bluuebunny/crossref_metadata_2025_split huggingface dataset:
Linear adiabatic dynamics generated by operators with continuous spectrum. I
Authors: Buslaev Vladimir S. , Sulem Catherine 
Published: November 2019

We are interested in the asymptotic behavior of the solution to the Cauchy problem for the linear evolution equation iε ∂tψ=A(t)ψ, A(t)=A0+V(t), ψ(0)=ψ0, in the limit ε→0. A case of special interest is when the operator A(t) has continuous spectrum and the initial data ψ0 is, in particular, an improper eigenfunction of the continuous spectrum of A(0). Under suitable assumptions on A(t), we derive a formal asymptotic solution of the problem whose leading order has an explicit representation. A key ingredient is a reduction of the original Cauchy problem to the study of the semiclassical pseudo-differential operator ℳ=M(t, iε ∂t) with compact operator-valued symbol M(t, E)=V1(t)(A0−EI)−1V2(t), V(t)=V2(t)V1(t), and an asymptotic analysis of its spectral properties. We illustrate our approach with a detailed presentation of the example of the Schrödinger equation on the axis with the δ-function potential: A(t)=−∂xx+α(t)δ(x).
Diamagnetic expansions for perfect quantum gases II: Uniform bounds
Authors: Briet Philippe , Cornean Horia D. , Louis Delphine 
Published: November 2019

Consider a charged, perfect quantum gas, in the effective mass approximation, and in the grand-canonical ensemble. We prove in this paper that the generalized magnetic susceptibilities admit the thermodynamic limit for all admissible fugacities, uniformly on compacts included in the analyticity domain of the grand-canonical pressure. The problem and the proof strategy were outlined in MPRF 11 (2005), 177–188. In J. Math. Phys. 47 (2006), 083511 we proved in detail the pointwise thermodynamic limit near z=0. The present paper is the last one of this series, and contains the proof of the uniform bounds on compacts needed in order to apply Vitali's Convergence Theorem.
A reduced model for spatially structured predator–prey systems with fast spatial migrations and slow demographic evolutions
Authors: Castella François , Hoffbeck Jean-Philippe , Lagadeuc Yvan 
Published: November 2019

We consider a spatially structured predator–prey model where fast migrations occur inside a given spatial domain, while slow predator–prey interactions prescribe the demographic evolution. The unknowns of our model are the numbers of predators and prey at each time t and each site x of the domain. In the idealized limit where migrations are infinitely fast, we show one can approximate the global dynamics using the mere two unknowns corresponding to the total number of preys and predators, irrespective of their respective spatial repartition. Besides, the error term induced by this approximation can be made exponentially small with respect to the natural asymptotic parameter. In doing so, we completely characterize how migrations do modify both the qualitative and quantitative properties of the global demography. Our analysis relies on a convenient version of the central manifold theorem, in conjunction with a spectral gap estimate on the involved migration operator.
Uniqueness and multiplicity results for N-Laplace equation with critical and singular nonlinearity in a ball
Authors: Giacomoni J. , Prashanth S. , Sreenadh K. 
Published: November 2019

Let B1 be the unit open ball with center at the origin in RN, N≥2. We consider the following quasilinear problem depending on a real parameter λ>0: −ΔNu=λ f(u), u>0 in Ω, u=0 on ∂Ω,   (Pλ) where f(t) is a nonlinearity that grows like etN/N−1 as t→∞ and behaves like tα, for some α∈(−∞, 0), as t→0+. More precisely, we require f to satisfy assumptions (A1) and (A2) listed in Section 1. For such a general nonlinearity we show that if λ>0 is small enough, (Pλ) admits at least one weak solution (in the sense of distributions). We further study the question of uniqueness and multiplicity of solutions to (Pλ) when Ω=B1 under additional structural conditions on f (see assumptions (A3)–(A8) in Section 2). Using shooting methods and asymptotic analysis of ODEs, under the additional assumptions (A3)–(A5), we prove uniqueness of solution to (Pλ) for all λ>0 small whereas under (A6), (A7) or (A8), we show multiplicity of solutions to (Pλ) for all λ>0 in a maximal interval. These results clearly show that the borderline between uniqueness and multiplicity is given by the growth condition lim inf t→∞h(t)teεt1/(N−1)=∞ ∀ε>0.
Stochastic homogenization of quasilinear PDEs with a spatial degeneracy
Authors: Delarue François , Rhodes Rémi 
Published: November 2019

