Asymptotic behavior of solutions to a class of nonlocal non‐autonomous diffusion equations

In this paper, we consider the nonlocal non‐autonomous evolution problems where Ω is a bounded smooth domain in , N≥1, β is a positive constant, the coefficient a is a continuous bounded function on , and K is an integral operator with symmetric kernel , being J a non‐negative function continuously differentiable on and . We prove the existence of global pullback attractor, and we exhibit a functional to evolution process generated by this problem that decreases along of solutions. Assuming the parameter β is small enough, we show that the origin is locally pullback asymptotically stable. Copyright © 2014 John Wiley & Sons, Ltd.     

Pullback attractors of 2D incompressible Navier‐Stokes‐Voight equations with delay

In this paper, the 2D Navier‐Stokes‐Voight equations with 3 delays in is considered. By using the Faedo‐Galerkin method, Lions‐Aubin lemma, and Arzelà‐Ascoli theorem, we establish the global well‐posedness of solutions and the existence of pullback attractors in H¹.     

Uniform attractor for non‐autonomous Boussinesq‐type equation with critical nonlinearity

In this paper, we study the longtime dynamics of the non‐autonomous Boussinesq‐type equation with critical nonlinearity, and time‐dependent external forcing, which is translation bounded but not translation compact. We prove the existence of a uniform attractor in . Copyright © 2015 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 well posedness of 3D‐NSE in Fourier–Lei–Lin spaces

In this paper, we prove a global well posedness of the three‐dimensional incompressible Navier–Stokes equation under an initial data, which belong to the non‐homogeneous Fourier–Lei–Lin space for σ ? ? 1 and if the norm of the initial data in the Lei–Lin space is controlled by the viscosity. Copyright © 2016 John Wiley & Sons, Ltd.     

Two types of upper semi‐continuity of bi‐spatial attractors for non‐autonomous stochastic p‐Laplacian equations on

We consider the long time behavior of solutions for the non‐autonomous stochastic p‐Laplacian equation with additive noise on an unbounded domain. First, we show the existence of a unique ‐pullback attractor, where q is related to the order of the nonlinearity. The main difficulty existed here is to prove the asymptotic compactness of systems in both spaces, because the Laplacian operator is nonlinear and additive noise is considered. We overcome these obstacles by applying the compactness of solutions inside a ball, a truncation method and some new techniques of estimates involving the Laplacian operator. Next, we establish the upper semi‐continuity of attractors at any intensity of noise under the topology of . Finally, we prove this continuity of attractors from domains in the norm of , which improves an early result by Bates et al.(2001) who studied such continuity when the deterministic lattice equations were approached by finite‐dimensional systems, and also complements Li et al. (2015) who discussed this approximation when the nonlinearity f(·,0) had a compact support. Copyright © 2017 John Wiley & Sons, Ltd.     

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.     

Non‐autonomous homogeneous rational difference equations of degree one: convergence and monotone solutions for second and third order case

Consider the non‐autonomous equations: where and also These are some non‐autonomous homogeneous rational difference equations of degree one. A reduction in order is consi‐dered. Convergence and monoton character of positive solutions are studied. Copyright © 2013 John Wiley & Sons, Ltd.     

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.     

Time‐harmonic and asymptotically linear Maxwell equations in anisotropic media

This paper is focused on following time‐harmonic Maxwell equation: where is a bounded Lipschitz domain, is the exterior normal, and ω is the frequency. The boundary condition holds when Ω is surrounded by a perfect conductor. Assuming that f is asymptotically linear as , we study the above equation by improving the generalized Nehari manifold method. For an anisotropic material with magnetic permeability tensor and permittivity tensor , ground state solutions are established in this paper. Applying the principle of symmetric criticality, we find 2 types of solutions with cylindrical symmetries in particular for the uniaxial material.     

Pullback attractors for a nonautonomous damped wave equation with infinite delays in weighted space

In this paper, we investigate the existence of solutions for a damped wave equation with infinite delays in the weighted space and , respectively. In addition, we study the existence of pullback attractor for the process associated to the problem by a direct and simple compactness method.     

