A class of vector‐valued dilation‐and‐modulation frames on the half real line

Vector‐valued frames were first introduced under the name of superframes by Balan in the context of signal multiplexing and by Han and Larson from the mathematical aspect. Since then, the wavelet and Gabor frames in have interested many mathematicians. The space models vector‐valued causal signal spaces because of the time variable being nonnegative. But it admits no nontrivial shift‐invariant system and thus no wavelet or Gabor frame since is not a group by addition (not as ). Observing that is a group by multiplication, we, in this paper, introduce a class of multiplication‐based dilation‐and‐modulation ( ) systems, and investigate the theory of frames in . Since is not closed under the Fourier transform, the Fourier transform does not fit . We introduce the notion of Θ_a transform in , and using Θ_a‐transform matrix method, we characterize frames, Riesz bases, and dual frames in and obtain an explicit expression of duals for an arbitrary given frame. An example theorem is also presented.     

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.     

Existence and uniqueness of solutions of sequential nonlinear fractional difference equations with three‐point fractional sum boundary conditions

In this paper, we consider a discrete fractional boundary value problem of the form: where 0 < α,β≤1, 1 < α + β≤2, λ and ρ are constants, γ > 0, , is a continuous function, and E_βx(t) = x(t + β ? 1). The existence and uniqueness of solutions are proved by using Banach's fixed point theorem. An illustrative example is also presented. Copyright © 2014 John Wiley & Sons, Ltd.     

Regular attractor for damped KdV‐Burgers equations on R

We study the well‐posedness and dynamic behavior for the KdV‐Burgers equation with a force on R . We establish L ^p ?L ^q estimates of the evolution , as an application we obtain the local well‐posedness. Then the global well‐posedness follows from a uniform estimate for solutions as t goes to infinity. Next, we prove the asymptotical regularity of solutions in space and by the smoothing effect of . The regularity and the asymptotical compactness in L ² yields the asymptotical compactness in by an interpolation arguement. Finally, we conclude the existence of an globalattractor.     

Multilinear and multiparameter spectral multipliers on stratified groups

As far as we know, the study of multilinear spectral multipliers on nilpotent Lie groups is a very new research work. There is even no study of Hörmander‐type multiplier theorem for multilinear and multiparameter spectral multipliers on nilpotent Lie groups. In this paper, on product spaces of stratified groups G = G₁ × ⋯ × G_M, we prove Hörmander‐type multiplier theorems for multilinear and multiparameter spectral multipliers from to L^r(G) with , from to with , and from to L^r(·)(G) with or for all ℓ = 1,…,N.     

Dynamic behavior of a stochastic SIRS epidemic model with media coverage

The main purpose of this paper is to explore the global behavior of a stochastic SIRS epidemic model with media coverage. The value of this research has 2 aspects: for one thing, we use Markov semigroup theory to prove that the basic reproduction number can be used to control the dynamics of stochastic system. If , the stochastic system has a disease‐free equilibrium, which implies the disease will die out with probability one. If , under the mild extra condition, the stochastic differential equation has an endemic equilibrium, which is globally asymptotically stable. For another, it is known that environment fluctuations can inhibit disease outbreak. Although the disease is persistent when R₀ > 1 for the deterministic model, if , the disease still dies out with probability one for the stochastic model. Finally, numerical simulations were carried out to illustrate our results, and we also show that the media coverage can reduce the peak of infective individuals via numerical simulations.     

Existence and concentration of bound states for a Kirchhoff type problem with potentials vanishing or unbounded at infinity

In this paper, we study the existence and concentration behavior of positive solutions for the following Kirchhoff type equation: where ɛ is a positive parameter, a and b are positive constants, and 3<p<5. Let denotes the ground energy function associated with , , where is regard as a parameter. Suppose that the potential V(x) decays to zero at infinity like |x|^−α with 0<α≤2, we prove the existence of positive solutions u_ɛ belonging to for vanishing or unbounded K(x) when ɛ > 0 small. Furthermore, we show that the solution u_ɛ concentrates at the minimum points of as ɛ→0⁺.     

A quasilinear chemotaxis–haptotaxis model: The roles of nonlinear diffusion and logistic source

In this paper, we study the following quasilinear chemotaxis–haptotaxis system (?) in a bounded smooth domain under zero‐flux boundary conditions, where the nonlinearities D ,S ₁, and S ₂ are supposed to generalize the prototypes with , and f ∈C ¹([0,+∞ ) × [0,+∞ )) satisfies with r > 0 and b > 0. If the nonnegative initial data u ₀(x )∈W ^1,∞ (Ω),v ₀(x )∈W ^1,∞ (Ω), and for some α ∈(0,1), it is proved that

1. For n = 1, if and then (?) has a unique nonnegative classical solution, which is globally bounded.
2. For n = 2, if and then (?) has a unique nonnegative classical solution, which is globally bounded.
3. For n ≥3, if and then (?) has a unique nonnegative classical solution, which is globally bounded.

Copyright © 2016 John Wiley & Sons, Ltd.     

Transition matrices for quantum waveguides with impurities

We consider the electron propagation in a cylindrical quantum waveguide where D is a bounded domain in described by the Dirichlet problem for the Schrödinger operator where x=(x₁, x₂), , is the transversal confinement potential, and is the impurity potential.  We construct the left and right transition matrices and give an numerical algorithm for their calculations based on the spectral parameter power series method.     