We investigate stochastic homogenization for some degenerate quasilinear parabolic PDEs. The underlying nonlinear operator degenerates along the space variable, uniformly in the nonlinear term: the degeneracy points correspond to the degeneracy points of a reference diffusion operator on the random medium. Assuming that this reference diffusion operator is ergodic, we can prove the homogenization property for the quasilinear PDEs, by means of the first-order approximation method. The (nonlinear) limit operator need not be nondegenerate. Concrete examples are provided.
On continuous dependence of solution of parabolic equations on coefficients
Authors: Kim Hyejin 
Published: November 2019

It is well known that solutions of classical initial-boundary problems for second-order parabolic equations depend continuously on the coefficients if the coefficients converge to their limits in a strong enough topology. In case of one spatial variable, we consider the question of the weakest possible topology providing convergence of the solutions. The problem is closely related to weak convergence of corresponding diffusion processes. Continuous Markov processes corresponding to the Feller operators DvDu arise, in general, as limiting processes. Solutions of the parabolic equations converge, in general, to the solutions of corresponding initial-boundary problems for the limiting operator DvDu.
H-convergence of singular solutions of some Dirichlet problems with terms of order one
Authors: Boccardo Lucio , Casado-Diaz Juan 
Published: November 2019

We study the H-convergence of boundary value problems of the type  −div(Mε(x)∇uε)=−div(uεEε(x))+f(x), inΩ,  uε=0 on∂Ω if Eε∈(L2(Ω))N.
Asymptotic expansions of Legendre series coefficients for functions with endpoint singularities
Authors: Sidi Avram 
Published: November 2019

Let Σ∞n=0en[f]Pn(x) be the Legendre expansion of a function f(x) on (−1, 1). In this work, we derive an asymptotic expansion as n→∞ for en[f], assuming that f∈C∞(−1, 1), but may have arbitrary algebraic-logarithmic singularities at one or both endpoints x=±1. Specifically, we assume that f(x) has asymptotic expansions of the forms  f(x)~Σ∞s=0Usl(log(1−x))(1−x)αs as x→1−,  f(x)~Σ∞s=0Vs(log(1+x))(1+x)βs as x→−1+, where Us(y) and Vs(y) are some polynomials in y. Here, αs and βs are in general complex and Rαs, Rβs>−1. An important special case is that in which Us(y) and Vs(y) are constant polynomials; for this case, the asymptotic expansion of en[f] assumes the form  en[f]~Σs=0αs∉Z+∞Σ∞i=0asihαs+i+1/2+(−1)n Σs=0βs∉Z+∞Σ∞i=0bsihβs+i+1/2 as n→∞, where h=(n+1/2)−2, Z+={0, 1 , 2, …}, and asi and bsi are constants independent of n.
On the spectrum of the Neumann problem for Laplace equation in a domain with a narrow slit
Authors: Gadyl'shin Rustem R. , Il'in Arlen M. 
Published: November 2019

The Neumann problem in two-dimensional domain with a narrow slit is studied. The width of the slit is a small parameter 0<ε�1. The complete asymptotic expansion with respect to ε for the eigenvalue of the perturbed problem converging to a simple eigenvalue of the limiting problem is constructed by means of the method of the matching asymptotic expansions. It is shown that the regular perturbation theory can formally be applied in a natural way up to terms of order ε2. However, the result obtained in that way is false. The correct result can be obtained only by means of inner asymptotic expansion. We also show that the eigenvalue of the perturbed problem can be both more and less than the eigenvalue of the limiting problem subject to the position and geometry of the slit.
Periodic constant-sign solutions for systems of Hill's equations
Authors: Agarwal Ravi P. , O'Regan Donal , Wong Patricia J.Y. 
Published: November 2019

