bluuebunny/crossref_metadata_2025 huggingface dataset:
Boundary controllability of the one‐dimensional wave equation with rapidly oscillating density
Authors: Castro C.
Published: December 2024

We consider the one‐dimensional wave equation with periodic density $\rho$ of period $\varepsilon\to0$ in a bounded interval. By a counterexample due to Avellaneda, Bardos and Rauch we know that the exact controllability property does not hold uniformly as $\varepsilon\to0$ when the control acts on one of the extremes of the interval. The reason is that the eigenfunctions with wavelength of the order of $\varepsilon$ may have a singular behavior so that their total energy cannot be uniformly estimated by the energy observed on one of the extremes of the interval. We give partial controllability results for the projection of the solutions over the subspaces generated by the eigenfunctions with wavelength larger and shorter than $\varepsilon$ . Both results are sharp.

We use recent results on the asymptotic behavior of the spectrum with respect to the oscillation parameter $\varepsilon$ , the theory of nonharmonic Fourier series and the Hilbert uniqueness method (HUM).
Long time semiclassical approximation of quantum flows: A proof of the Ehrenfest time
Authors: Bambusi Dario , Graffi Sandro , Paul Thierry
Published: December 2024

Let $\mathcal{H}$ be a holomorphic Hamiltonian of quadratic growth on $\mathbb{R}^{2n}$ , $b$ a holomorphic exponentially localized observable, $H,B$ the corresponding operators on $L^2(\mathbb{R}^n)$ generated by Weyl quantization, and $U(t)=\exp{\mathrm{i}Ht/\hbar}$ . It is proved that the $L^2$ norm of the difference between the Heisenberg observable $B_t=U(t)BU(-t)$ and its semiclassical approximation of order $N-1$ is majorized by $K^N N^{(6n+1)N}\hbar^{-4/9}(-\hbar\log\hbar)^N$ for $t\in [0,T_n(\hbar)]$ , where $T_n(\hbar):=-2\log\hbar/[\alpha(6n+3)(N-1)]$ and $\alpha:=\Vert\mathrm{Hess}_{(x,\xi)}\,\mathcal{H}\Vert$ . Choosing a suitable $N(\hbar)$ the error is majorized by $C\hbar^{\log\vert\log\hbar\vert}$ , $0\leq t\leq \vert\log\hbar\vert/\log\vert\log\hbar\vert$ (here $K$ and $C$ are explicit constants independent of $N,\hbar$ ).
Estimates in probability of the residual between the random and the homogenized solutions of one‐dimensional second‐order operator
Authors: Bourgeat A. , Piatnitski A.
Published: December 2024

The work is giving estimations of the discrepancy between solutions of the initial and the homogenized problems for a one‐dimensional second‐order elliptic operators with random coefficients satisfying strong or uniform mixing conditions. We obtain several sharp estimates in terms of the corresponding mixing coefficient.
Large time behavior in $\mathbb{R}^N$ for linear parabolic equations with periodic coefficients
Authors: Ortega Jaime H. , Zuazua Enrique
Published: December 2024

In this paper we study the asymptotic behavior of solutions of linear parabolic equations in $\mathbb{R}^N$ with periodic coefficients and $L^1$ initial data, as $t\rightarrow \infty$ . It was already known that, in a first approximation, solutions behave as the fundamental solution of the homogenized system. We use the Bloch waves decomposition to obtain a complete expansion of the solutions as $t\rightarrow \infty$ .
Solution of irregular problems by the asymptotic method
Authors: Yakubov Sasun
Published: December 2024

We investigate the Regge problem in Sobolev spaces with a weight. Moreover, conditions of proved theorems are weaker than those in the author’s paper published in the Journal of Mathematical Analysis and Applications 220 (1998), pp. 224–249. We prove an isomorphism, coerciveness with a defect with respect to the spectral parameter and 2‐fold completeness of root functions of the Regge problem. By the suggested method one can investigate both irregular and regular problems.
Two‐scale convergence for thin domains and its applications to some lower‐dimensional models in fluid mechanics
Authors: Marušić Sanja , Marušić‐Paloka Eduard
Published: December 2024

Inspired by the similar ideas from the homogenization theory, in this paper we introduce the notion of two‐scale convergence for thin domains that allow lower‐dimensional approximations. We prove the compactness theorem, analogous to the one in homogenization theory. Using those results we derive the lower‐dimensional models for potential flow in thin (possibly degenerated) pipe, the degenerated Reynold’s equation for viscous flow in degenerated thin domain and the 1‐dimensional approximation for the non‐Newtonian (power‐law) flow in a thin pipe.
Attractors for a Bénard type problem with constraints
Authors: Lezaun Mikel
Published: December 2024

In this paper, we deal with the dynamic of a model for the mass and heat transfer by a gaseous phase with an instantaneous chemical reaction, which is modeled by means of a subdifferential. To solve this problem, we use a penalization method and we prove that a maximal attractor describes the long‐time behaviour of the weak solutions. Furthermore, we prove the upper semicontinuity in $L^{2}$ of the family of attractors related to the penalization approach of this problem.
Wave equation in domains with many small obstacles
Authors: Toader Rodica
Published: December 2024

