bluuebunny/crossref_metadata_2025 huggingface dataset:
Overdamping and energy decay for abstract wave equations with strong damping
Authors: Goldstein Gisèle Ruiz , Goldstein Jerome A. , Reyes Guillermo
Published: November 2019

In Quart. Appl. Math. 71 (2013), 183–199, the authors find sharp exponential rates for the energy decay of nontrivial solutions to the abstract telegraph equation

utt+2aut+S2u=0,

where S is a strictly positive self-adjoint operator in a (complex) Hilbert space and a is a positive constant. The aim of this paper is a further extension of these results by considering equations of the form

utt+2F(S)ut+S2u=0,

where the damping term involves the action of the positive self-adjoint operator F(S). The main assumption on the continuous function F :(0,+∞)→(0,+∞) is that g(x)=F(x)−x changes sign only once, being positive close to zero. We obtain sharp estimates of the form

E(t)≤Ce−2αt,

where α>0 depends on the relative position of the bottom of the spectrum of S and the point where g vanishes, as well as on the specific behavior of F on the spectrum of S. The general result is then applied to some particular classes of functions F. We also provide a number of applications.
Weyl-type laws for fractional p-eigenvalue problems
Authors: Iannizzotto Antonio , Squassina Marco
Published: November 2019

We prove an asymptotic estimate for the growth of variational eigenvalues of fractional p-Laplacian eigenvalue problems on a smooth bounded domain.
Asymptotic behavior of elliptic nonlocal equations set in cylinders
Authors: Yeressian Karen
Published: November 2019

In this paper the asymptotic behavior of elliptic nonlocal equations in cylindrical domains is studied. These equations are associated with symmetric pure jump Lévy stochastic processes and are generalisation of equations involving the fractional Laplacian operator. First we consider the existence and uniqueness of weak solutions in appropriate Hilbert spaces associated with the Lévy measure and then study the asymptotic behavior in cylinders.
Pullback attractors for a non-autonomous Cahn–Hilliard–Navier–Stokes system in 2D
Authors: Tachim Medjo T.
Published: November 2019

This article studies the pullback asymptotic behavior of solutions for a non-autonomous Cahn–Hilliard–Navier–Stokes (CH–NS) system in a two-dimensional domain. We prove the existence of pullback attractors 𝒜VM in VM (the velocity has the H1-regularity) and 𝒜YM in YM (the velocity has the L2-regularity). Then we verify the regularity of the pullback attractors by proving that 𝒜VM=𝒜YM, which implies the pullback asymptotic smoothing effect of the model in the sense that the solutions eventually become more regular than the initial data.
The necessary conditions for the existence of local Ginzburg–Landau minimizers with prescribed degrees on the boundary
Authors: Misiats Oleksandr
Published: November 2019

We study the minimization problem for simplified Ginzburg–Landau functional in doubly connected domain. This minimization problem is a subject to “semi-stiff” boundary conditions: |u|=1 and prescribed degrees p and q on the outer and inner boundaries respectively. Following the work of Berlyand and Rybalko [J. Eur. Math. Soc. 12 (2010), 1497–1531], we additionally prescribe the degree in the bulk (approximate bulk degree) to be d. The work [J. Eur. Math. Soc. 12 (2010), 1497–1531] established the sufficient conditions on the existence of Ginzburg–Landau minimizers, given in terms of p, q and d. The present work complements the result of [J. Eur. Math. Soc. 12 (2010), 1497–1531] by providing the necessary conditions for the existence of nontrivial (nonconstant) minimizers.
Fractal powers in Serrin's swirling vortex solutions
Authors: Bělík Pavel , Dokken Douglas P. , Scholz Kurt , Shvartsman Mikhail M.
Published: November 2019

We consider a modification of the fluid flow model for a tornado-like swirling vortex developed by Serrin [Phil. Trans. Roy. Soc. London, Series A, Math & Phys. Sci. 271(1214) (1972), 325–360], where velocity decreases as the reciprocal of the distance from the vortex axis. Recent studies, based on radar data of selected severe weather events [Mon. Wea. Rev. 133(9) (2005), 2535–2551; Mon. Wea. Rev. 128(7) (2000), 2135–2164; Mon. Wea. Rev. 133(1) (2005), 97–119], indicate that the angular momentum in a tornado may not be constant with the radius, and thus suggest a different scaling of the velocity/radial distance dependence.

Motivated by this suggestion, we consider Serrin's approach with the assumption that the velocity decreases as the reciprocal of the distance from the vortex axis to the power b with a general b>0. This leads to a boundary-value problem for a system of nonlinear differential equations. We analyze this problem for particular cases, both with nonzero and zero viscosity, discuss the question of existence of solutions, and use numerical techniques to describe those solutions that we cannot obtain analytically.
Asymptotics of delay differential equations via polynomials
Authors: Wang Xiang-Sheng
Published: November 2019

In this paper, we introduce an innovative and systematic technique to study delay differential equations via polynomials. First, we review an intrinsic relation between delay differential equations and polynomials. From this relation, we obtain long time behaviors of the solutions to delay differential equations via asymptotic analysis of the corresponding polynomials. Moreover, we derive asymptotic formulas and upper bounds for the intrinsic growth rate of delay differential equations, as well as a Gronwall-type inequality for delay differential inequalities.
A model for crack growth with branching and kinking
Authors: Racca Simone
Published: November 2019

