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.     

On the existence and uniqueness of time periodic solutions for a semilinear heat equation in the whole space

This paper is concerned with the existence and uniqueness of time periodic solutions in the whole‐space for a heat equation with nonlinear term. The nonlinear term we considered is of this type, |u |^{q ? 1}u + f (x ,t ), with , N > 2. We show that there exists a unique time periodic solution when the source term f is small. In fact, is a critical exponent; when , there is no time periodic solution. Copyright © 2016 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.     

Rotating periodic solutions for second‐order dynamical systems with singularities of repulsive type

In this paper, we study the following second‐order dynamical system: where c ?0 is a constant, and . When g admits a singularity at zero of repulsive type without the restriction of strong force condition, we apply the coincidence degree theory to prove that the system admits nonplanar collisionless rotating periodic solutions taking the form u (t  + T ) = Q u (t ), with T  > 0 and Q an orthogonal matrix under the assumption of Landesman–Lazer type. Copyright © 2016 John Wiley & Sons, Ltd.     

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.     

Pseudo asymptotically periodic solutions for Volterra integro‐differential equations

In this paper, we propose a new class of functions called pseudo ‐asymptotically ω‐periodic function in the Stepanov sense and explore its properties in Banach spaces including composition results. Furthermore, the existence and uniqueness of the pseudo ‐asymptotically ω‐periodic mild solutions to Volterra integro‐differential equations is investigated. Applications to integral equations arising in the study of heat conduction in materials with memory are shown. Copyright © 2014 John Wiley & Sons, Ltd.     

Extending LSQR methods to solve the generalized Sylvester‐transpose and periodic Sylvester matrix equations

It is well known that the least‐squares QR‐factorization (LSQR) algorithm is a powerful method for solving linear systems Ax = b and unconstrained least‐squares problem min_x | | Ax ? b | | . In the paper, the LSQR approach is developed to obtain iterative algorithms for solving the generalized Sylvester‐transpose matrix equation the minimum Frobenius norm residual problem and the periodic Sylvester matrix equation Numerical results are given to illustrate the effect of the proposed algorithms. Copyright © 2013 John Wiley & Sons, Ltd.     

Spatially almost periodic solutions for active scalar equations

We are concerned with a family of dissipative active scalar equation on . By using similar methods from the previous paper of Y. Giga et al. (see Introduction below), we construct a unique real, spatially almost periodic mild solution θ of 1.1 satisfying 1.11 . In this paper, we consider some countable sum‐closed frequency sets (see Remark 1.1 ). We show that the property of the solution is rather different from Chae et al 1 and obtain that with some initial data θ₀ for all t≥0, and 0≤α≤ω, where ω is a fixed constant. Furthermore, arranging the elements of a countable sum‐closed frequency set F_δ as in Remark 1.3 , we have for any 0≤α≤ω that belongs to , where F_δ is defined in 1.4 or 1.5 .     

Asymptotical stability of almost periodic solution for an impulsive multispecies competition–predation system with time delays on time scales

In this paper, we consider the almost periodic dynamics of an impulsive multispecies Lotka–Volterra competition system with time delays on time scales. By establishing some comparison theorems of dynamic equations with impulses and delays on time scales, a permanence result for the model is obtained. Furthermore, based on the permanence result, by studying the Lyapunov stability theory of impulsive dynamic equations on time scales, we establish the existence and uniformly asymptotic stability of a unique positive almost periodic solution of the system. Finally, we give an example to show the feasibility of our main results, and our example also shows that the continuous time system and its corresponding discrete time system have the same dynamics. Our results of this paper are completely new even if for both the case of the time scale and the case of the time scale . Copyright © 2017 John Wiley & Sons, Ltd.     

New results on the existence of periodic solutions for a higher‐order Li énard type p‐Laplacian differential equation

In this paper, by using the continuation theorem of coincidence degree theory, we consider the higher‐order Li énard type p‐Laplacian differential equation as follows Some new results on the existence of periodic solutions for the previous equation are obtained, which generalize and enrich some known results to some extent from the literature. Copyright © 2011 John Wiley & Sons, Ltd.     