We study the behaviour of solutions of relaxed wave equations with Dirichlet boundary conditions corresponding to a γ‐convergent sequence of measures. The model case is that of a sequence of domains with many small obstacles. Convergence results are proved for the solutions on finite time intervals.
Robustness of boundary feedback controls for a flexible beam with respect to perturbation
Authors: Liu Wei‐Jiu
Published: December 2024

In this paper we test the robustness of a nonlinear boundary feedback control for a flexible beam with respect to perturbation. We show that additional dynamics of perturbation at one of the ends of the beam, as long as they are strictly passive, will not destabilize the controlled beam.
Focusing on a line for nonlinear Schrödinger equations in $\mathbb{R}^{2}$
Authors: Carles Rémi
Published: December 2024

This paper is devoted to the study of a semi‐classical NLS equation with a small parameter ε in two space dimensions, with oscillating data that are highly oscillating in one direction only, with the aim of modelling geometric optics with a caustic consisting of a line in $\mathbb{R}^{2}$ . We prove that the phenomena encountered are typically one‐dimensional. In order to describe the results, we introduce spaces that make it possible to define scattering operators with a parameter.
Null controllability of linear and semilinear heat equations in thin domains
Authors: Teresa Luz de , Zuazua Enrique
Published: December 2024

We consider the linear heat equation with potential in a n‐dimensional thin cilinder Ωε=Ω×(0,ε) where Ω is a bounded open smooth set of $\mathbb{R}^{n-1}$ with n≥2 and ε is a small parameter. We study the null controllability problem when the control acts in a cylindrical region ωε=ω×(0,ε), where ω⊂Ω is an open and non‐empty subset of Ω. We prove that, under appropriate boundary conditions, for a suitable class of potentials the heat equation is uniformly null controllable as ε→0. We also prove the convergence of the controls to a null control for the n−1‐dimensional heat equation in Ω. Similar results are proved for the semilinear heat equation with globally Lipschitz nonlinearities.
Asymptotic analysis of some control problems
Authors: Arada Nadir , El Fekih Henda , Raymond Jean‐Pierre
Published: December 2024

We study Dirichlet boundary control problem for semilinear parabolic equations, with pointwise state constraints. By penalizing the Dirichlet boundary condition, we define a family of control problems with Robin boundary conditions. We study the asymptotic behavior of solutions of these penalized problems, when the penalty parameter tends to infinity. We also prove that optimality conditions for the Dirichlet boundary control problem can be obtained by passage to the limit in the optimality conditions of the penalized problem.
Semi‐classical analysis for the eigenvalues of a Schrödinger operator with magnetic field
Authors: Aramaki Junichi
Published: December 2024

We discuss semi‐classical analysis for the eigenvalues of the Schrödinger operator on ${\mathbb{R}}$ d with magnetic and electoric potentials. We consider the case where the electric potential vanishes precisely of even order at the wells. Our final aim is to clarify that when the operator has a non‐degenerate eigenvalue and then to get the complete asymptotics of the eigenvalue in the semi‐classical sense. Moreover, we give an example which seems to be new.
Around the best constants for the Sobolev trace map from W1,1(Ω) into L1(∂Ω)
Authors: Motron Mélissa
Published: December 2024

In this article, we prove that the best constant for the Sobolev trace map from W1,1( $\mathbb{R}$ N−1× $\mathbb{R}$ +) into L1( $\mathbb{R} $ N−1×{0}) is 1. We deduce from it that when Ω is a bounded open set whose boundary is piecewise C1, the best first constant for the Sobolev trace map γ from W1,1(Ω) into L1(∂Ω) is also 1, in the sense that: for any ε>0, there exists Bε>0 such that for any u in W1,1(Ω), ∫∂Ω|u|≤(1+ε)∫Ω|∇u|+Bε∫Ω|u|.

Independently, we prove that the best second constant for Sobolev trace map from W1,1(Ω) into L1(∂Ω) is |∂Ω|/|{Ω}| (where |∂Ω| denotes the (N−1)‐dimensional measure of ∂Ω and |{Ω}| the N‐dimensional measure of Ω) when Ω is a connected open bounded set; in other words, there exists A>0 such that for any u in W1,1(Ω): ∫∂Ω|u|≤A∫Ω|∇u|+ $\dfrac{|\partial{\Omega}|}{|{\Omega}|}$ ∫Ω|u|.
Competition between the surface and the boundary layer energies in a Ginzburg–Landau model of a liquid crystal composite
Authors: Berlyand Leonid , Khruslov Evgen
Published: December 2024

We consider a nonlinear homogenization problem for a Ginzburg–Landau 3D model with a surface energy term, in a liquid crystalline medium with inclusions. We show that the presence of the inclusions can be accounted for by an effective potential that can be viewed as an effective external field. Our main objective is to compute the contribution of the surface and bulk energies into this potential.

We introduce a small parameter ε such that the average distances between the inclusions are of the order of ε, the inclusions sizes are of the order of εα, α>1, and the coefficient in front of the surface energy term, is of the order of εβ.

We found that the parametric half‐plane {(α,β): α>1, −∞<β+∞} is partitioned into two parts by a polygonal line, consisting of two linear parts. We show that, on the first part, the surface energy dominates and, on the second part, the boundary layer energy takes over. We focus our attention on the junction (critical or transitional) point, where both the bulk energy of thin layers around the inclusions and the surface energy provide finite contributions into the effective potential. We present explicit formulas for computing the effective potential on the polygonal line and at the critical point in terms of specific surface energy and specific boundary layer energy. We also show that in the domain to the right of the polygonal line, the potential is zero and discuss the homogenized limit in the remaining part of the half plane.

