Operators with rough singular kernels

For α>0, we study the singular integral operators TΩ,α and the Marcinkiewicz integral operator μΩ,α. The kernels of these operators behave like |y|^−n−α near y=0, and contain a distribution Ω on the unit sphere Sn−1. We prove that if Ω∈Hr(Sn−1)(r=(n−1)/(n−1+α)) satisfying certain cancellation condition, then both TΩ,α and μΩ,α can be extend to be the bounded operators from the Sobolev space to the Lebesgue space Lp(Rn). The result improves and extends some known results.     

Boundedness of rough singular integral operators on the Triebel-Lizorkin spaces

We consider the singular integral operator T with kernel K(x)=Ω(x)/n|x| and prove its boundedness on the Triebel-Lizorkin spaces provided that Ω satisfies a size condition which contains the case Ω∈Lr(Sn−1), r>1.     

On hyper-singular integral operators with variable kernels

Let n?2, Sn−1 be the unit sphere in Rn. For 0?α<1, m∈N₀, 1<p?2, and Ω∈L^∞(Rn)×Hr(Sn−1) with (where Hr is the Hardy space if r?1 and Hr=Lr if 1<r<∞), we study the singular integral operator, for r?1, defined by

     

Local existence of solutions to some degenerate parabolic equation associated with the p-Laplacian

The existence of local (in time) solutions of the initial-boundary value problem for the following degenerate parabolic equation: ut(x,t)−Δpu(x,t)−|u|^q−2u(x,t)=f(x,t), (x,t)∈Ω×(0,T), where 2?p<q<+∞, Ω is a bounded domain in RN, is given and Δp denotes the so-called p-Laplacian defined by Δpu:=∇⋅(|∇u|^p−2∇u), with initial data u₀∈Lr(Ω) is proved under r>N(q−p)/p without imposing any smallness on u₀ and f. To this end, the above problem is reduced into the Cauchy problem for an evolution equation governed by the difference of two subdifferential operators in a reflexive Banach space, and the theory of subdifferential operators and potential well method are employed to establish energy estimates. Particularly, Lr-estimates of solutions play a crucial role to construct a time-local solution and reveal the dependence of the time interval [0,T₀] in which the problem admits a solution. More precisely, T₀ depends only on Lr|u₀| and f.     

Necessary and sufficient conditions for boundedness of some commutators with weighted Lipschitz functions

In this paper we show that b∈Lipβ,μ if and only if the commutator [b,T] of the multiplication operator by b and the singular integral operator T is bounded from Lp(μ) to Lq(μ1−q), where 1<p<q<∞, 0<β<1 and 1/q=1/p−β/n. Also we will obtain that b∈Lipβ,μ if and only if the commutator [b,Iα] of the multiplication operator by b and the fractional integral operator Iα is bounded from Lp(μ) to Lr(μ1−(1−α/n)r), where 1<p<∞, 0<β<1 and 1/r=1/p−(β+α)/n with 1/p>(β+α)/n.     

Boundedness of rough oscillatory singular integral on Triebel-Lizorkin spaces

We give the boundedness on Triebel-Lizorkin spaces for oscillatory singular integral operators with polynomial phases and rough kernels of the form eiP(x)Ω(x)|x|⁻ⁿ, where Ω∈Llog⁺L(Sn−1) is homogeneous of degree zero and satisfies certain cancellation condition.     

Isolated singularities of solutions to quasi-linear elliptic equations with absorption

We study the problem of removability of isolated singularities for a general second-order quasi-linear equation in divergence form −divA(x,u,∇u)+a₀(x,u)+g(x,u)=0 in a punctured domain Ω?{0}, where Ω is a domain in Rn, n?3. The model example is the equation −Δpu+gu|u|^p−2+u|u|^q−1=0, q>p−1>0, p<n. Assuming that the lower-order terms satisfy certain non-linear Kato-type conditions, we prove that for all point singularities of the above equation are removable, thus extending the seminal result of Brezis and Véron.     

A multiply intersecting Erd?s-Ko-Rado theorem — The principal case

Let m(n,k,r,t) be the maximum size of satisfying |F₁∩?∩F_r|≥t for all F₁,…,F_r∈F. We prove that for every p∈(0,1) there is some r₀ such that, for all r>r₀ and all t with 1≤t≤⌊(p^1−r−p)/(1−p)⌋−r, there exists n₀ so that if n>n₀ and p=k/n, then . The upper bound for t is tight for fixed p and r.     

Nonlinear dissipative wave equations with space-time dependent potential

We study the long time behavior of solutions for damped wave equations with absorption. These equations are generally accepted as models of wave propagation in heterogeneous media with space-time dependent friction a(t,x)u_t and nonlinear absorption |u|^p−1u (Ikawa (2000) [17]). We consider 1<p<(n+2)/(n−2) and separable a(t,x)=λ(x)η(t) with λ(x)∼(1+|x|)^−α and η(t)∼(1+t)^−β satisfying conditions (A1) or (A2) which are given. The main results are precise decay estimates for the energy, L² and L^p+1 norms of solutions. We also observe the following behavior: if α∈[0,1), β∈(−1,1) and 0<α+β<1, there are three different regions for the decay of solutions depending on p; if α∈(−∞,0) and β∈(−1,1), there are only two different regions for the decay of the solutions depending on p.     

