A new Beale–Kato–Majda criteria for the 3D magneto‐micropolar fluid equations in the Orlicz–Morrey space

In this work, we are interested in the study of regularity for the three‐dimensional magneto‐micropolar fluid equations in Orlicz–Morrey spaces. If the velocity field satisfies or the gradient field of velocity satisfies then we show that the solution remains smooth on [0,T]. In view of the embedding with 2 < p < 3 ∕ r and P > 1, we see that our result extends the result of Yuan and that of Gala. Copyright © 2012 John Wiley & Sons, Ltd.     

Infinitely many solutions for a Hardy–Sobolev equation involving critical growth

In this paper, by an approximating argument, we obtain infinitely many solutions for the following Hardy–Sobolev equation with critical growth: provided N > 6 + t, where and Ω is an open bounded domain in , which contains some points x₀ = (0,z₀). Copyright © 2013 John Wiley & Sons, Ltd.     

The Decay of mass for a nonlinear fractional reaction–diffusion equation

We study the large time behavior of non‐negative solutions to the nonlinear fractional reaction–diffusion equation ?_tu = ? t^σ( ? Δ)^{α ∕ 2}u ? h(t)u^p (α ∈ (0,2]) posed on and supplemented with an integrable initial condition, where σ ≥ 0, p > 1, and h : [0, ∞ ) → [0, ∞ ). Defining the mass , under certain conditions on the function h, we show that the asymptotic behavior of the mass can be classified along two cases as follows:

• if , then there exists M _∞ ∈ (0, ∞ ) such that ;
• if , then .

Copyright © 2014 John Wiley & Sons, Ltd.     

Existence of positive solutions to a coupled system of fractional differential equations

We consider the following system of fractional differential equations where is the Riemann‐Liouville fractional derivative of order α,f,g : [0,1] × [0, ∞ ) × [0, ∞ ) → [0, ∞ ). Sufficient conditions are provided for the existence of positive solutions to the considered problem. Copyright © 2014 John Wiley & Sons, Ltd.     

Maxwell meets Korn: A new coercive inequality for tensor fields in with square‐integrable exterior derivative

For a bounded domain with connected Lipschitz boundary, we prove the existence of some c > 0, such that holds for all square‐integrable tensor fields , having square‐integrable generalized “rotation” tensor fields and vanishing tangential trace on ?Ω, where both operations are to be understood row‐wise. Here, in each row, the operator curl is the vector analytical reincarnation of the exterior derivative d in . For compatible tensor fields T, that is, T = ? v, the latter estimate reduces to a non‐standard variant of Korn's first inequality in , namely for all vector fields , for which ? v_n,n = 1, … ,N, are normal at ?Ω. Copyright © 2012 John Wiley & Sons, Ltd.     

A remark on the logarithmically improved regularity criterion for the micropolar fluid equations in terms of the pressure

In this paper, we study the regularity criterion of weak solutions of the three dimensional micropolar fluid flows. It is proved that if the pressure satisfies where denotes the critical Besov space, then the weak solution (u,w) becomes a regular solution on (0,T]. This regularity criterion can be regarded as log in time improvements of the standard Serrin's criteria established before. Copyright © 2011 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.     

Multiple solutions for a modified Kirchhoff‐type equation in RN

In this paper, we study the following modified Kirchhoff‐type equations of the form: where a > 0, b ≥ 0, and . Under appropriate assumptions on V (x) and h(x,u), some existence results for positive solutions, negative solutions, and sequence of high energy solutions are obtained via a perturbation method. Copyright © 2014 John Wiley & Sons, Ltd.     

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.     

Solutions for subquadratic fractional Hamiltonian systems without coercive conditions

In this paper, we are concerned with the existence of infinitely many solutions for the following fractional Hamiltonian systems (FHS) where α ∈ (1 ∕ 2,1), , , is a symmetric and positive definite matrix for all , , and ? W is the gradient of W at u. The novelty of this paper is that, assuming L is bounded in the sense that there are constants 0 < τ₁ < τ₂ < + ∞ such that τ₁ | u | ² ≤ (L(t)u,u) ≤ τ₂ | u | ² for all and W is of subquadratic growth as | u | → + ∞ , we show that (FHS) possesses infinitely many solutions via the genus properties in the critical theory. Recent results in [Z. Zhang and R. Yuan, Variational approach to solutions for a class of fractional Hamiltonian systems, Math. Methods Appl. Sci., DOI:10.1002/mma.2941] are generalized and significantly improved. Copyright © 2014 John Wiley & Sons, Ltd.     

Improvement of convergence rates for a parabolic system of chemotaxis in the whole space

We are interested in the asymptotic behavior of solutions towards a parabolic system of chemotaxis in , n ≥ 1. It was proved in the previous results that decaying solutions converge to the heat kernel in at the rate t^{?n(1 ? 1/p)/2 ? 1/2} as t → ∞. Our aim in this paper is to improve the convergence rates by taking into account the center of mass of such solutions. Copyright © 2011 John Wiley & Sons, Ltd.     

Analytic invariant curves for an iterative equation related to mosquito model

In this paper, we study the existence of analytic invariant curves of a iterative equation which from mosquito model. By constructing an invertible analytic solution g(x) of an auxiliary equation of the form invertible analytic solutions of the form g(αg ^{? 1}(x)) for the original iterative functional equation are obtained. Besides the hyperbolic case 0 < | α | < 1, we focus on those α on the unit circle S¹, that is, | α | = 1. We discuss not only those α at resonance, that is at a root of the unity, but also those α near resonance under the Brjuno condition. Copyright © 2013 John Wiley & Sons, Ltd.     