Our proof is based on the quasisolutions method, which incorporates some classical ideas of Stokes in hydrodynamics, as well as variational energy techniques previously developed in the study of linear homogenization problems.
Mathematical foundations of the time reversal mirror
Authors: Bardos Claude , Fink Mathias
Published: December 2024

In the present paper a mathematical analysis of the “time reversal mirror” (cf. [4,9,10]) is given. As a first step of a more detailed program, the emphasis is put on phenomena which are described by the genuine acoustic equation with Dirichlet or “impedance” boundary conditions. An ideal situation is first considered then relation between the question of the local decay of energy and the accuracy of the method are exploited. The positive effect of the ergodicity is explained and eventually comparison with control theory approach is considered.
Dirichlet boundary stabilization of the wave equation
Authors: Ammari Kais
Published: December 2024

We consider the stabilization problem for a wave equation. In the case when the geometric control assumption, see [3], is not satisfied, we prove that the energy of system decay with a logarithmic rate for all initial data in the domain of the infinitesimal generator of evolution equation. The technique used consists in deducing the decay estimate from an observability inequality for the associated undamped problem, via sharp regularity results.
Stability of solutions to the Burgers equation
Authors: Aassila Mohammed
Published: December 2024

In this paper we study the boundary stabilization of the Burgers equation. We prove that the closed‐loop system is globally H1‐stable and H3‐stable and well‐posed. Furthermore, we show that the delayed Burgers' equation is exponentially stable if the delay parameter is sufficiently small. We also give an explicit estimate of the delay parameter in term of the viscosity and the initial condition.
Micropolar effects in the coupling of a thin film past a porous medium
Authors: Bayada G. , Chambat M. , Gamouana S.R.
Published: December 2024

We study the coupling of a porous medium and an adjacent thin film flow, taking account of non‐Newtonian effects induced by additives in the fluid. The starting point is the micropolar Stokes system. A critical relation between η, the thin gap height size and ε, the characteristic size of the pores in the porous medium is exhibited. Asymptotic equations are obtained including the continuity of the pressure at the interface. The influence of the rheological micropolar parameters is discussed and various asymptotic behaviours are identified: the micropolar effects are found to be significant in both media, or only in the porous medium, or can be neglected everywhere.
Global existence and global nonexistence of solutions to a wave equation with nonlinear damping and source terms
Authors: Aassila Mohammed
Published: December 2024

In this paper we study the global existence and nonexistence problems for the Klein–Gordon equation with nonlinear damping and nonlinear perturbation.
Asymptotic behaviour for a model of flexible cable with tip masses
Authors: Conrad Francis , O'Dowd Geoffrey , Saouri Fatima‐Zahra
Published: December 2024

We consider a model of an overhead crane where the boundary feedback takes into account the velocity, but not the position. In this noncoercive case, we prove that asymptotically the system converges to a rest position which depends on the initial data. The case of linear and (more sophisticated) nonlinear feedbacks is presented.
Extinction and non‐extinction for viscous Hamilton–Jacobi equations in $\mathbb{R}$N
Authors: Benachour S. , Laurençot Ph. , Schmitt D. , Souplet Ph.
Published: December 2024

Extinction in finite time and non‐compactness of the support are investigated for non‐negative classical solutions to the Cauchy problem ut−Δu+|∇u|p=0 when p∈(0,1). The occurrence of these phenomena is shown to depend on the behaviour of the initial data u0 for large values of x. In particular, we obtain the optimal decay rate of u0 at infinity to ensure finite‐time extinction of u. The case of periodic boundary conditions is also considered. Finally the optimality of temporal L∞‐decay estimates obtained previously is discussed.
Asymptotic expansion of small solutions to the Landau–Ginzburg type equations
Authors: Hayashi Nakao , Kaikina Elena I. , Naumkin Pavel I.
Published: December 2024

We obtain the large time asymptotic expansion of small solutions to the Landau–Ginzburg type equations [Formula: see text] where α,β∈C,Rα>0, R(β/ $\sqrt{2|\alpha |^{2}+\alpha ^{2}}$ )≥0, under the condition that the initial data ϕ are sufficiently small in a suitable weighted norm and the mean value ∫ϕ(x) dx≠0.
A penalized Robin approach for solving a parabolic equation with nonsmooth Dirichlet boundary conditions
Authors: Ben Belgacem F. , El Fekih H. , Raymond J.P.
Published: December 2024

We study the approximation of a parabolic equation with a nonsmooth Dirichlet boundary condition by equations with Robin boundary conditions. We give the convergence rate, in some suitable anisotropic Sobolev spaces, of the penalized solution towards the Dirichlet solution when the penalty parameter ε>0 becomes small. Some numerical tests, presented at the end of the paper, illustrate the theoretical estimates.
Singular perturbations in nonlinear boundary value problems
Authors: Shi Yufeng
Published: December 2024