We study an evolution model for fractured elastic materials in the 2-dimensional case, for which the crack path is not assumed to be known a priori. We introduce some general assumptions on the structure of the fracture sets suitable to remove the restrictions on the regularity of the crack sets and to allow for kinking and branching to develop. In addition we define the front of the fracture and its velocity. By means of a time-discretization approach, we prove the existence of a continuous-time evolution that satisfies an energy inequality and a stability criterion. The energy balance also takes into account the energy dissipated at the front of the fracture. The stability criterion is stated in the framework of Griffith's theory, in terms of the energy release rate, when the crack grows at least at one point of its front.
Multi-scale asymptotic expansion for a singular problem of a free plate with thin stiffener
Authors: Rahmani Leila , Vial Grégory
Published: November 2019

In this paper, we consider a partially clamped plate which is stiffened on a portion of its free boundary. Our aim is to build an asymptotic expansion of the displacement, solution of the Kirchhoff–Love model, with respect to the thickness of the stiffener. Due to the mixed boundary conditions, singularities appear, obstructing the construction of the terms of the asymptotic expansion in the same way as if the plate was surrounded by the stiffener on its whole boundary. Using a splitting into regular and singular parts, we are able to formulate an asymptotic expansion involving profiles which allow to take into account the singularities.
Moments for strong solutions of the 2D stochastic Navier–Stokes equations in a bounded domain
Authors: Kukavica Igor , Vicol Vlad
Published: November 2019

We address a question posed by Glatt-Holtz and Ziane in Advances in Differential Equations 14 (2009), 567–600, regarding moments of strong pathwise solutions to the Navier–Stokes equations in a two-dimensional bounded domain 𝒪. We prove that Eφ(‖u(t)‖H1(𝒪)2)<∞ for any deterministic t>0, where φ(x)=log (1+log (1+x)). Such moment bounds may be used to study statistical properties of the long time behavior of the equation. In addition, we obtain algebraic moment bounds on compact subdomains 𝒪0 of the form Eφε(‖u(t)‖H1(𝒪0)2)<∞, where φε(x)=(1+x)(1−ε)/2, for any deterministic t>0 and any ε>0.
On the ℛ-boundedness of the solution operators in the study of the compressible viscous fluid flow with free boundary conditions
Authors: Götz Dario , Shibata Yoshihiro
Published: November 2019

In this paper, we consider a generalized resolvent problem for the linearization system of the Navier–Stokes equations describing some free boundary problem of a compressible barotropic viscous fluid flow without taking the surface tension into account. We prove the existence of the ℛ-bounded solution operators, which drives not only the generation of analytic semigroup but also the maximal Lp–Lq regularity by means of Weis' operator valued Fourier multiplier theorem for the corresponding time dependent problem that enable us to prove a local in time existence theorem of the free boundary problem for a compressible barotropic viscous fluid flow in the Lp in time and Lq space setting (cf. Annali dell Universita di Ferrara 60 (2014), 55–89). The results in this paper were given in the PhD thesis [Three topics in fluid dynamics: viscoelastic, generalized Newtonian, and compressible fluids, 2012, TU Darmstadt] by the first author under supervision of the second author. Here we present a slightly different method of deriving a concrete form of solutions to the model problem. In this paper, one of the essential points is to show the invertibility of a 2×2 Lopatinski matrix function. The corresponding system in [Three topics in fluid dynamics: viscoelastic, generalized Newtonian, and compressible fluids, 2012, TU Darmstadt] is a 3×3 matrix, so that the method presented here is slightly simpler.
Microstructure models for composites with imperfect interface via the periodic unfolding method
Authors: Ene Horia , Timofte Claudia
Published: November 2019

Using the periodic unfolding method, we shall analyze the asymptotic behavior of the solution of a system of coupled partial differential equations appearing in the modeling of an elasticity problem in a periodic structure formed by two interwoven and connected components with imperfect contact at the interface. Our setting proves to be relevant for analyzing contact problems for multiphase composites with imperfect interfaces.
Corrigendum
Authors: Capdeboscq Yves
Published: November 2019

Related article: http://iospress.metapress.com/content/elt156k8607uh516/
Diffusion through a composite structure with a high contrasting diffusivity
Authors: Sili Ali
Published: November 2019

We consider the stationary diffusion equation in a periodic composite medium made of two components Mε and Bε having very different diffusivity, the ratio between the coefficients of the diffusion in that structure being 1/αε2, where ε is the size of the period and αε which represents the amplitude of the diffusion in the inclusions Bε is a decreasing sequence towards zero. We show that the inclusions Bε work on the macroscopic diffusion as holes. In particular for scalings 0<αε�ε corresponding to very weak diffusion in Bε, the volume fraction of the material at the limit is the well known one in the case of holes.
A nonlinear general Neumann problem involving two critical exponents
Authors: Hadiji Rejeb , Yazidi Habib
Published: November 2019

We discuss the existence of solutions to the following nonlinear problem involving two critical Sobolev exponents