We consider the system of Hill's equations u″i(t)+ai(t)ui(t)=Fi(t, u1(t), u2(t), …, un(t)), 1≤i≤n, where ai and Fi are periodic in t, and the non-linearities Fi(t, x1, x2, …, xn) can be singular at xj=0 where j∈{1, 2, …, n}. Criteria are offered for the existence of periodic constant-sign solutions, i.e., θiui(t)≥0 for each 1≤i≤n, where θi∈{1, −1} is fixed. The main tool used is Schauder's fixed point theorem. We also include examples to illustrate the usefulness of the results obtained.
Global Strichartz estimates for the wave equation with a time-periodic non-trapping metric
Authors: Kian Yavar 
Published: November 2019

We obtain global Strichartz estimates for the solution u of the wave equation ∂t2u−divx(a(t, x)∇xu)=0 with time-periodic metric a(t, x) equal to 1 outside a compact set with respect to x. We assume a(t, x) is a non-trapping perturbation and, moreover, we suppose that there are no resonances zj∈C with |zj|≥1.
Dynamic properties of almost monochromatic standing waves
Authors: Elbaz Claude 
Published: November 2019

For a scalar field propagating at light velocity c, we show that kinematic and dynamic properties of almost monochromatic standing waves, with frequency Ω(x, t)=ω±δΩ(x, t), where ω is constant, and δΩ(x, t)�ω, are formally identical with mechanical properties of matter. They are both described by equations with the same mathematical structure. The energy conservation stems from stability in time, while the variational principle stems from stability in space. In classical mechanics of a particle, the relativist equations correspond to the geometrical optics approximation as ω→∞. The quantum mechanical equations correspond to the wave optics approximation, in which wave homogeneous Fourier relations are replaced by the material Heisenberg relations.
Stabilization of the linear Kawahara equation with localized damping
Authors: Vasconcellos Carlos F. , da Silva Patricia N. 
Published: November 2019

In this erratum of the article: “Stabilization of the linear Kawahara equation with localized damping”, published in Asymptotic Analysis 58(4) (2008), 229–252, DOI 10.3233ASY-2008-0895, we shall fix a mistake which was done when we proved the existence and we defined the countable set of the lengths L where the decay of the energy fails, a set named 𝒩, see Lemma 2.1, p. 235. We considered the equation defined by λu0+u0‴+u0'''''=0, which is wrong since the parameter η in the Kawahara equation was completely forgotten and it makes changing in the result. In fact, η is a negative constant and the correct equation is λu0+u0‴+ηu0'''''=0. Then, as η<0 we can prove, in this case, that the set 𝒩 is empty. Therefore, we obtain that the energy associated with linear Kawahara equation decays exponentially for all lengths L, even in absence of the damping. In the new proof for Lemma 2.1, remain the same idea and the same arguments of the original proof, but the signal of parameter η produces different final result.
Decay estimates to equilibrium for some asymptotically autonomous semilinear evolution equations
Authors: Ben Hassen Imen 
Published: November 2019

We estimate the rate of decay of the difference between a solution and its limiting equilibrium for the nonautonomous first-order problem u˙+ℳu=g(t),  t∈R+ , and for the second-order problem u¨+u˙+ℳu=g(t),  t∈R+, where ℳ is a gradient operator and g is a function which tends to 0 when t tends to ∞.
Nonconvergent radial solutions of semilinear elliptic equations
Authors: Kwong Man Kam , Wong Solomon Wai-Him 
Published: November 2019