A class of nonlinear two‐point boundary value problems with singular perturbations are studied. We show asymptotic behavior of the solutions of the perturbed systems and derive asymptotic expansions for the solutions of the perturbed systems, which can approximate the solutions of the perturbed systems as accurately as desired.
On a model boundary value problem for Laplacian with frequently alternating type of boundary condition
Authors: Borisov Denis I.
Published: December 2024

Model two‐dimensional singular perturbed eigenvalue problem for Laplacian with frequently alternating type of boundary condition is considered. Complete two‐parametrical asymptotics for the eigenelements are constructed.
On a nonlinear elliptic system from Maxwell–Chern–Simons vortex theory
Authors: Ricciardi Tonia
Published: December 2024

We define an abstract nonlinear elliptic system, admitting a variational structure, and including the vortex equations for some Maxwell–Chern–Simons gauge theories as special cases. We analyze the asymptotic behavior of its solutions, and we provide a general simplified framework for the asymptotics previously derived in those special cases. As a byproduct of our abstract formulation, we also find some new qualitative properties of solutions.
A sharp asymptotic for the lengths of certain scattering rays in the exterior of two convex domains
Authors: Stoyanov Luchezar
Published: December 2024

Billiard trajectories in the exterior of two strictly convex domains in the plane are considered incoming from a given source S and arriving at a given target P. An asymptotic with an exponentially small error term for the sequence of travelling times of these trajectories is obtained involving the distance between the two domains and the curvatures at the ends of the shortest segment connecting the domains.
Existence of resonances for matrix Schrödinger operators
Authors: Nédélec L.
Published: December 2024

We study the resonances of matrix Schrödinger operators, in the semi‐classical limit, motivated by the Born–Oppenheimer approximation. Precisely, we are interested in a lower bound on the number of resonances near a point. We obtain the lower bound Ch−1|ln h|−3/2 for n=3.
Disappearance of interfaces for the porous medium equation with variable density and absorption
Authors: Reyes Guillermo , Sánchez Ariel
Published: December 2024

We improve previous results on the qualitative properties of solutions to the Cauchy problem for the equation ρ(x)ut=(um)xx−c0up, where 1<m
0 with nonnegative, compactly supported initial data. Precisely, we prove that if the density ρ(x) decays faster than |x|−k with k>k*=2(p−1)/(p−m), then the interfaces disappear in finite time for any initial data in the above class. We also weaken conditions previously imposed on the density function.
Asymptotic analysis of linearly viscoelastic shells
Authors: Fushan Li
Published: December 2024

By applying formal asymptotic analysis and Laplace transformation, we obtain two‐dimensional model system of linearly viscoelastic “membrane” shell from three‐dimensional equations. Then we prove that the scaled displacement of three‐dimensional linearly viscoelastic shell converges to one of the model problem.
Modeling of a thin film passing a thin porous medium
Authors: Bayada G. , Benhaboucha N. , Chambat M.
Published: December 2024

We have studied the asymptotic behaviour of a coupling between a thin film of fluid and an adjacent thin porous medium. This situation appears in boundary lubrication problems. We show that there is a critical value between the size of the microstructure of the porous medium, the free fluid gap and the thickness of the porous medium. Moreover it is shown that an actual geometry can always be described by that critical value for which a modified Reynolds equation is proved. Numerical calculations show the difference between our model and two other models proposed in mechanical literature.

Résumé. On étudie le comportement asymptotique d'un écoulement fluide constité d'une couche poreuse mince adjacente à un milieu fluide mince. Cette géométrie est caractéristique d'un écoulement lubrifié “limite”. On met en évidence l'existence d'un rapport critique entre la taille de la microstructure du milieu poreux et les deux épaisseurs, rapport pour lequel une équation de Reynolds modifiée est obtenue. De plus il est montré qu'on peut toujours pour une géométrie réelle donnée se placer dans ce cas critique. Enfin, on présente des simulations numériques qui mettent en évidence les différences entre le modèle présenté ici et deux autres utilisés en mécanique.
Sharp conditions for the classical asymptotic behaviour near a point for solutions to quasilinear elliptic systems
Authors: Kozlov Vladimir , Maz'ya Vladimir
Published: December 2024

It is well known that distributional solutions of an elliptic equation with constant coefficients behave asymptotically near an interior point as sums of polynomials and linear combinations of derivatives of a fundamental solution. We consider a class of quasilinear elliptic systems and give mild conditions ensuring the same asymptotic behaviour. The sharpness of our conditions is illustrated by examples. The results are obtained as corollaries of a general theorem on the asymptotics of solutions to nonlinear ordinary differential equations in Banach spaces.
Asymptotic solution to the Dirichlet problem for a two‐dimensional Riccatti's type equation near a corner point
Authors: Maz'ya V. , Slutskiǐ A.S.
Published: December 2024

The Dirichlet problem for a quasilinear elliptic equation of the second order with quadratic nonlinearity in the first derivatives is considered in a plane domain with a corner point. An asymptotic solution which has a strong singularity at this point is constructed.
Locally periodic homogenization
Authors: Benchérif‐Madani A. , Pardoux É.
Published: December 2024

In this paper, two linear second PDEs are homogenized. The coefficients are supposed to be locally periodic, Lipschitz and bounded. Compared to our previous work [1], we provide a new and simpler proof and weaken the hypotheses of the main theorem. We use both probabilistic and analytic arguments.
Geometric tools of the adiabatic complex WKB method
Authors: Fedotov Alexander , Klopp Frédéric
Published: December 2024

The paper is devoted to the description of the main geometric and analytic tools of a complex WKB method for adiabatic problems. We illustrate their use by numerous examples.