−div (p(x)∇u)=β|u|2*−2u+f(x,u)   in Ω,

u≢0   in Ω,

∂u/∂ν=Q(x)|u|2*−2u   on ∂Ω,

where β≥0, Q is continuous on ∂Ω, p∈H1(Ω) is continuous and positive in Ω¯ and f is a lower-order perturbation of |u|2*−1 with f(x,0)=0.
On a Φ-Kirchhoff multivalued problem with critical growth in an Orlicz–Sobolev space
Authors: Figueiredo Giovany M. , Santos Jefferson A.
Published: November 2019

This paper is concerned with the multiplicity of nontrivial solutions in an Orlicz–Sobolev space for a nonlocal problem with critical growth, involving N-functions and theory of locally Lipschitz continuous functionals. More precisely, in this paper, we study a result of multiplicity to the following multivalued elliptic problem:

−M(∫ΩΦ(|∇u|) dx)ΔΦu∈∂F(·,u)+αh(u)   in Ω,

u∈W01LΦ(Ω),

where Ω⊂RN is a bounded smooth domain, N≥3, M is a continuous function, Φ is an N-function, h is an odd increasing homeomorphism from R to R, α is positive parameter, ΔΦ is the corresponding Φ-Laplacian and ∂F(·,t) stands for Clarke generalized of a function F linked with critical growth. We use genus theory to obtain the main result.
Inverse scattering at high energies for the multidimensional Newton equation in a long range potential
Authors: Jollivet Alexandre
Published: November 2019

We define scattering data for the Newton equation in a potential V∈C2(Rn,R), n≥2, that decays at infinity like r−α for some α∈(0,1]. We provide estimates on the scattering solutions and scattering data and we prove, in particular, that the scattering data at high energies uniquely determine the short range part of the potential up to the knowledge of the long range tail of the potential. The Born approximation at fixed energy of the scattering data is also considered. We then change the definition of the scattering data and consider also inverse scattering in other asymptotic regimes. These results were obtained by developing the inverse scattering approach of Novikov [Ark. Mat. 37 (1999), 141–169].
Existence of solutions for a class of singular elliptic systems with convection term
Authors: Alves Claudianor O. , Moussaoui Abdelkrim
Published: November 2019

We show the existence of positive solutions for a class of singular elliptic systems with convection term. The approach combines sub- and supersolution method with the pseudomonotone operator theory and perturbation arguments involving singular terms.
A full dispersion asymptotic model to the water wave equations with surface tension
Authors: Obrecht Caroline
Published: November 2019

By a modulational analysis, we derive a Benney–Roskes type asymptotic model to the water wave equation with surface tension. In contrast to the classical Benney–Roskes system, this model preserves – at least for high frequencies – the dispersion relation of the full water wave equations. The formal derivation of the model is completed by some considerations on its consistency with the water wave equations.
Interface effects on the effective behavior of some Kelvin–Voigt viscoelastic heterogeneous bodies
Authors: Licht C. , Orankitjaroen S.
Published: November 2019

The effective behavior of a solid made from a periodic distribution of inclusions in a matrix is investigated. Inclusions and matrix are linearly elastic or viscoelastic of Kelvin–Voigt type (and possibly rigid for the inclusions) while the link between them can be pure adhesion or viscous friction with bilateral contact or involve a very thin viscoelastic layer. As long as one constituent is viscoelastic, the effective behavior is no longer of Kelvin–Voigt type but with memory.
The effective permeability of fractured porous media subject to the Beavers–Joseph contact law
Authors: Gruais Isabelle , Poliševski Dan , Stanescu Florentina-Alina
Published: November 2019

We are interested in the asymptotic behaviour of a fluid flow contained in a microscopic periodic distribution of fissures perturbating a porous medium where the Darcy law is valid, when the coupling between both systems is modeled by the Beavers–Joseph interface condition. As the small period of the distribution tends to zero, the interface condition is preserved on a microscopic scale under the additional assumption that the permeability coefficients behave like the squared period of the distribution which is also the squared size of the fissures. Moreover, the resulting pressure is purely macroscopic unlike the velocity field which also depends on the microscopic variable.
Resonance asymptotics for asymptotically hyperbolic manifolds with warped-product ends
Authors: Borthwick David , Philipp Pascal
Published: November 2019

We study the spectral theory of asymptotically hyperbolic manifolds with ends of warped-product type. Our main result is an upper bound on the resonance counting function with a geometric constant expressed in terms of the respective Weyl constants for the core of the manifold and the base manifold defining the ends.
On the metastable behavior of solutions to a class of parabolic systems
Authors: Strani Marta
Published: November 2019

In this paper we describe the metastable behavior of solutions to a class of parabolic systems. In particular, we improve some results contained in SIAM J. Math. Anal. 45(5) (2013), 3084–3113, by using different techniques to describe the slow motion of the internal layers. Numerical simulations illustrate the results.
Asymptotic behavior of the solutions of a degenerating elliptic equation in a thin multidomain
Authors: Prakash Ravi , Sili Ali
Published: November 2019

We study the behavior of the solutions uε of a degenerating elliptic equation in the framework of the dimension reduction problem coupled with an oscillating boundary. Under a suitable choice of the scalings on the source term and on the diffusivity coefficients, we prove that the limit problem is a nonlocal problem involving only the diffusion through the upper part of the domain.
Existence and localization of solutions for nonlocal fractional equations
Authors: Molica Bisci Giovanni , Repovš Dušan
Published: November 2019

