Multiplicity of positive solutions to boundary blow‐up problem with variable exponent and sign‐changing weights

In this paper, we consider the elliptic boundary blow‐up problem where Ω is a bounded smooth domain of are positive continuous functions supported in disjoint subdomains Ω₊,Ω_? of Ω, respectively, a ₊ vanishes on the boundary of satisfies p (x )≥1 in Ω,p (x ) > 1 on ? Ω and , and ε is a parameter. We show that there exists ε ^?>0 such that no positive solutions exist when ε > ε ^?, while a minimal positive solution u _ε exists for every ε ∈(0,ε ^?). Under the additional hypotheses that is a smooth N ? 1‐dimensional manifold and that a ₊,a _? have a convenient decay near Γ, we show that a second positive solution v _ε exists for every ε ∈(0,ε ^?) if , where N ^?=(N + 2)/(N ? 2) if N > 2 and if N = 2. Our results extend that of Jorge Garcá‐Melián in 2011, where the case that p > 1 is a constant and a ₊>0 on ? Ω is considered. Copyright © 2016 John Wiley & Sons, Ltd.     

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.     

On the fractional Schrödinger problem with non‐symmetric potential

We study the semilinear equation where 0 < s < 1, , V(x) is a sufficiently smooth non‐symmetric potential with , and ? > 0 is a small number. Letting U be the radial ground state of (?Δ)^sU + U ? U^p=0 in , we build solutions of the form for points ?_j,j = 1,?,m, using a Lyapunov–Schmidt variational reduction. 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.     

Boundedness in a higher‐dimensional chemotaxis system with porous medium diffusion and general sensitivity

This paper deals with the following chemotaxis system: in a bounded domain with smooth boundary under no‐flux boundary conditions, where satisfies for all with l ?2 and some nondecreasing function on [0,∞ ). Here, f (v )∈C ¹([0,∞ )) is nonnegative for all v ?0. It is proved that when , the system possesses at least one global bounded weak solution for any sufficiently smooth nonnegative initial data. This extends a recent result by Wang (Math. Methods Appl. Sci. 2016 39 : 1159–1175) which shows global existence and boundedness of weak solutions under the condition . Copyright © 2017 John Wiley & Sons, Ltd.     

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.     

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.     

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.     

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.     

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.     

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.     

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.     

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.     

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.     

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.     

On the Riemann‐Hilbert boundary value problem for generalized analytic functions in the framework of variable exponent spaces

Let Γ be a simple closed curve that bounds the finite domain D , z =z (ζ )=z (r e ^{i ?} ) be the conformal mapping of the circle {ζ :|ζ |<1} onto the domain D . Furthermore, let the functions A (z ), B (z ) be given on D and U ^{s ,2}(A ;B ;D ) be the set of regular solutions of the equation . We call the Smirnov class E ^{p (t )}(A ;B ;D ) the set of those generalized functions W in D for which where p (t ) is a positive measurable function on Γ. We consider the Riemann‐Hilbert problem: Define a function W (z ) from the class E ^{p (t )}(A ;B ;D ) for which the equality, is fulfilled almost everywhere on Γ. It is assumed that Γ is a piecewise‐smooth curve without external peaks; , p is Log Hölder continuous and the function belongs to the class A (p (t );Γ), which is the generalization of the well‐known Simonenko class A (p ;Γ), where p is constant. The solvability conditions are established, and solutions are constructed.     

Lifespan for a semilinear pseudo‐parabolic equation

This paper deals with the blow‐up solution to the following semilinear pseudo‐parabolic equation in a bounded domain , which was studied by Luo (Math Method Appl Sci 38(12):2636‐2641, 2015) with the following assumptions on p: and the lifespan for the initial energy J(u₀)<0 is considered. This paper generalizes the above results on the following two aspects:

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Well‐posedness and unique continuation property for the generalized Ostrovsky equation with low regularity

In this paper, we investigate the initial value problem (IVP henceforth) associated with the generalized Ostrovsky equation as follows: with initial data in the modified Sobolev space . Using Fourier restriction norm method, Tao's [k,Z]?multiplier method and the contraction mapping principle, we show that the local well‐posedness is established for the initial data with (k = 2) and is established for the initial data with (k = 3). Using these results and conservation laws, we also prove that the IVP is globally well‐posed for the initial data with s = 0(k = 2,3). Finally, using complex variables technique and Paley–Wiener theorem, we prove the unique continuation property for the IVP benefited from the ideas of Zhang ZY. et al., On the unique continuation property for the modified Kawahara equation, Advances in Mathematics (China), http://advmath.pku.edu.cn/CN/10.11845/sxjz.2014078b . Copyright © 2015 John Wiley & Sons, Ltd.     

On the periodicity of solutions of max‐type difference equation

This paper is devoted to study the periodic nature of the solution of the following max‐type difference equation: where the initial conditions x_?2,x_?1,x₀ are arbitrary positive real numbers and is a periodic sequence of period two. Copyright © 2014 John Wiley & Sons, Ltd.