Résumé. L'article est consacré à la description des principaux outils géométriques et analytiques d'une méthode WKB complexe pour des problèmes adiabatiques. Nous illustrons leur utilisation par de nombreux exemples.
Asymptotic formula for eigenvalues of simple pendulum problems
Authors: Shibata Tetsutaro
Published: December 2024

We consider the nonlinear eigenvalue problem −Δu=λf(u), u>0 in BR, u=0 on ∂BR, where BR is a ball with radius R>0 and λ>0 is a parameter. Under the appropriate conditions of f, it is known that for a given 0<ε<1, there exists (λ,u)=(λ(ε),uε) satisfying the equation with ∫BRF(uε(x)) dx=|BR|F(u0)(1−ε), where F(u)=∫0uf(s) ds and u0>0 is the smallest zero of f in R+. Furthermore, uε(x)→u0 (x∈BR) and λ(ε)→∞ as ε→0. This concept of parametrization of solution pair by a new parameter ε is based on the variational structure of the equation. We establish the asymptotic formulas for λ(ε) as ε→0 with the ‘optimal’ estimate of the second term.
Error estimate and unfolding for periodic homogenization
Authors: Griso Georges
Published: December 2024

This paper deals with the error estimate in problems of periodic homogenization. The methods used are those of the periodic unfolding. We give the upper bound of the distance between the unfolded gradient of a function belonging to H1(Ω) and the space ∇xH1(Ω)⌖∇yL2(Ω;H1per(Y)). These distances are obtained thanks to a technical result presented in Theorem 2.3: the periodic defect of a harmonic function belonging to H1(Y) is written with the help of the norms H1/2 of its traces differences on the opposite faces of the cell Y. The error estimate is obtained without any supplementary hypothesis of regularity on correctors.
Exact controllability for wave equations with an equivalued boundary on a shrinking “hole”
Authors: Du Zhuoran , Yan Jinhai
Published: December 2024

In this paper we obtain the exact controllability for wave equations with an equivalued boundary on a “hole” in a bounded domain, and prove that when the “hole” shrinks to a point, the HUM solutions converge in a suitable sense.
Bloch approximation in homogenization on bounded domains
Authors: Conca C. , Orive R. , Vanninathan M.
Published: December 2024

The classical problem of homogenization deals with elliptic operators in periodically oscillating media of small period ε>0 and the asymptotic behavior of solution uε of boundary value problems associated with such operators. In a previous work [Conca et al., SIAM J. Math. Anal. 33 (2002), 1166–1198], the authors introduced what is called Bloch approximation which provided energy norm approximation for the solution in $\mathbb{R}^{N}$ . This paper continues with the above development and proposes a modified Bloch approximation. This function takes into account boundary effects. Its connection with the first order classical correctors is also established with the corresponding error estimate. All the proofs are worked out entirely in the Fourier–Bloch space.
Asymptotic properties of the differential equation h3(h″+h′)=1
Authors: Asch J. , Benguria R.D. , Št'ovíček P.
Published: December 2024

We derive the complete asymptotic series, as t→+∞, for a general solution h(t) of the nonlinear differential equation h3(h″+h′)=1. The equation originates from a physical model related to the Hall effect.
An inverse boundary value problem for harmonic differential forms
Authors: Joshi M.S. , Lionheart W.R.B.
Published: December 2024

We show that the full symbol of the Dirichlet to Neumann map of the k‐form Laplace's equation on a Riemannian manifold (of dimension greater than 2) with boundary determines the full Taylor series, at the boundary, of the metric. This extends the result of Lee and Uhlmann for the case k=0. The proof avoids the computation of the full symbol by using the calculus of pseudo‐differential operators parametrized by a boundary normal coordinate and recursively calculating the principal symbol of the difference of boundary operators.
The negative discrete spectrum of a class of two‐dimensional Schrödinger operators with magnetic fields
Authors: Laptev A. , Safronov O.
Published: December 2024

We obtain an asymptotic formula for the number of negative eigenvalues of a class of two‐dimensional Schrödinger operators with small magnetic fields. This number increases as a coupling constant of the magnetic field tends to zero.
Parametrix for a hyperbolic initial value problem with dissipation in some region
Authors: Stolk Christiaan C.
Published: December 2024

We construct parametrices for initial value problems of the form [Formula: see text] where $(z,x)\in \mathbb{R} \times \mathbb{R} ^{n}$ , A(z,x,Dx) is a family of order 1 pseudodifferential operators with homogeneous real principal symbol a(z,x,ξ), and B(z,x,Dx) is a family of order γ>0 pseudodifferential operators with non‐negative homogeneous real principal symbol b(z,x,ξ). The parametrix is a family of pseudodifferential operators when A=0, and a Fourier integral operator with real phase function if A≠0. A priori this leads to symbols of type $(\rho,\delta)=(1-\frac{\gamma}{2},\frac{\gamma}{2})$ , which limits our construction to γ<1, and leads to operators with a complicated symbol calculus in the case γ=1. With an additional assumption on B we obtain symbols of type $(\rho,\delta)=(1-\frac{\gamma}{L},\frac{\gamma}{L})$ , for some L≥2. The assumption implies in particular that the first L−1 derivatives of b vanish where b=0. Parametrices for (*) are constructed for the case when 2γ<L.
Asymptotic self‐similarity for a simplified model for radiating gases
Authors: Laurençot Philippe
Published: December 2024