This work is devoted to the study of the existence of at least one weak solution to nonlocal equations involving a general integro-differential operator of fractional type. As a special case, we derive an existence theorem for the fractional Laplacian, finding a nontrivial weak solution of the equation

(−Δ)su=h(x)f(u) in Ω,

u=0 in Rn\Ω, where h∈L∞+(Ω)\{0} and f :R→R is a suitable continuous function. These problems have a variational structure and we find a nontrivial weak solution for them by exploiting a recent local minimum result for smooth functionals defined on a reflexive Banach space. To make the nonlinear methods work, some careful analysis of the fractional spaces involved is necessary.
Upper semicontinuity of global attractors for damped wave equations
Authors: Wang Yonghai , Zhong Chengkui
Published: November 2019

In this paper, we mainly prove the upper semicontinuity of global attractors for a damped wave equations with perturbations.
Asymptotic behavior of solutions to a degenerate parabolic equation with a gradient source term
Authors: Zhou Shuangshuang , Zheng Sining
Published: November 2019

In this paper, we investigate positive solutions of the degenerate parabolic equation not in divergence form: ut=upΔu+uq|∇u|2−ur, subject to null Dirichlet boundary condition. We study the existence of global solutions and the large time behavior for them. The main effort is paid to obtain uniform asymptotic profiles for decay solutions, under various dominations of the nonlinear diffusion or absorption. It is shown that the large time property of the solution u behaves just like (1+(r−1)t)(−1/r−1) if the decay is governed by the nonlinear absorption with 1<r
p+1=q+2; and (iii) −ΔW=W1−p if r,q+2>p+1.
Semiclassical measure for the solution of the Helmholtz equation with an unbounded source
Authors: Royer Julien
Published: November 2019

We study the high frequency limit for the dissipative Helmholtz equation when the source term concentrates on a submanifold of Rn. We prove that the solution has a unique semi-classical measure, which is precisely described in terms of the classical properties of the problem. This result is already known when the micro-support of the source is bounded, we now consider the general case.
Resonance widths in a case of multidimensional phase space tunneling
Authors: Grigis Alain , Martinez André
Published: November 2019

Motivated by the study of resonances for molecular systems in the Born–Oppenheimer approximation, we consider a semiclassical 2×2 matrix Schrödinger operator of the form

P=−h2ΔI2+diag (xn−μ,τV2(x))+hR(x,hDx),

where μ and τ are two small positive constants, V2 is real-analytic and admits a nondegenerate minimum at 0, and R=(rj,k(x,hDx))1≤j,k≤2 is a symmetric off-diagonal 2×2 matrix of first-order differential operators with analytic coefficients. Then, denoting by e1 the first eigenvalue of −Δ+〈τV2″(0)x,x〉/2, and under some ellipticity condition on r1,2=r2,1*, we show that, for any μ sufficiently small, and for 0<τ≤τ(μ) with some τ(μ)>0, the unique resonance ρ of P such that ρ=τV2(0)+(e1+r2,2(0,0))h+𝒪(h2) (as h→0+) satisfies

Im ρ=−h3/2f(h,ln 1/h)e−2S/h,

where f(h,ln 1/h)~Σ0≤m≤ℓfℓ,mhℓ(ln 1/h)m is a symbol with f0,0>0, and S is the imaginary part of the complex action along some convenient closed path containing (0,0) and consisting of a union of complex nul-bicharacteristics of p1:=ξ2−xn−μ and p2:=ξ2+τV2(x) (broken instanton). This broken instanton is described in terms of the outgoing and incoming complex Lagrangian manifolds associated with p2 at the point (0,0), and their intersections with the characteristic set p1−1(0) of p1.
Nonnegative solutions for a class of singular parabolic problems involving p-Laplacian
Authors: de Bonis Ida , Giachetti Daniela
Published: November 2019

We deal with the existence of nonnegative solutions to parabolic problems which are singular in the u variable whose model is

ut−Δpu=f(x,t)(1/uθ+1) in Ω×(0,T),

u(x,t)=0 on ∂Ω×(0,T),

u(x,0)=u0(x) in Ω.

Here Ω is a bounded open subset of RN, N≥2, 0<T<+∞, θ>0, Δpu=div (|∇u|p−2∇u) with p>1.

As far as the data are concerned, we assume f(x,t)∈Lr(0,T;Lm(Ω)), with 1/r+N/pm<1, f(x,t)≥0 a.e. in Ω×(0,T) and u0(x)≥0 a.e. in Ω.