For many known examples of semilinear elliptic equations Δu+f(u)=0 in RN (N>1), a bounded radial solution u(r) converges to a constant as r→∞. Maier, in 1994, constructed, for N=2, an equation with a nonconvergent radial solution. Some necessary conditions for the existence of a nonconvergent solution were given by Maier, and later extended by Iaia. These conditions point out that, for N>2, equations with nonconvergent solutions are rather rare. A nonconvergent solution must oscillate between two constant values c1<c2 and f must vanish at either c1 or c2. In the neighborhood of one of these points, f must fluctuate wildly in an unusual way that excludes almost all common functions. In this paper, we give a further improvement of the above result with an alternative, simpler proof. The proof depends on an elementary, but nonobvious property of an initial value problem.
Toda system and cluster phase transition layers in an inhomogeneous phase transition model
Authors: Wei Juncheng , Yang Jun 
Published: November 2019

We consider the following singularly perturbed elliptic problem ε2Δũ+(ũ−a(y˜))(1−ũ2)=0 in Ω,  ∂ũ/∂ν=0 on ∂Ω, where Ω is a bounded domain in R2 with smooth boundary, −1<a(y˜)<1, ε is a small parameter, ν denotes the outward normal of ∂Ω. Assume that Γ={y˜∈Ω: a(y˜)=0} is a simple closed and smooth curve contained in Ω in such a way that Γ separates Ω into two disjoint components Ω+={y˜∈Ω: a(y˜)<0} and Ω−={y˜∈Ω: a(y˜)>0} and ∂a/∂ν0>0 on Γ, where ν0 is the outer normal of Ω+, pointing to the interior of Ω−. For any fixed integer N=2m+1≥3, we will show the existence of a clustered solution uε with N-transition layers near Γ with mutual distance O(ε|log ε|), provided that ε stays away from a discrete set of values at which resonance occurs. Moreover, uε approaches 1 in Ω− and −1 in Ω+. Central to our analysis is the solvability of a Toda system.
A lower semicontinuity result in SBD for surface integral functionals of Fracture Mechanics
Authors: Gargiulo Giuliano , Zappale Elvira 
Published: November 2019

A lower semicontinuity result is proved in the space of special vector fields with bounded deformation for a fracture energetic model of the type ∫JuΨ([u], νu) dℋN−1,  [u]·νu≥0, ℋN−1 -a.e. on Ju. A representation of the energy density Ψ, which ensures lower semicontinuity, is also given.
Limit cases in an elliptic problem with a parameter-dependent boundary condition
Authors: García-Melián Jorge , Rossi Julio D. , Sabina de Lis José C. 
Published: November 2019

In this work we discuss existence, uniqueness and asymptotic profiles of positive solutions to the quasilinear problem  −Δpu+a(x)up−1=−ur in Ω,  |∇u|p−2 ∂u/∂ν=λup−1 on ∂Ω for λ∈R, where r>p−1>0, a∈L∞(Ω). We analyze the existence of solutions in terms of a principal eigenvalue, and determine their asymptotic behavior both when r→p−1 and when r→∞.
Spectral properties of combinatorial Schrödinger operators on infinite weighted graphs
Authors: Akkouche Sofiane 
Published: November 2019

Let H(λ)=−Δ+λb be the combinatorial Schrödinger operator on an infinite connected graph G with a potential b and a non-negative coupling constant λ. When b≡0, it is well known that σ(−Δ)⊂[0,2]. When b≢0, let s(−Δ+λb):=inf σ(−Δ+λb) and M(−Δ+λb):=sup σ(−Δ+λb) be the bounds of the spectrum of the Schrödinger operator. One of the aims of this paper is to study the influence of the potential b on the bounds of the spectrum of −Δ. More precisely, we give a condition on the potential b such that s(−Δ+λb) is strictly positive for λ small enough. We obtain a similar result for the top of the spectrum. We also prove that for a recurrent bipartite weighted graph, the only Schrödinger operator H=−Δ+b such that σ(H)⊂[0,2] is the Laplacian −Δ. We study independently the asymptotic behaviour of the function λ↦s(−Δ+λb) for non-negative potentials b. We prove that s(∞):=lim λ→+∞s(−Δ+λb)<+∞ if and only if inf x∈Vb(x)=0 and we characterize this limit in the general case. As an application, we give an exact value of this limit for certain Jacobi matrices.
On a homogenization technique for singular integrals
Authors: Anza Hafsa Omar , Leghmizi Mohamed Lamine , Mandallena Jean-Philippe 
Published: November 2019