Convergence of integrable and bounded entropy solutions to a simplified model for radiating gases towards the source‐type solution to the viscous Burgers equation is shown by a rescaling method. Temporal decay estimates for zero‐mass solutions are also established.
Quantum resonances without analyticity
Authors: Cancelier Claudy , Martinez André , Ramond Thierry
Published: December 2024

We propose a definition for the resonances of Schrödinger operators with slowly decaying 𝒞∞ potentials without any analyticity assumption. Our definition is based on almost analytic extensions for these potentials, and we describe a systematic way to build such an extension that coincide with the function itself whenever it is analytic. That way, if the potential is dilation analytic, our resonances are the usual ones. We show that our resonances with negative real part are exactly the eigenvalues of the operator. We also prove that our definition coincides with the usual ones in the case of smooth exponentially decaying potentials.

Then we consider semiclassical results. We show that, if the trapped set for some energy E is empty, there is no resonance in any complex vicinity of E of size O(hlog (1/h)). Finally, we investigate the semiclassical shape resonances and generalize some results of Helffer and Sjöstrand.
Neumann problem in a perforated layer (sieve)
Authors: Nazarov Sergueï A. , Thäter Gudrun
Published: December 2024

The objective of our paper is to construct the asymptotic representation of solutions to our problem at infinity. In doing so the main difficulty is the justification of the formal procedure presented at the beginning. Then, based on the asymptotics, we calculate integral representations for the involved coefficients and prove the Fredholm property of our problem in step-weighted spaces providing at the same moment kernel and cokernel.
Schrödinger operator with a junction of two 1-dimensional periodic potentials
Authors: Korotyaev Evgeny
Published: December 2024

The spectral properties of the Schrödinger operator Tty=−y″+qty in $\[$L^{2}(\mathbb{R})$$ are studied, with a potential qt(x)=p1(x), x<0, and qt(x)=p(x+t), x>0, where p1,p are periodic potentials and $\[$t\in \mathbb{R}$$ is a parameter of dislocation. Under some conditions there exist simultaneously gaps in the continuous spectrum of T0 and eigenvalues in these gaps. The main goal of this paper is to study the discrete spectrum and the resonances of Tt. The following results are obtained: (i) In any gap of Tt there exist 0,1 or 2 eigenvalues. Potentials with 0, 1 or 2 eigenvalues in the gap are constructed. (ii) The dislocation, i.e., the case p1=p is studied. If t→0, then in any gap in the spectrum there exist both eigenvalues (≤2) and resonances (≤2) of Tt which belong to a gap on the second sheet and their asymptotics as t→0 are determined. (iii) The eigenvalues of the half-solid, i.e., p1=constant, are also studied. (iv) We prove that for any even 1-periodic potential p and any sequences {dn}1∞, where dn=1 or dn=0 there exists a unique even 1-periodic potential p1 with the same gaps and dn eigenvalues of T0 in the n-th gap for each n≥1.
Nonlinear problems with boundary blow-up: a Karamata regular variation theory approach
Authors: Cîrstea Florica Corina , Rădulescu Vicenţiu
Published: December 2024

We study the uniqueness and expansion properties of the positive solution of the logistic equation Δu+au=b(x)f(u) in a smooth bounded domain Ω, subject to the singular boundary condition u=+∞ on $\curpartial \varOmega $ . The absorption term f is a positive function satisfying the Keller–Osserman condition and such that the mapping f(u)/u is increasing on (0,+∞). We assume that b is non-negative, while the values of the real parameter a are related to an appropriate semilinear eigenvalue problem. Our analysis is based on the Karamata regular variation theory.
Distributional asymptotics and p-adic Tauberian and Shannon–Kotelnikov theorems
Authors: Khrennikov A.Yu. , Shelkovich V.M.
Published: December 2024

The notion of the quasi-asymptotics adapted to the case of p-adic distributions (generalized functions) is introduced. p-Adic analogs of Tauberian theorems for distributions are proved. We show that some properties of Vladimirov's pseudo-differential operator Dα are connected with a prototype of the Tauberian type theorem (with respect to distributional asymptotic). We also prove the p-adic version of the Shannon–Kotelnikov theorem.
Convergence of formal solutions of some nonlinear differential systems at an irregular singularity
Authors: Rezaoui Med-Salem
Published: December 2024

Sibuya was concerned in [Lecture Notes in Math., Vol. 810, 1980, pp. 135–140] with studying analyticity of formal solutions for nonlinear differential systems xp+1 $\[$\frac{\mathrm{d}u}{\mathrm{d}x}$$ =E(x,y,u), where E is an analytic function in its arguments, near the origin. We are aiming in this work, to generalize the previous Sibuya's paper to nonlinear differential systems A(x) $\[$\frac{\mathrm{d}u}{\mathrm{d}x}$$ =E(x,y,u), where A(x) is an N×N matrix, whose entries are holomorphic near the origin.