We consider also the case where the right-hand side depends on the gradient of the solution. In this last case the model of the right-hand side is F(x,t,u,∇u)=(f(x,t)+D|∇u|q)/uθ, with θ>0, D>0, 1<q<p and f(x,t) as before.
On the asymptotic spatial behaviour of the solutions of the nerve system
Authors: Leseduarte M.C. , Quintanilla R.
Published: November 2019

In this paper we investigate the asymptotic spatial behavior of the solutions for several models for the nerve fibers. First, our analysis deals with the coupling of two parabolic equations. We prove that, under suitable assumptions on the coefficients and the nonlinear function, the decay is similar to the one corresponding to the heat equation. A limit case of this system corresponds to the coupling of a parabolic equation with an ordinary differential equation. In this situation, we see that for suitable boundary conditions the solution ceases to exist for a finite value of the spatial variable. Next two sections correspond to the coupling of a hyperbolic/parabolic and hyperbolic/ordinary differential problems. For the first one we obtain that the decay is like an exponential of a second degree polynomial in the spatial variable. In the second one, we prove a similar behaviour to the one corresponding to the wave equation. In these two sections we use in a relevant way an exponentially weighted Poincaré inequality which has been revealed very useful in several thermal and mechanical problems. This kind of results have relevance to understand the propagation of perturbations for nerve models.
A note on the transition from diffusion with the flow to diffusion against the flow, for first passage times in singularly perturbed drift–diffusion models
Authors: Knessl Charles , Yao Haishen
Published: November 2019

We consider some singularly perturbed ODEs and PDEs that correspond to the mean first passage time T until a diffusion process exits a domain Ω in Rn. We analyze the limit of small diffusion relative to convection, and assume that in a part of Ω the convection field takes the process toward the exit boundary. In the remaining part the flow does not hit the exit boundary, instead taking the process toward a stable equilibrium point inside Ω. Thus Ω is divided into a part where the diffusion is with the flow and a complementary part where the diffusion is against the flow. We study such first passage problems asymptotically and, in particular, determine how T changes as we go between the two parts of the domain. We shall show that the mean first passage time may be exponentially large even in the part of Ω that is with the flow, and where a typical sample path of the process hits the exit boundary on much shorter time scales.
Some remarks on the radius of spatial analyticity for the Euler equations
Authors: Cappiello Marco , Nicola Fabio
Published: November 2019

We consider the Euler equations on Td with analytic data and prove lower bounds for the radius of spatial analyticity ε(t) of the solution using a new method based on inductive estimates in standard Sobolev spaces. Our results are consistent with similar previous results proved by Kukavica and Vicol, but give a more precise dependence of ε(t) on the radius of analyticity of the initial datum.
Persistence of regularity for the viscous Boussinesq equations with zero diffusivity
Authors: Hu Weiwei , Kukavica Igor , Ziane Mohammed
Published: November 2019

We address the global regularity for the 2D Boussinesq equations with positive viscosity and zero diffusivity. We prove that for data (u0,ρ0) in Hs×Hs−1, where 1<s<2, the persistence of regularity holds, i.e., the solution (u(t),ρ(t)) exists and belongs to Hs×Hs−1 for all positive t. Given the existing results, this provides the persistence of regularity for all s≥0. In addition, we address the Hs×Hs persistence and establish it for all s>1.
Asymptotic analysis of the perturbed Poisson–Boltzmann equation on unbounded domains
Authors: Ma Manjun , Ou Chunhua
Published: November 2019

We study the existence, uniqueness and asymptotic expansions to perturbed Poisson–Boltzmann equations on an unbounded domain in R2 or R3. First, a shooting method is applied to prove the existence and uniqueness of the exact solution. For the approximation to the regularly perturbed Poisson–Boltzmann equation, the solution via the classical method fails. We develop a novel approximate solution in terms of generalized asymptotic expansions. For the singularly perturbed problem, we show that a formula of asymptotic expansions with a boundary layer near the left end point provides a valid approximation. All our results are proved rigorously.
Boundedness of global solutions of a p-Laplacian evolution equation with a nonlinear gradient term
Authors: Attouchi Amal
Published: November 2019

We investigate the boundedness and large time behavior of solutions of the Cauchy–Dirichlet problem for the one-dimensional degenerate parabolic equation with gradient nonlinearity:

ut=(|ux|p−2ux)x+|ux|q in (0,∞)×(0,1),q>p>2.

We prove that: either ux blows up in finite time, or u is global and converges in W1,∞(0,1) to the unique steady state. This in particular eliminates the possibility of global solutions with unbounded gradient. For that purpose a Lyapunov functional is constructed by the approach of Zelenyak.
Effective equation for a system of mechanical oscillators in an acoustic field
Authors: Cacciapuoti Claudio , Figari Rodolfo , Posilicano Andrea
Published: November 2019

We consider a one-dimensional evolution problem modeling the dynamics of an acoustic field coupled with a set of mechanical oscillators. We analyze solutions of the system of ordinary and partial differential equations with time-dependent boundary conditions describing the evolution in the limit of a continuous distribution of oscillators.
Strong invariance for switching diffusions
Authors: Mei Hongwei , Yin George
Published: November 2019

This work is concerned with switching diffusions in which the switching depends on the diffusion process. It has been shown in the literature that if such a process is positive recurrent then it is ergodic. This paper further develops a law of iterated logarithms for the estimates on the aforementioned convergence rate. The law of iterated logarithms gives us a sharp bound on the estimation error sequence.
On the large argument asymptotics of the Lommel function via Stieltjes transforms
Authors: Nemes Gergő
Published: November 2019

The aim of this paper is to investigate the known large argument asymptotic series of the Lommel function by Stieltjes transform representations. We obtain a number of properties of this asymptotic expansion, including explicit and realistic error bounds, and an exponentially improved asymptotic expansion. An interesting consequence related to the large argument asymptotic series of the Struve function is also proved.
Singular perturbations for a problem of cardiac images tracking
Authors: Benmansour Khadidja , Pousin Jérôme
Published: November 2019