Boundedness of solutions to parabolic–elliptic Keller–Segel systems with signal‐dependent sensitivity

This paper deals with the parabolic–elliptic Keller–Segel system with signal‐dependent chemotactic sensitivity function, under homogeneous Neumann boundary conditions in a smooth bounded domain , with initial data satisfying u₀ ≥ 0 and . The chemotactic sensitivity function χ(v) is assumed to satisfy The global existence of weak solutions in the special case is shown by Biler (Adv. Math. Sci. Appl. 1999; 9:347–359). Uniform boundedness and blow‐up of radial solutions are studied by Nagai and Senba (Adv. Math. Sci. Appl. 1998; 8:145–156). However, the global existence and uniform boundedness of classical nonradial solutions are left as an open problem. This paper gives an answer to the problem. Namely, it is shown that the system possesses a unique global classical solution that is uniformly bounded if , where γ > 0 is a constant depending on Ω and u₀. Copyright © 2014 John Wiley & Sons, Ltd.     

Multiple solutions of a quasilinear elliptic system involving the nonlinear boundary conditions on exterior domain

In this paper, we consider the multiplicity results of nontrivial nonnegative solutions of the quasilinear p‐Laplacian system with the nonlinear boundary conditions: (0.1) where Ω is a smooth exterior domain in is the outward normal derivative on the boundary Γ = ?Ω, and . By the Nehari manifold and variational methods, we prove that the problem (0.1) has at least two nontrivial nonnegative solutions when the pair of the parameters (λ,μ) belongs to a certain subset of . Copyright © 2012 John Wiley & Sons, Ltd.     

Existence of positive solutions for singular impulsive differential equations with integral boundary conditions

In this paper, we study the existence of positive solutions for singular impulsive differential equations with integral boundary conditions where the nonlinearity f(t,u,v) may be singular at u = 0 and v = 0. The proof is based on the theory of Leray–Schauder degree, together with a truncation technique. Some recent results in the literature are generalized and improved. Copyright © 2014 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.     

The Bergman–Sce transform for slice monogenic functions

In a recent paper, we showed that the classical Bergman theory admits two possible formulations for the class of slice regular functions with quaternionic values. In the so called formulation of the first kind, we provide a Bergman kernel which is defined on and is a reproducing kernel. In the so called formulation of the second kind, we use the Representation Formula for slice regular functions to define a second Bergman kernel; this time the kernel is still defined on U, but the integral representation of f is based on an integral computed only on and the integral does not depend on , (here denotes the sphere unit of purely imaginary quaternions, and represents the complex plane with imaginary unit I). In this paper, we extend the second formulation of the Bergman theory to the case of slice monogenic functions and we focus our attention on the so‐called Bergman–Sce transform. This integral transform is defined by using the Bergman kernel and the Sce mapping theorem and associates to every slice monogenic function f, an axially monogenic function . Copyright © 2011 John Wiley & Sons, Ltd.     

On a nonlinear wave equation with boundary damping

This paper is concerned with the existence and decay of solutions of the mixed problem for the nonlinear wave equation with boundary conditions Here, Ω is an open bounded set of with boundary Γ of class C²; Γ is constituted of two disjoint closed parts Γ₀ and Γ₁ both with positive measure; the functions μ(t), f(s), g(s) satisfy the conditions μ(t) ≥ μ₀ > 0, f(s) ≥ 0, g(s) ≥ 0 for t ≥ 0, s ≥ 0 and h(x,s) is a real function where x ∈ Γ₁, ν(x) is the unit outward normal vector at x ∈ Γ₁ and α, β are non‐negative real constants. Assuming that h(x,s) is strongly monotone in s for each x ∈ Γ₁, it is proved the global existence of solutions for the previous mixed problem. For that, it is used in the Galerkin method with a special basis, the compactness approach, the Strauss approximation for real functions and the trace theorem for nonsmooth functions. The exponential decay of the energy is derived by two methods: by using a Lyapunov functional and by Nakao's method. Copyright © 2013 John Wiley & Sons, Ltd.     

Simultaneous reconstruction of the source term and the surface heat transfer coefficient

We study the problem of identifying unknown source terms in an inverse parabolic problem, when the overspecified (measured) data are given in form of Dirichlet boundary condition u(0,t) = h(t) and , where is an arbitrarily prescribed subregion. The main goal here is to show that the gradient of cost functional can be expressed via the solutions of the direct and corresponding adjoint problems. We prove Hölder continuity of the cost functional and derive the Lipschitz constant in the explicit form via the given data. On the basis of the obtained results, we propose a monotone iteration process. Copyright © 2014 John Wiley & Sons, Ltd.     

On the ‐boundedness of solution operator families of the generalized Stokes resolvent problem in an infinite layer

In this paper, we prove the ‐boundedness of solution operator families of the generalized Stokes resolvent problem in an infinite layer with resolvent parameter , where , and our boundary conditions are nonhomogeneous Neumann on upper boundary and Dirichlet on lower boundary. We want to emphasize that we can choose 0 < ? < π ∕ 2 and γ₀ > 0 arbitrarily, although usual parabolic theorem tells us that we must choose a large γ₀ > 0 for given 0 < ? < π ∕ 2. We also prove the maximal L_p ? L_q regularity theorem of the nonstationary Stokes problem as an application of the ‐boundedness. The key of our approach is to apply several technical lemmas to the exact solution formulas of a resolvent problem. The formulas are obtained through the solutions of the ODEs, in the Fourier space, driven by the partial Fourier transform with respect to tangential space variable . Copyright © 2014 John Wiley & Sons, Ltd.