We study homogenization by Γ-convergence of functionals of type ∫ΩW(x/ε,∇ϕ(x)) dx, where Omega⊂RN is a bounded open set, ϕ∈W1,p(Omega;Rm) and p>1, when the 1-periodic integrand W :RN×Mm×N→[0,+∞] is not of p-polynomial growth. Our homogenization technique can be applied when m=N and W has a singular behavior of type W(x,ξ)→+∞ as det ξ→0. However, our technique is not consistent with the constraint W(x,ξ)=+∞ if and only if det ξ≤0.
Convergence estimates for abstract second-order singularly perturbed Cauchy problems with Lipschitzian nonlinearities
Authors: Perjan Andrei , Rusu Galina 
Published: November 2019

We study the behavior of solutions to the problem ε(u″ε(t)+A1uε(t))+u′ε(t)+A0uε(t)+B(uε(t))=fε(t), t∈(0,T], uε(0)=u0ε, u′ε(0)=u1ε, in the Hilbert space H as ε→0, where A1, A0 are two linear self-adjoint operators and B is a Lipschitzian operator.
On the uniqueness of solution Cauchy's inverse problem for the equation Δu=au+b
Authors: Bezrodnykh S.I. , Demidov A.S. 
Published: November 2019

The inverse Cauchy problem Δu(x)=au(x)+b≥0  for x∈ω,  and  u=0,   ∂u/∂ν=Φ   on γ is shown as having a unique solution within a wide class of simply connected domains ω⋐R2 with smooth boundary γ. Here a and b are real numbers to be determined, and Φ is a given function which is normalized by the condition ∫γΦ ds=1.
Boundary blow up solutions for fractional elliptic equations
Authors: Felmer Patricio , Quaas Alexander 
Published: November 2019

In this article we study existence of boundary blow up solutions for some fractional elliptic equations including (−Δ)αu+up=f in Ω, u=g on Ωc, lim x∈Ω,x→∂Ωu(x)=∞, where Ω is a bounded domain of class C2, α∈(0,1) and the functions f :Ω→R and g :RN\Ω−→R are continuous. We obtain existence of a solution u when the boundary value g blows up at the boundary and we get explosion rate for u under an additional assumption on the rate of explosion of g. Our results are extended for an ample class of elliptic fractional nonlinear operators of Isaacs type.
Models for growth of heterogeneous sandpiles via Mosco convergence
Authors: Bocea M. , Mihăilescu M. , Pérez-Llanos M. , Rossi J.D. 
Published: November 2019

In this paper we study the asymptotic behavior of several classes of power-law functionals involving variable exponents pn(·)→∞, via Mosco convergence. In the particular case pn(·)=np(·), we show that the sequence {Hn} of functionals Hn:L2(RN)→[0,+∞] given by Hn(u)=∫RNλ(x)n/np(x)|∇u(x)|np(x) dx   if u∈L2(RN)∩W1,np(·)(RN), +∞  otherwise, converges in the sense of Mosco to a functional which vanishes on the set u∈L2(RN): λ(x)|∇u|p(x)≤ 1 a.e. x∈RN and is infinite in its complement. We also provide an example of a sequence of functionals whose Mosco limit cannot be described in terms of the characteristic function of a subset of L2(RN). As an application of our results we obtain a model for the growth of a sandpile in which the allowed slope of the sand depends explicitly on the position in the sample.
Permeability of interfaces with alternating pores in parabolic problems
Authors: Andreucci Daniele , Bellaveglia Dario 
Published: November 2019