The result presented in this article is an adaptation of the gradient flow technique applied to a periodic problem combined with a singular perturbation method. The motivation for considering such a question originates from problems of cardiac images segmentation and of tracking cardiac images in medical image analysis. The ‘elastic deformable template’ model introduced previously for image segmentation is improved and adapted to the image tracking problem. A result of convergence is proved, for a periodic linear elastic model, the elastic tensor of which vanishes, when the parameter associated to the quasi-static technique goes to infinity.
Sharp interface limit for the Cahn–Larché system
Authors: Abels Helmut , Schaubeck Stefan
Published: November 2019

We prove rigorously the convergence of the Cahn–Larché system, which is a Cahn–Hilliard system coupled with the system of linearized elasticity, to a modified Hele–Shaw problem as long as a classical solution of the latter system exists. By matched asymptotic expansion we construct approximate solutions and estimate the difference between the true and the approximate solutions.
Homogenization of linear hyperbolic stochastic partial differential equation with rapidly oscillating coefficients: The two scale convergence method
Authors: Mohammed Mogtaba , Sango Mamadou
Published: November 2019

In this paper we establish new homogenization results for stochastic linear hyperbolic equations with periodically oscillating coefficients. We first use the multiple expansion method to drive the homogenized problem. Next we use the two scale convergence method and Prokhorov's and Skorokhod's probabilistic compactness results. We prove that the sequence of solutions of the original problem converges in suitable topologies to the solution of a homogenized stochastic hyperbolic problem with constant coefficients. We also prove a corrector result.
Analytic continuation of Borel sum of formal solution of semilinear partial differential equation
Authors: Yoshino Masafumi
Published: November 2019

We study the Borel summability and the analytic behavior of the Borel sum of a formal solution of first-order semilinear system with a singular perturbative parameter. By virtue of the representation formula of the Borel sum of a formal series solution expanded in terms of a parameter, we show that the analytic continuation of the Borel sum with respect to the parameter to a regular point in a singular direction coincides with the solution of the initial value problem expanded in the space variable. We also show that a similar phenomenon occurs outside the origin of the independent variables.
Asymptotic equivalence of the discrete variational functional and a rate-large-deviation-like functional in the Wasserstein gradient flow of the porous medium equation
Authors: Duong Manh Hong
Published: November 2019

In this paper, we study the Wasserstein gradient flow structure of the porous medium equation restricted to q-Gaussians. The JKO-formulation of the porous medium equation gives a variational functional Kh, which is the sum of the (scaled-) Wasserstein distance and the internal energy, for a time step h. We prove that, for the case of q-Gaussians on the real line, Kh is asymptotically equivalent, in the sense of Γ-convergence as h tends to zero, to a rate-large-deviation-like functional. The result explains why the Wasserstein metric as well as the combination of it with the internal energy play an important role.
Basis properties in a problem of a nonhomogeneous string with damping at the end
Authors: Gomilko Alexander , Rzepnicki Łukasz
Published: November 2019

This paper is concerned with the equation of a nonhomogeneous string of length one with one end fixed and the other one damped with a parameter h∈C. This problem can be rewritten as an abstract Cauchy problem for a densely defined closed operator iAh acting on an appropriate energy Hilbert space H. Under assumptions that the density function of the string ρ∈W21[0,1] is strictly positive and has ρ(1)≠h2 (if h∈R), we prove that the set of root vectors of Ah form a basis with parentheses in H. We show that with the additional condition

∫01ω12(ρ′,τ)/τ2 dτ<∞,

where ω1 is the integral modulus of continuity, the root vectors of the operator Ah form a Riesz basis in H.
Asymptotics of spectral gaps of 1D Dirac operator whose potential is a linear combination of two exponential terms
Authors: Anahtarci Berkay , Djakov Plamen
Published: November 2019

The one-dimensional Dirac operator [Formula: see text] considered on $[0,\uppi ]$ with periodic or antiperiodic boundary conditions, has discrete spectra. For large enough |n|, n∈Z, there are two (counted with multiplicity) eigenvalues λn−, λn+ (periodic if n is even, or antiperiodic if n is odd) such that |λn±−n|<1/2.

We study the asymptotics of spectral gaps γn=λn+−λn− in the case P(x)=ae−2ix+Ae2ix,  Q(x)=be−2ix+Be2ix, where a, A, b, B are any complex numbers. We show, for large enough m, that γ±2m=0 and [Formula: see text]
Homogenization and correctors of a class of elliptic problems in perforated domains
Authors: Chourabi Imen , Donato Patrizia
Published: November 2019

In this paper, we study the homogenization and the correctors for a class of linear elliptic problems in a periodically perforated domain when the oscillating matrix field also depend on a weakly converging sequence. We prescribe a Dirichlet condition on the exterior boundary and a nonhomogeneous nonlinear Robin condition on the boundary of the holes. Using the periodic unfolding method, we first derive the homogenized problem, then we study the convergence of the energy of the solution and the related corrector, which are the main results of this paper. As a particular case, we obtain a corrector result for the Laplacian with a linear nonhomogeneous Neumann condition on the hole, in the case of a nonzero Neumann data with a zero average. This remained an open problem since the corresponding homogenized results given in [Asymptotic Anal. 1 (1988), 115–138].
Anisotropic thermo-elasticity in 2D. Part I: A unified treatment
Authors: Reissig Michael , Wirth Jens
Published: November 2019