Nonlinear diffusion equations as asymptotic limits of Cahn‐Hilliard systems on unbounded domains via Cauchy's criterion

This paper develops an abstract theory for subdifferential operators to give existence and uniqueness of solutions to the initial‐boundary problem P for the nonlinear diffusion equation in an unbounded domain ( ), written as which represents the porous media, the fast diffusion equations, etc, where β is a single‐valued maximal monotone function on , and T>0. In Kurima and Yokota (J Differential Equations 2017; 263:2024‐2050 and Adv Math Sci Appl 2017; 26:221‐242) existence and uniqueness of solutions for P were directly proved under a growth condition for β even though the Stefan problem was excluded from examples of P . This paper completely removes the growth condition for β by confirming Cauchy's criterion for solutions of the following approximate problem _ε with approximate parameter ε>0: which is called the Cahn‐Hilliard system, even if ( ) is an unbounded domain. Moreover, it can be seen that the Stefan problem excluded from Kurima and Yokota (J Differential Equations 2017; 263:2024‐2050 and Adv Math Sci Appl 2017; 26:221‐242) is covered in the framework of this paper.     

Global existence and asymptotic behavior of classical solutions for a 3D two‐species chemotaxis‐Stokes system with competitive kinetics

This paper considers the 2‐species chemotaxis‐Stokes system with competitive kinetics under homogeneous Neumann boundary conditions in a 3‐dimensional bounded domain with smooth boundary. Both chemotaxis‐fluid systems and 2‐species chemotaxis systems with competitive terms were studied by many mathematicians. However, there have not been rich results on coupled 2‐species–fluid systems. Recently, global existence and asymptotic stability in the above problem with (u·∇)u in the fluid equation were established in the 2‐dimensional case. The purpose of this paper is to give results for global existence, boundedness, and stabilization of solutions to the above system in the 3‐dimensional case when is sufficiently small.     

Global stability for the fractional Navier‐Stokes equations in the Fourier‐Herz space

We consider global stability for the fractional incompressible Navier‐Stokes equations in a 3‐D critical Fourier‐Herz space. By introducing a weighted norm space and using Fourier localization technique, the stability of mild solutions with small initial perturbation is established. With the Friedrichs method, the stability of weak solutions is proved under arbitrary large initial perturbation.     

Periodicity and non‐negativity of solutions for nonlinear neutral differential equations with variable delay via fixed point theorems

We use a modification of Krasnoselskii's fixed point theorem introduced by Burton to show the periodicity and non‐negativity of solutions for the nonlinear neutral differential equation with variable delay We invert this equation to construct the sum of a compact map and a large contraction, which is suitable for applying the modification of Krasnoselskii's theorem. Copyright © 2015 John Wiley & Sons, Ltd.     

About the notion of non‐T‐resonance and applications to topological multiplicity results for ODEs on differentiable manifolds

By using topological methods, mainly the degree of a tangent vector field, we establish multiplicity results for T‐periodic solutions of parametrized T‐periodic perturbations of autonomous ODEs on a differentiable manifold M. In order to provide insights into the key notion of T‐resonance, we consider the elementary situations and . Doing so, we provide more comprehensive analysis of those cases and find improved conditions. Copyright © 2015 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.     

Bilinear estimate on tent‐type spaces with application to the well‐posedness of fluid equations

We introduce a class of tent‐type spaces and establish a Poisson extension result of Triebel–Lizorkin spaces . As an application, we get the well‐posedness of Navier–Stokes equations and magnetohydrodynamic equations with initial data in critical Triebel–Lizorkin spaces , . Copyright © 2016 John Wiley & Sons, Ltd.     

Existence of ground state sign‐changing solutions for p‐Laplacian equations of Kirchhoff type

In this paper, we prove the existence of ground state sign‐changing solutions for the following class of elliptic equation: where , and K(x) are positive continuous functions. Firstly, we obtain one ground state sign‐changing solution u_b by using some new analytical skills and non‐Nehari manifold method. Furthermore, the energy of u_b is strictly larger than twice that of the ground state solutions of Nehari type. We also establish the convergence property of u_b as the parameter b↘0. Copyright © 2017 John Wiley & Sons, Ltd.     

Max‐type system of difference equations with positive two‐periodic sequences

This paper deals with the form and the periodicity of the solutions of the max‐type system of difference equations where , and are positive two‐periodic sequences and initial values x₀, x _{? 1}, y₀, y _{? 1} ∈ (0, + ∞ ). Copyright © 2014 John Wiley & Sons, Ltd.