Large time periodic solution to the coupled chemotaxis‐Stokes model

In this paper, we deal with the time periodic problem for the coupled chemotaxis‐fluid model with logistic growth term. We prove the existence of large time periodic solution in spatial dimension . Furthermore, we also show that if the time periodic source g and the potential force belong to , the solution is also a classical solution.     

Ground state solutions for the nonlinear Schrödinger‐Poisson systems with sum of periodic and vanishing potentials

We study the existence of ground state solutions for the following Schrödinger‐Poisson equations: where is the sum of a periodic potential V_p and a localized potential V_loc and f satisfies the subcritical or critical growth. Although the Nehari‐type monotonicity assumption on f is not satisfied in the subcritical case, we obtain the existence of a ground state solution as a minimizer of the energy functional on Nehari manifold. Moreover, we show that the existence and nonexistence of ground state solutions are dependent on the sign of V_loc.     

Global strong solutions for 3D viscous incompressible heat conducting Navier‐Stokes flows with the general external force

In this paper, we consider the initial boundary value problem for the nonhomogeneous heat–conducting fluids with non‐negative density and the general external force. We prove that there exists a unique global strong solution to the 3D viscous nonhomogeneous heat–conducting Navier‐Stokes flows if is suitably small.     

Homoclinic solutions in periodic difference equations with mixed nonlinearities

In this paper, by using critical point theory in combination with periodic approximations, we obtain some new sufficient conditions on the nonexistence and existence of homoclinic solutions for a class of periodic difference equations. Unlike the existing literatures that always assume that the nonlinear terms are only either superlinear or asymptotically linear at , but superlinear at 0, our nonlinear term can mix superlinear nonlinearities with asymptotically linear ones at both and 0. To the best of our knowledge, this is the first time to consider the homoclinic solutions of this class of difference equations with mixed nonlinearities. Our results are necessary in some sense, and extend and improve some existing ones even for some special cases. Copyright © 2015 John Wiley & Sons, Ltd.     

Global well‐posedness and scattering of a generalized nonlinear fourth‐order wave equation

In this paper, we study the global well‐posedness and scattering theory of the solution to the Cauchy problem of a generalized fourth‐order wave equation where if d ?4, and if d ?5. The main strategy we use in this paper is concentration‐compactness argument, which was first introduced by Kenig and Merle to handle the scattering problem vector so as to control the momentum. 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.     

Analyticity for a class of nonlocal Kuramoto‐Sivashinsky equations arising in interfacial electrohydrodynamics

In this article, we study the analyticity properties of solutions of the nonlocal Kuramoto‐Sivashinsky equations, defined on 2π‐periodic intervals, where ν is a positive constant; μ is a nonnegative constant; p is an arbitrary but fixed real number in the interval [3,4); and is an operator defined by its symbol in Fourier space, with be the Hilbert transform. We establish spatial analyticity in a strip around the real axis for the solutions of such equations, which possess universal attractors. Also, a lower bound for the width of the strip of analyticity is obtained.     

Global dynamics of a SEIR model with information dependent vaccination and periodically varying transmission rate

This paper is devoted to the investigation of the global dynamics of a SEIR model with information dependent vaccination. The basic reproduction number is derived for the model, and it is shown that gives the threshold dynamics in the sense that the disease‐free equilibrium is globally asymptotically stable and the disease dies out if , while there exists at least one positive periodic solution and the disease is uniformly persistent when . Further, we give the approximation formula of . This answers the concerns presented in [B. Buonomo, A. d'Onofrio, D. Lacitignola, Modeling of pseudo‐rational exemption to vaccination for SEIR diseases, J. Math. Anal. Appl. 404 (2013) 385–398]. Copyright © 2014 John Wiley & Sons, Ltd.     

Existence of ground state solutions for Kirchhoff‐type problems involving critical Sobolev exponents

In this paper, we study the existence of ground state solutions for a Kirchhoff‐type problem in involving critical Sobolev exponent. With the help of Nehari manifold and the concentration‐compactness principle, we prove that problem admits at least one ground state solution.     

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.