In this note we develop tools and techniques for the treatment of anisotropic thermo-elasticity in two space dimensions. We use a diagonalisation technique to obtain properties of the characteristic roots of the full symbol of the system in order to prove Lp–Lq decay rates for its solutions.
Multiple highly oscillatory shock waves
Authors: Peng Yahong , Wang Ya-Guang
Published: November 2019

This paper is devoted to the rigorous theory of nonlinear geometric optics for multiple shocks to a general N×N conservation law in one space variable. For the problem of multiple weak shocks perturbed by small amplitude, high-frequency oscillatory waves, we obtain that the leading profiles of oscillatory shocks are solution to an integro-differential system with free boundaries, the leading terms of shock fronts don't oscillate, and oscillations only appear in the leading terms shock speeds.
Finite elements scheme in enriched subspaces for singularly perturbed reaction–diffusion problems on a square domain
Authors: Jung Chang-Yeol
Published: November 2019

In this article, we discuss reaction-diffusion problems which produce ordinary boundary layers and elliptic corner layers. Using the classical polynomial Q1-finite elements spaces enriched with the so-called boundary layer elements which absorb the singularities due to the boundary and corner layers we are able to attain high numerical accuracies. We essentially obtain ε-uniform approximation errors in a weighted energy norm with significant simplifications in the numerical implementations; here we do not use mesh refinements.
Uniform decay rates for second-order hyperbolic equations with variable coefficients
Authors: Hamchi Ilhem
Published: November 2019

In this paper, we consider a second-order hyperbolic equations with variable coefficients and linear lower-order term. We use a new main multiplier and a suitable nonlinear version of a compactness uniqueness argument (Acta. Math. Sin. – Engl. Ser. 20(6) (2004), 1057–1072) to obtain the exponential stabilization of this system where no condition of smallness is needed.
On the long term spatial segregation for a competition–diffusion system
Authors: Squassina Marco
Published: November 2019

We investigate the long term behavior for a class of competition–diffusion systems of Lotka–Volterra type for two competing species in the case of low regularity assumptions on the data. Due to the coupling that we consider the system cannot be reduced to a single equation yielding uniform estimates with respect to the inter-specific competition rate parameter. Moreover, in the particular but meaningful case of initial data with disjoint support and Dirichlet boundary data which are time-independent, we prove that as the competition rate goes to infinity the solution converges, along with suitable sequences, to a spatially segregated state satisfying some variational inequalities.
On the stabilization of the elasticity system by the boundary
Authors: Khenissi Moez , Vodev Georgi
Published: November 2019

We obtain free of resonances regions for the elasticity system in the exterior of a strictly convex body in R3 with dissipative boundary conditions under some natural assumptions on the behaviour of the geodesics on the boundary. To do so, we use the properties of the parametrix of the Neumann operator constructed in Duke Math. J. 78 (1995), 677–714. As a consequence, we obtain time decay estimates for the local energy of the solutions of the corresponding mixed boundary value problems.
Asymptotic analysis of the scattering of a time-harmonic electromagnetic wave by a perfectly conducting metal coated with a thin dielectric shell
Authors: Bendali A. , Lemrabet K.
Published: November 2019

Our purpose is to investigate the effect of a thin dielectric coating on the scattering of a time-harmonic electromagnetic wave by a perfectly conducting metal. An asymptotic expansion of the solution of a model problem is performed relatively to the thickness of the dielectric shell. A suitable scaling as well as an adapted writing of the curl operator inside the thin shell make it possible to achieve an asymptotic expansion of the electromagnetic field at any order. Special stability estimates for Maxwell's equations relatively to the thickness of the dielectric shell then yield optimal error estimates outside the obstacle.
Homogenization of Dirichlet optimal control problems with exact partial controllability constraints
Authors: Kogut P.I. , Leugering G.
Published: November 2019

We consider an elliptic distributed quadratic optimal control problem with exact controllability constraints on a part of the domain which, in turn, is parametrized by a small parameter ε. The quadratic tracking type functional is defined on the remaining part of the domain. We thus consider a family of optimal control problems with state equality constraints. The purpose of this paper is to study the asymptotic limit of the optimal control problems as the parameter ε tends to zero. The analysis presented is in the spirit of the direct approach of the calculus of variations. This is achieved in the framework of relaxed problems. We finally apply the procedure to an optimal control problem on a perforated domain with holes of critical size. It is shown that a strange term in the terminology of Cioranescu and Murat (Prog. Nonlinear Diff. Eq. Appl., Vol. 31, Birkhäuser-Verlag, Boston, 1997, pp. 49–93) appears in the limiting homogenized problem.
Asymptotic behavior of threshold and sub-threshold solutions of a semilinear heat equation
Authors: Poláčik Peter , Quittner Pavol
Published: November 2019