Three rapidly convergent parareal solvers with application to time‐dependent PDEs with fractional Laplacian

Time‐dependent PDEs with fractional Laplacian ( ? Δ)^α play a fundamental role in many fields and approximating ( ? Δ)^α usually leads to ODEs' system like u ^′ (t ) + A u (t ) =  g (t ) with A  = Q ^α , where is a sparse symmetric positive definite matrix and α  > 0 denotes the fractional order. The parareal algorithm is an ideal solver for this kind of problems, which is iterative and is characterized by two propagators and . The propagators and are respectively associated with large step size ΔT and small step size Δt , where ΔT  = J Δt and J ?2 is an integer. If we fix the ‐propagator to the Implicit‐Euler method and choose for some proper Runge–Kutta (RK) methods, such as the second‐order and third‐order singly diagonally implicit RK methods, previous studies show that the convergence factors of the corresponding parareal solvers can satisfy and , where σ (A ) is the spectrum of the matrix A . In this paper, we show that by choosing these two RK methods as the ‐propagator, the convergence factors can reach , provided the one‐stage complex Rosenbrock method is used as the ‐propagator. If we choose for both and , the complex Rosenbrock method, we show that the convergence factor of the resulting parareal solver can also reach . Numerical results are given to support our theoretical conclusions. Copyright © 2017 John Wiley & Sons, Ltd.     

Real‐variable characterizations of Musielak–Orlicz–Hardy spaces associated with Schrödinger operators on domains

Let n≥3, Ω be a strongly Lipschitz domain of and L_Ω:=?Δ+V a Schrödinger operator on L²(Ω) with the Dirichlet boundary condition, where Δ is the Laplace operator and the nonnegative potential V belongs to the reverse Hölder class for some q₀>n/2. Assume that the growth function satisfies that ?(x,·) is an Orlicz function, (the class of uniformly Muckenhoupt weights) and its uniformly critical lower type index , where and μ₀∈(0,1] denotes the critical regularity index of the heat kernels of the Laplace operator Δ on Ω. In this article, the authors first show that the heat kernels of L_Ω satisfy the Gaussian upper bound estimates and the Hölder continuity. The authors then introduce the ‘geometrical’ Musielak–Orlicz–Hardy space via , the Hardy space associated with on , and establish its several equivalent characterizations, respectively, in terms of the non‐tangential or the vertical maximal functions or the Lusin area functions associated with L_Ω. All the results essentially improve the known results even on Hardy spaces with p∈(n/(n + δ),1] (in this case, ?(x,t):=t^p for all x∈Ω and t∈[0,∞)). Copyright © 2016 John Wiley & Sons, Ltd.     

Liouville‐type theorem for the steady compressible Hall‐MHD system

In this paper, we prove a Liouville‐type theorem for the steady compressible Hall‐magnetohydrodynamics system in Π, where Π is whole space or half space . We show that a smooth solution (ρ, u , B ,P) satisfying 1/C₀<ρ<C₀, , and B ∈L^9/2(Π) for some constant C₀>0 is indeed trivial. This generalizes and improves 2 results of Chae.     

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 .     

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.     

A nonhomogeneous boundary value problem for the Kuramoto–Sivashinsky equation in a quarter plane

We study the initial boundary value problem for the one‐dimensional Kuramoto–Sivashinsky equation posed in a half line with nonhomogeneous boundary conditions. Through the analysis of the boundary integral operator, and applying the known results of the Cauchy problem of the Kuramoto–Sivashinsky equation posed on the whole line , the initial boundary value problem of the Kuramoto–Sivashinsky equation is shown to be globally well‐posed in Sobolev space for any s >?2. Copyright © 2017 John Wiley & Sons, Ltd.     

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.     

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.     

Characterizations of Sobolev spaces associated to operators satisfying off‐diagonal estimates on balls

Let be a metric measure space of homogeneous type and L be a one‐to‐one operator of type ω on for ω ∈[0, π /2). In this article, under the assumptions that L has a bounded H _∞ ‐functional calculus on and satisfies (p _L , q _L ) off‐diagonal estimates on balls, where p _L ∈[1, 2) and q _L ∈(2, ∞ ], the authors establish a characterization of the Sobolev space , defined via L ^{α /2}, of order α ∈(0, 2] for p ∈(p _L , q _L ) by means of a quadratic function S _{α , L} . As an application, the authors show that for the degenerate elliptic operator L _w : =? w  ^? 1div(A ?) and the Schrödinger type operator with a ∈(0, ∞ ) on the weighted Euclidean space with A being real symmetric, if n ?3, with q ∈[1, 2], , p ∈(1, ∞ ) and with , then, for all , , where the implicit equivalent positive constants are independent of f , denotes the class of Muckenhoupt weights, the reverse Hölder class, and D (L _w ) and the domains of L _w and , respectively. Copyright © 2016 John Wiley & Sons, Ltd.     

Boundary value problem for p−Laplacian Caputo fractional difference equations with fractional sum boundary conditions

In this paper, we consider a discrete fractional boundary value problem of the form where 0 < α,β≤1, 1 < α + β≤2, 0 < γ≤1, , ρ is a constant, and denote the Caputo fractional differences of order α and β, respectively, is a continuous function, and ?_p is the p‐Laplacian operator. The existence of at least one solution is proved by using Banach fixed point theorem and Schaefer's fixed point theorem. Some illustrative examples are also presented. Copyright © 2015 John Wiley & Sons, Ltd.     

Ground state of solutions for a class of fractional Schrödinger equations with critical Sobolev exponent and steep potential well

In this paper, we study the following fractional Schrödinger equations: (1) where (?△)^α is the fractional Laplacian operator with , 0≤s ≤2α , λ >0, κ and β are real parameter. is the critical Sobolev exponent. We prove a fractional Sobolev‐Hardy inequality and use it together with concentration compact theory to get a ground state solution. Moreover, concentration behaviors of nontrivial solutions are obtained when the coefficient of the potential function tends to infinity.