The heat equation with singular nonlinearity and singular initial data

We study the existence, uniqueness and regularity of positive solutions of the parabolic equation ut−Δu=a(x)uq+b(x)up in a bounded domain and with Dirichlet's condition on the boundary. We consider here a∈Lα(Ω), b∈Lβ(Ω) and 0<q?1<p. The initial data u(0)=u₀ is considered in the space Lr(Ω), r?1. In the main result (0<q<1), we assume a,b?0 a.e. in Ω and we assume that u₀?γdΩ for some γ>0. We find a unique solution in the space .     

Partial autocorrelation functions of the fractional ARIMA processes with negative degree of differencing

Let be a fractional ARIMA(p,d,q) process with partial autocorrelation function α(·). In this paper, we prove that if d∈(−1/2,0) then |α(n)|∼|d|/n as n→∞. This extends the previous result for the case 0<d<1/2.     

The asymptotic behaviour of the unique solution for the singular Lane-Emden-Fowler equation

By Karamata regular variation theory and constructing comparison functions, we show the exact asymptotic behaviour of the unique classical solution near the boundary to a singular Dirichlet problem −Δu=k(x)g(u), u>0, x∈Ω, u_|∂Ω=0, where Ω is a bounded domain with smooth boundary in RN; g∈C¹((0,∞),(0,∞)), , for each ξ>0, for some γ>0; and for some α∈(0,1), is nonnegative on Ω, which is also singular near the boundary.     

Asymptotic behavior of positive solutions of a singular nonlinear Dirichlet problem

Let Ω be a C1,1-bounded domain in Rn for n?2. In this paper, we are concerned with the asymptotic behavior of the unique positive classical solution to the singular boundary-value problem Δu+a(x)u−σ=0 in Ω, u|∂Ω=0, where σ?0, a is a nonnegative function in , 0<α<1 and there exists c>0 such that . Here λ?2, μk∈R, ω is a positive constant and δ(x)=dist(x,∂Ω).     

Separable solutions of some quasilinear equations with source reaction

We study the existence of singular solutions to the equation −div(|Du|^p−2Du)=|u|^q−1u under the form u(r,θ)=r−βω(θ), r>0, θ∈SN−1. We prove the existence of an exponent q below which no positive solutions can exist. If the dimension is 2 we use a dynamical system approach to construct solutions.     

Boundary regularity for viscosity solutions to degenerate elliptic PDEs

Sufficient conditions are given for the solutions to the (fully nonlinear, degenerate) elliptic equation F(x,u,Du,D²u)=0 in Ω to satisfy |u(x)−u(y)|?Cα|x−y| for some α∈(0,1) when x∈Ω and y∈∂Ω.     

Existence and multiplicity of solutions for quasilinear nonhomogeneous problems: An Orlicz-Sobolev space setting

We study the boundary value problem −div(log(1+q|∇u|)|∇u|^p−2∇u)=f(u) in Ω, u=0 on ∂Ω, where Ω is a bounded domain in RN with smooth boundary. We distinguish the cases where either f(u)=−λ|u|^p−2u+|u|^r−2u or f(u)=λ|u|^p−2u−|u|^r−2u, with p, q>1, p+q<min{N,r}, and r<(Np−N+p)/(N−p). In the first case we show the existence of infinitely many weak solutions for any λ>0. In the second case we prove the existence of a nontrivial weak solution if λ is sufficiently large. Our approach relies on adequate variational methods in Orlicz-Sobolev spaces.     

An even-order three-point boundary value problem on time scales

We study the even-order dynamic equation (−1)ⁿx^(Δ∇)n(t)=λh(t)f(x(t)), t∈[a,c] satisfying the boundary conditions x^(Δ∇)i(a)=0 and x^(Δ∇)i(c)=βx^(Δ∇)i(b) for 0?i?n−1. The three points a,b,c are from a time scale , where 0<β(b−a)<c−a for b∈(a,c), β>0, f is a positive function, and h is a nonnegative function that is allowed to vanish on some subintervals of [a,c] of the time scale.     

Super-exponential decay and holomorphic extensions for semilinear equations with polynomial coefficients

We show that all eigenfunctions of linear partial differential operators in Rn with polynomial coefficients of Shubin type are extended to entire functions in Cn of finite exponential type 2 and decay like exp(−²|z|) for |z|→∞ in conic neighbourhoods of the form |Imz|?γ|Rez|. We also show that under semilinear polynomial perturbations all nonzero homoclinics keep the super-exponential decay of the above type, whereas a loss of the holomorphicity occurs, namely we show holomorphic extension into a strip {z∈Cn||Imz|?T} for some T>0. The proofs are based on geometrical and perturbative methods in Gelfand-Shilov spaces. The results apply in particular to semilinear Schrödinger equations of the form

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