We study asymptotic behavior of global positive solutions of the Cauchy problem for the semilinear parabolic equation ut=Δu+up in RN, where p>1+2/N, p(N-2)≤N+2. The initial data are of the form u(x, 0)=αϕ(x), where ϕ is a fixed function with suitable decay at |x|=∞ and α>0 is a parameter. There exists a threshold parameter α* such that the solution exists globally if and only if α≤α*. Our main results describe the asymptotic behavior of the solutions for α∈(0, α*] and in particular exhibit the difference between the behavior of sub-threshold solutions (α<α*) and the threshold solution (α=α*).
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).
Effective model for the cathode catalyst layer in fuel cells
Authors: Mihailovici Monika , Schweizer Ben
Published: November 2019

We study a pore scale model for the reactive layer of a fuel cell and derive effective equations. The cathode catalytic layer has a complex micro-structure to facilitate the transport of ion-gases and electrons. The reaction takes place at the surface of catalyst particles that are small compared to the pore space dimensions. The surface reaction law incorporates an exponential nonlinearity. The mathematical treatment is based on maximum principles and introduces a measure valued variant of compensated compactness.
Semiclassical scattering amplitude at the maximum of the potential
Authors: Alexandrova Ivana , Bony Jean-François , Ramond Thierry
Published: November 2019

We study the scattering amplitude for Schrödinger operators at a critical energy level, which is a unique non-degenerate maximum of the potential. We do not assume that the maximum point is non-resonant and use results by Bony, Fujiié, Ramond and Zerzeri to analyze the contributions of the trapped trajectories. We prove a semiclassical expansion of the scattering amplitude and compute its leading term. We show that it has different orders of magnitude in specific regions of phase space. We also prove upper and lower bounds for the resolvent in this setting.
Compact decoupling for thermoelasticity in irregular domains
Authors: Hassi E. Ait Ben , Bouslous H. , Maniar L.
Published: November 2019

In this paper, we prove the compactness of the difference between the thermoelasticity semigroup and its decoupled one on some irregular domains. This will be achieved by proving the norm continuity of this difference and the compactness of the difference between the resolvent of their generators and using Theorem 2.3 in Semigroup Forum 65 (2002), 58–70. An application to a thermoelastic system on a concrete irregular domain is given.
Darcy's law for a compressible thermofluid
Authors: Meirmanov Anvarbek
Published: November 2019

A linear system of differential equations describing a joint motion of a thermoelastic porous body with a sufficiently large Lamé's constants (absolutelty rigid body) and a thermofluid, occupying porous space, is considered. The rigorous justification, under various conditions imposed on physical parameters, is fulfilled for homogenization procedures as the dimensionless size of the pores tends to zero, while the porous body is geometrically periodic. As the results we derive Darcy's system of filtration or acoustic equations for thermofluid, depending on ratios between physical parameters. The proofs are based on Nguetseng's two-scale convergence method of homogenization in periodic structures.
Semiclassical analysis for Hartree equation
Authors: Carles Rémi , Masaki Satoshi
Published: November 2019

We justify WKB analysis for Hartree equation in space dimension at least three, in a régime which is supercritical as far as semiclassical analysis is concerned. The main technical remark is that the nonlinear Hartree term can be considered as a semilinear perturbation. This is in contrast with the case of the nonlinear Schrödinger equation with a local nonlinearity, where quasilinear analysis is needed to treat the nonlinearity.
Anisotropic thermo-elasticity in 2D. Part II: Applications
Authors: Wirth Jens
Published: November 2019

This note deals with concrete applications of the general treatment of anisotropic thermo-elasticity developed in the first part, Asympt. Anal. 57 (2008). We give dispersive decay rates for solutions to the type-1 system of thermo-elasticity for certain types of anisotropic media.
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.
Spectral asymptotic analysis in algebras of generalized functions
Authors: Delcroix Antoine , Marti Jean-André , Oberguggenberger Michael
Published: November 2019

We introduce a new type of local and microlocal asymptotic analysis in algebras of generalized functions, based on the presheaf properties of those algebras and on the properties of their elements with respect to a regularizing parameter. Contrary to the more classical frequential analysis based on the Fourier transform, we can describe a singular asymptotic spectrum which has good properties with respect to nonlinear operations. In this spirit we give several examples of propagation of singularities through nonlinear operators.
Dynamics of the nonclassical diffusion equations
Authors: Sun Chunyou , Yang Meihua
Published: November 2019

We consider the dynamical behavior of the nonclassical diffusion equation with critical nonlinearity for both autonomous and nonautonomous cases. For the autonomous case, we obtain the existence of a global attractor when the forcing term only belongs to H−1, this result simultaneously resolves a problem in Acta Mathematicae Applicatae Sinica 18 (2002), 273–276 related to the critical exponent. For the nonautonomous case, assumed that the time-dependent forcing term is translation bounded instead of translation compact, we first prove the asymptotic regularity of solutions, then the existence of a compact uniform attractor together with its structure and regularity has been obtained; finally, we show the existence of (nonautonomous) exponential attractors.
The Lévy–Fokker–Planck equation: Φ-entropies and convergence to equilibrium
Authors: Gentil Ivan , Imbert Cyril
Published: November 2019

In this paper, we study a Fokker–Planck equation of the form ut=ℐ[u]+div(xu), where the operator ℐ, which is usually the Laplacian, is replaced here with a general Lévy operator. We prove by the entropy production method the exponential decay in time of the solution to the steady state of the associated stationary equation.