We study a parabolic problem set in a domain divided by a perforated interface. The pores alternate between an open and a closed state, periodically in time. We consider the asymptotics of the solution for vanishingly small size of the pores and time period. The interface condition prevailing in the limit is a linear relation between the flux (on either side) and the jump of the limiting solution across the interface. More exactly this behaviour only takes place when the relative sizes of the relevant geometrical and temporal parameters are connected by suitable relations. With respect to the stationary version of this problem, which is known in the literature, we demonstrate the appearance of a new admissible asymptotic standard. More in general, we describe the precise interplay between the geometrical and temporal parameters leading to the quoted interface condition. This work represents a preliminary mathematical investigation of a model of selective transport of chemical species through biological membranes.
Asymptotic expansion of the scattering matrix associated with matrix Schrödinger operator
Authors: Baklouti Hamadi , Abdeljeilil Slaheddine Ben 
Published: November 2019

In this paper, we study the scattering theory for a 2×2 matrix Schrödinger operator P=−h2d2/dx2I2+V(x)+hR(x,hDx) on L2(R)⌖L2(R), where V(x) is a real diagonal matrix, the eigenvalues of which are never equal. Under some assumptions of analyticity and decay at infinity of V, we describe the asymptotic behavior of the scattering matrix S=(sij)1≤i,j≤4 associated with P when the semi-classical parameter h goes to zero. Moreover, we obtain the estimate ‖S12‖+‖S21‖=O(e−δ/h), where S12 and S21 are the two off-diagonal elements of S and δ>0 is a constant which is explicitly related to the behavior of V(x) in the complex domain.
The dominant asymptotic term of solution to the Hamer equation
Authors: Soloviev A.A. 
Published: November 2019

It is shown that the dominant asymptotic term of solution to the problem ut+uux=K*u−u, u|t=0=u0, where K(x)=(1/2)e−|x|, is a function u0(x,t)=2ζ(x,t)(1−∫−∞xζ(s,t) ds)−1 with ζ(x,t)=M0(h)ex2/4t/√4πt, M0(h)=∫−∞∞h(s) ds and h(x)=(1/2)u0(x)exp {−(1/2)∫−∞xu0(s) ds}, and the function u0(x,t) coincides with the principal asymptotic term of solution to the Burgers equation with initial function u0.
On a Vlasov–Euler system for 2D sprays with gyroscopic effects
Authors: Moussa Ayman , Sueur Franck 
Published: November 2019

In this paper we introduce a PDE system which aims at describing the dynamics of a dispersed phase of particles moving into an incompressible perfect fluid, in two space dimensions. The system couples a Vlasov-type equation and an Euler-type equation: the fluid acts on the dispersed phase through a gyroscopic force whereas the latter contributes to the vorticity of the former. First we give a Dobrushin-type derivation of the system as a mean-field limit of a PDE system which describes the dynamics of a finite number of massive pointwise particles moving into an incompressible perfect fluid. This last system is itself inferred from the paper “On the motion of a small body immersed in a two-dimensional incompressible perfect fluid”, accepted for publication in Bulletin de la SMF where the system for one massive pointwise particle was derived as the limit of the motion of a solid body when the body shrinks to a point with fixed mass and circulation. Then we deal with the well-posedness issues including the existence of weak solutions. Next we exhibit the Hamiltonian structure of the system and finally, we study the behavior of the system in the limit where the mass of the particles vanishes.
Asymptotic equipartition of operator-weighted energies in damped wave equations
Authors: Goldstein Jerome A. , Reyes Guillermo 
Published: November 2019

We prove an asymptotic energy equipartition result for abstract damped wave equations of the form utt+2F(S)ut+S2u=0, where S is a strictly positive self-adjoint operator and the damping operator F(S) is “small”. This means that under certain assumptions, the ratio of suitably modified kinetic and potential energies, K˜(t)/P˜(t), tends to 1 as t→∞ for all nonzero solutions u(t) of the equation. Here, K˜(t) and P˜(t) are conveniently weighted versions of the usual kinetic and potential energies of the associated undamped equation. Previous results, concerning the undamped case and the scalar-damped one, are particular cases. We propose an extension of the concepts of hyperbolicity and unitarity that allows one to consider the equipartition property in a more general setting. Some examples involving PDEs, as well as pseudo-differential equations, are given.