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⁺.     

Multiplicity and symmetry results for a nonlinear Schrödinger equation with non‐local regional diffusion

In this paper, we are interested in the nonlinear Schrödinger equation with non‐local regional diffusion (1) where 0 < α < 1 and is a variational version of the regional Laplacian, whose range of scope is a ball with radius ρ(x) > 0. The novelty of this paper is that, assuming f is of subquadratic growth as |u|→+∞, we show that 1 possesses infinitely many solutions via the genus properties in critical point theory. Furthermore, if f(x,u) = γa(x)|u|^{γ ? 1}, where is a nonincreasing radially symmetric function, then the solution of 1 is radially symmetric. Copyright © 2015 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.     

Global bounded weak solutions to a degenerate quasilinear chemotaxis system with rotation

This paper deals with the quasilinear Keller–Segel system with rotation where is a bounded domain with smooth boundary, D(u) is supposed to be sufficiently smooth and satisfies D(u)≥D₀u^{m ? 1}(m≥1) and D(u)≤D₁(u + 1)^{K ? m}u^{m ? 1}(K≥1) for all u≥0 with some positive constants D₀ and D₁, and f(u) is assumed to be smooth enough and non‐negative for all u≥0 and f(0) = 0, while S(u,v,x) = (s_ij)_{n × n} is a matrix with and with l≥2, where is nondecreasing on [0,∞). It is proved that when , the system possesses at least one global and bounded weak solution for any sufficiently smooth non‐negative initial data. Copyright © 2015 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.     

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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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.     

On a class of coupled Schrödinger systems with critical Sobolev exponent growth

Consider the following two critical nonlinear Schrödinger systems: (0.1) (0.2) where is a smooth bounded domain, N≥3,?λ(Ω) < λ₁,λ₂<0,μ₁,μ₂>0,α,β≥1 with α + β = 2^?,γ ≠ 0,λ(Ω) is the first eigenvalue of ?Δ with the Dirichlet boundary condition and For N = 3,λ₁=λ₂,γ > 0 small, we obtain the existence of positive least energy solution of 0.1 and 0.2 . For N≥5,γ > 0, the existence of positive least energy solution of 0.2 is established. For N≥5,γ ≠ 0, we prove that 0.1 possesses a positive least energy solution. The limit behavior of the positive least energy solutions when γ→?∞ and phase separation for 0.1 are also considered. Copyright © 2015 John Wiley & Sons, Ltd.     

The existence of ‐bounded solution operators of the thermoelastic plate equation with Dirichlet boundary conditions

We consider the linearized thermoelastic plate equation with the Dirichlet boundary condition in a general domain Ω, given by with the initial condition u|_(t=0)=u₀, u_t|_(t=0)=u₁, and θ|_(t=0)=θ₀ in Ω and the boundary condition u=∂_νu=θ=0 on Γ, where u=u(x,t) denotes a vertical displacement at time t at the point x=(x₁,⋯,x_n)∈Ω, while θ=θ(x,t) describes the temperature. This work extends the result obtained by Naito and Shibata that studied the problem in the half‐space case. We prove the existence of ‐bounded solution operators of the corresponding resolvent problem. Then, the generation of C₀ analytic semigroup and the maximal L_p‐L_q‐regularity of time‐dependent problem are derived.     

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.     

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.     

The dynamics of the 3D NLS with the convolution term

We consider the nonlinear Schrödinger equation (NLSH) with the convolution combined term (CNLSH) (CNLSH) in the energy space . We firstly use a variational approach to give a dichotomy of scattering and blow up for the radial solution with the energy below the threshold, which is given by the ground state W for the energy‐critical NLS: iu_t+Δu=?|u|⁴u. The basic strategy is the concentration‐compactness arguments from Kenig and Merle. We overcome the main difficulties coming from the lack of scaling invariance and the non‐local property of the convolution term. Our result shows that the focusing, ‐critical term ?|u|⁴u plays the decisive role of the threshold of the scattering solution of (CNLSH) in the energy space. Copyright © 2015 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.     

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.     

Boundedness of solutions to a quasilinear parabolic–elliptic Keller–Segel system with logistic source

We study a quasilinear parabolic–elliptic Keller–Segel system involving a source term of logistic type u_t = ? ? (?(u) ? u) ? χ ? ? (u ? v) + g(u), ? Δv = ? v + u in Ω × (0,T), subject to nonnegative initial data and the homogeneous Neumann boundary condition in a bounded domain with smooth boundary, n ≥ 1, χ > 0, ? ≥ c₁s^p for s ≥ s₀ > 1, and g(s) ≤ as ? μs² for s > 0 with a,g(0) ≥ 0, μ > 0. There are three nonlinear mechanisms included in the chemotaxis model: the nonlinear diffusion, aggregation and logistic absorption. The interaction among the triple nonlinearities shows that together with the nonlinear diffusion, the logistic absorption will dominate the aggregation such that the unique classical solution of the system has to be global in time and bounded, regardless of the initial data, whenever , or, equivalently, , which enlarge the parameter range , or , required by globally bounded solutions of the quasilinear K‐S system without the logistic source. Copyright © 2013 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.     

The blow‐up criteria of smooth solutions to the generalized and ideal incompressible viscoelastic flow

In this paper, we prove two blow‐up criteria of smooth solution: one for the generalized incompressible Oldroyd model with fractional Laplacian velocity dissipation (?Δ)^αu in the space and one for the inviscid Oldroyd model. Assume that (u(t,x),F(t,x)) is a smooth solution to the generalized Oldroyd model in [0,T); then, the solution (u(t,x),F(t,x)) does not develop singularity until t = T provided . For the ideal impressible viscoelastic flow, it is shown that the smooth solution (u,F) can be extended beyond T if , which is an improvement of the result given by Hu and Hynd (A blowup criterion for ideal viscoelastic flow, J. Math. Fluid Mech., 15(2013), 431–437). Copyright © 2015 John Wiley & Sons, Ltd.     

Finite‐dimensional global attractor for a semi‐discrete fractional nonlinear Schrödinger equation

We consider a semi‐discrete in time Crank–Nicolson scheme to discretize a weakly damped forced nonlinear fractional Schrödinger equation u _t ?i (?Δ)^α u +i |u |²u +γ u =f for considered in the the whole space . We prove that such semi‐discrete equation provides a discrete infinite‐dimensional dynamical system in that possesses a global attractor in . We show also that if the external force is in a suitable weighted Lebesgue space, then this global attractor has a finite fractal dimension. Copyright © 2017 John Wiley & Sons, Ltd.     

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.     

Boundedness in a parabolic‐elliptic chemotaxis system with nonlinear diffusion and sensitivity and logistic source

In this paper, we study the zero‐flux chemotaxis‐system where Ω is a bounded and smooth domain of , n≥1, and where , k,μ>0 and α≤1. For any v≥0, the chemotactic sensitivity function is assumed to behave as the prototype χ(v)=χ₀/(1+av)², with a≥0 and χ₀>0. We prove that for any nonnegative and sufficiently regular initial data u(x,0), the corresponding initial‐boundary value problem admits a unique global bounded classical solution if α<1; indeed, for α=1, the same conclusion is obtained provided μ is large enough. Finally, we illustrate the range of dynamics present within the chemotaxis system in 1, 2, and 3 dimensions by means of numerical simulations.