a graph‐theoretical approach to the method of Lyapunov functionals. Our results show that distributed delay has no impact on the global stability of equilibria, and the results improve and generalize some known results. Copyright © 2016 John Wiley & Sons, Ltd.     

Mehler's formulas for the univariate complex Hermite polynomials and applications

We give 2 widest Mehler's formulas for the univariate complex Hermite polynomials , by performing double summations involving the products and . They can be seen as the complex analogues of the classical Mehler's formula for the real Hermite polynomials. The proof of the first one is based on a generating function giving rise to the reproducing kernel of the generalized Bargmann space of level m. The second Mehler's formula generalizes the one appearing as a particular case of the so‐called Kibble‐Slepian formula. The proofs we present here are direct and more simpler. Moreover, direct applications are given and remarkable identities are derived.     

Attractors for the strongly damped wave equation with p‐Laplacian

This paper is concerned with the initial‐boundary value problem for one‐dimensional strongly damped wave equation involving p‐Laplacian. For p > 2 , we establish the existence of weak local attractors for this problem in . Under restriction 2 < p < 4, we prove that the semigroup, generated by the considered problem, possesses a strong global attractor in , and this attractor is a bounded subset of . Copyright © 2017 John Wiley & Sons, Ltd.     

Ground state solutions for asymptotically periodic fractional Schrödinger–Poisson problems with asymptotically cubic or super‐cubic nonlinearities

In this paper, we consider the following fractional Schrödinger–Poisson problem: where s,t∈(0,1],4s+2t>3,V(x),K(x), and f(x,u) are periodic or asymptotically periodic in x. We use the non‐Nehari manifold approach to establish the existence of the Nehari‐type ground state solutions in two cases: the periodic one and the asymptotically periodic case, by introducing weaker conditions uniformly in with and with constant θ₀∈(0,1), instead of uniformly in and the usual Nehari‐type monotonic condition on f(x,τ)/|τ|³. Our results unify both asymptotically cubic or super‐cubic nonlinearities, which are new even for s=t=1. Copyright © 2017 John Wiley & Sons, Ltd.     

Lagrange‐type basis for multivariate uniform integrable tensor‐product Birkhoff interpolation

Birkhoff interpolation is the most general interpolation scheme. We study the Lagrange‐type basis for uniform integrable tensor‐product Birkhoff interpolation. We prove that the Lagrange‐type basis of multivariate uniform tensor‐product Birkhoff interpolation can be obtained by multiplying corresponding univariate Lagrange‐type basis when the integrable condition is satisfied. This leads to less computational complexity, which drops to from .     

On the existence of positive least energy solutions for a coupled Schrödinger system with critical exponent

In this paper, we consider the following coupled Schrödinger system with critical exponent: where is a smooth bounded domain, λ > 0,μ≥0, and . Under certain conditions on λ and μ, we show that this problem has at least one positive least energy solution. Copyright © 2016 John Wiley & Sons, Ltd.     

Multiple sign‐changing solutions for a class of quasilinear elliptic equations

In this paper, we study the following quasilinear Schrödinger equations: where Ω is a bounded smooth domain of , . Under some suitable conditions, we prove that this equation has three solutions of mountain pass type: one positive, one negative, and sign‐changing. Furthermore, if g is odd with respect to its second variable, this problem has infinitely many sign‐changing solutions. Copyright © 2016 John Wiley & Sons, Ltd.     

Tempered fractional differential equation: variational approach

In this paper, we are concerned with the existence of ground state solution for the following fractional differential equations with tempered fractional derivative: (FD) where α∈(1/2,1), λ>0, are the left and right tempered fractional derivatives, is the fractional Sobolev spaces, and . Assuming that f satisfies the Ambrosetti–Rabinowitz condition and another suitable conditions, by using mountain pass theorem and minimization argument over Nehari manifold, we show that (FD) has a ground state solution. Furthermore, we show that this solution is a radially symmetric solution. Copyright © 2017 John Wiley & Sons, Ltd.