Sibuya s'est intéressé dans [Lecture Notes in Math., Vol. 810, 1980, pp. 135–140] à l'analyticité des solutions formelles des systèmes différentiels non linéaires xp+1 $\[$\frac{\mathrm{d}u}{\mathrm{d}x}$$ =E(x,y,u), où E est une fonction analytique en ses arguments, au voisinage de l'origine. Nous nous intéressons dans ce travail, à généraliser le résultat de Sibuya aux systèmes différentiels non linéaires A(x) $\[$\frac{\mathrm{d}u}{\mathrm{d}x}$$ =E(x,y,u), où A(x) est une matrice carrée N×N, à coefficients holomorphes au voisinage de l'origine.
On the eigenvalues for slowly varying perturbations of a periodic Schrödinger operator
Authors: Marx Magali
Published: December 2024

In this paper, I consider one-dimensional periodic Schrödinger operators perturbed by a slowly decaying potential. In the adiabatic limit, I give an asymptotic expansion of the eigenvalues in the gaps of the periodic operator. When one slides the perturbation along the periodic potential, these eigenvalues oscillate. I compute the exponentially small amplitude of the oscillations.
The persistence of canards in 3-D singularly perturbed systems with two fast variables
Authors: Xie Feng , Han Maoan , Zhang Weijiang
Published: December 2024

In this paper, a class of three-dimensional singularly perturbed systems of ordinary differential equations with two fast variables are considered. Sufficient conditions for the existence of canards are obtained by asymptotic methods.
Asymptotic behavior for solutions of parabolic equations with natural growth terms and irregular data
Authors: Leonori Tommaso , Petitta Francesco
Published: December 2024

In this paper we deal with the asymptotic behavior of solution for quasilinear parabolic equations with absorbing lower-order terms whose model is [Formula: see text] with L1 data that do not depend on time. The main result states that, under suitable assumptions, the solution of the parabolic problem converges to the stationary solution.
The onset problem for a thin superconducting loop in a large magnetic field
Authors: Shieh Tien-Tsan , Sternberg Peter
Published: December 2024

We present a rigorous analysis of the eigenvalue problem associated with the onset of superconductivity for a thin domain in the presence of a large applied magnetic field. We prove the validity of the formal result of Richardson and Rubinstein (Proc. Roy. Soc. London A 455 (1999), 2549–2564) revealing that in this double limit of thin domain and large field, the appropriate Rayleigh quotient differs from the standard one for order 1 applied fields through the addition of a potential depending on the field.
Asymptotic analysis of linearly elastodynamic shallow shells with variable thickness
Authors: Guan Yan
Published: December 2024

The paper deals with the dynamic problem for linearly elastic shallow shells with variable thickness. The author shows that the solution of three-dimensional problem converges to the solution of two-dimensional shallow shell equations as the thickness of the shell goes to zero.
Structure of the short range amplitude for general scattering relations
Authors: Alexandrova Ivana
Published: December 2024

We consider scattering by short range perturbations of the semi-classical Laplacian. We prove that when a polynomial bound on the resolvent holds the scattering amplitude is a semi-classical Fourier integral operator associated to the scattering relation near a non-trapped ray. Compared to previous work, we allow the scattering relation to have more general structure.
Some properties of Sobolev spaces
Authors: Borghol Rouba
Published: December 2024

The aim of this paper is to present some new characterizations of the Sobolev spaces Wk,p(Ω) where 1<p<∞, k≥2 and Ω is a bounded smooth domain of $\mathbb{R}$ N. For p=1, we prove same results by replacing Wk,1(Ω) by the BVk(Ω) space. As an application we show how to recognize polynomial functions among locally integrable functions.
The far-field expansion of the Green's function in a 3-D optical waveguide
Authors: Alexandrov Oleg
Published: December 2024

In this paper we find the far-field expansion of the outgoing Green's function in a three-dimensional cylindrical optical waveguide. As model for the field propagation we use the Helmholtz equation. We compare our results with the related paper (J. Inst. Math. Appl. 21(3) (1978), 315–330).
On slender shells and related problems suggested by Torroja's structures
Authors: Díaz J.I. , Sanchez-Palencia E.
Published: December 2024

We study the rigidification phenomenon for several thin slender bodies or shells, with a small curvature in the transversal direction to the main length, for which the propagation of singularities through the characteristics is of parabolic type. The asymptotic behavior is obtained starting with the two-dimensional Love–Kirchoff theory of plates. We consider, in a progressive study, a starting basic geometry, we pass then to consider the “V-shaped” structure formed by two slender plates pasted together along two long edges forming a small angle between their planes and, finally, we analyze the periodic extension to a infinite slab. We introduce a scalar potential φ and prove that the equation and constrains satisfied by the limit displacements are equivalent to a parabolic higher-order equation for φ. We get some global informations on φ, some on them easely associated to the different momenta and others of a different nature. Finally, we study the associate obstacle problem and obtain a global comparison result between the third component of the displacements with and without obstacle.
On asymptotics of a tracer advected in a locally self-similar, correlated flow
Authors: Komorowski Tomasz , Ryzhik Lenya
Published: December 2024

In this paper we consider the motion of a tracer in a flow that is locally self-similar and whose correlations decay at infinity but at the rate that does not guarantee that the flow does not have “memory effect”. We show that when the field is Gaussian the appropriately regularized scaling limit of the trajectory is a super-diffusive fractional Brownian motion. This complements our previous result contained in Komorowski and Ryzhik (Nonlinearity, to appear).
Convergence to equilibrium for a Cahn–Hilliard model with the Wentzell boundary condition
Authors: Wu Hao
Published: December 2024

In this paper we consider a Cahn–Hilliard model endowed with the Wentzell boundary condition, which arises from the study of spinodal decomposition in binary mixtures confined to a bounded domain with permeable wall. Under the assumption that the nonlinearity is analytic with respect to the unknown dependent function, we prove the convergence of a global solution to an equilibrium as time goes to infinity by means of an extended Łojasiewicz–Simon type inequality with boundary term. Estimates of the convergence rate are also obtained.
Attractors for nonautonomous reaction–diffusion systems with symbols without strong translation compactness
Authors: Lu Songsong
Published: December 2024

The long time behavior of the solutions of a general reaction–diffusion system (RDS) that covers many examples, such as the RDS with polynomial nonlinearity and Ginzburg–Landau equation, is discussed. First, the existence of a compact uniform attractor 𝒜0 in H is proved without additional assumptions on the interaction functions. Then the structure of the attractor is obtained for a certain class of interaction functions without strong translation compactness. For instance, the interaction functions are not required to be uniformly continuous. Moreover, an interesting problem arises naturally from this paper.
Partial regularity for local minimizers of splitting-type variational integrals
Authors: Bildhauer M. , Fuchs M.
Published: December 2024

We consider local minimizers u: Rn⊃Ω→RN of anisotropic variational integrals of (p, q)-growth with exponents 2≤p≤q≤min {2+p, p n/(n−2)}. If the integrand is of splitting-type, then partial C1-regularity of u is established.
On some anisotropic singular perturbation problems
Authors: Chipot Michel
Published: December 2024

We investigate the asymptotic behavior of some anisotropic diffusion problems and give some estimates on the rate of convergence of the solution toward its limit. We also relate this type of elliptic problems to problems set in cylinder becoming unbounded in some directions and show how some information on one type leads to information for the other type and conversely.
Asymptotic behavior of weak solutions to nonlinear thermoviscoelastic systems with constant temperature boundary conditions
Authors: Shang Chanyu
Published: December 2024

This paper deals with the one-dimensional nonlinear thermoviscoelastic system subject to constant temperature boundary conditions, describing phase transitions in shape memory alloys. We shall prove the global existence and uniqueness of the weak solution for initial data (strain, velocity, absolute temperature) (u0, v0, θ0)∈L∞×W01,∞×H1. Furthermore, we investigate the asymptotic behavior as time tends to infinity and establish the following asymptotic properties for the weak solution: As t→∞,

v→0 in H1(0, 1),

θ→T0 in L∞(0, 1),

u(x, t)→u∞(x) a.e.,

where u∞∈L∞ is a stationary state satisfying

f1(u∞)T0+f2(u∞)=0, a.e., in [0, 1].

Here we work in a framework in which u belongs to L∞ to describe phase transitions between different configurations of crystal lattices.
A Foundation for Refining Concurrent Objects
Authors: Büchi Martin , Sekerinski Emil
Published: December 2024

We study the notion of class refinement in a concurrent object-oriented setting. Our model is based on a combination of action systems and classes. An action system describes the behavior of a concurrent, distributed, or interactive system in terms of the atomic actions that can take place during the execution of the system. Classes serve as templates for creating objects. To express concurrency with objects, we add actions to classes. We define class refinement based on trace refinement of action systems. Additionally, we give a simulation-based proof rule. We show that the easier to apply simulation rule implies the trace-based definition of class refinement. Class refinement embraces algorithmic refinement, data refinement, and atomicity refinement. Atomicity refinement allows us to split large atomic actions into several smaller ones. Thereby, it paves the way for more parallelism. We investigate the special case of atomicity refinement by early returns in methods.
Sharing Variables in Distributed Memory
Authors: Chrobot Stanislaw
Published: December 2024

In the classical shared-variable models, component processes reside on their processors and communicate by shared variables in memory shared by the processors. In this paper, we argue that shared memory is not necessary to share variables. Processes can share variables in local memories of processors if they travel among the processors. We present a formal distributed memory model in which a system can be decomposed into processes residing on processors and communicating by message passing or into processes travelling among processors and communicating by shared variables. We call this property communication dualism of distributed systems. We point out that the shared-memory monitor can be used in distributed memory and suggest that variable sharing is a convenient alternative to message passing. We also point out that a mobile agent is a kind of travelling process, but its prominent property, code mobility, is not related to shared-variable communication.
Dynamical Teams in Eco-Grammar Systems
Authors: Csuhaj-Varjú Erzsébet , Mitrana Victor
Published: December 2024

In this paper we investigate simple eco-grammar systems with dynamically formed teams of agents. Three natural conditions for constituting a team, based on the agents' current capability of activation, are considered. We study the relations of language classes of these simple eco-grammar systems to each other and to some other well-known classes of languages. Moreover, we prove that any recursively enumerable language can be obtained as the intersection of a regular language and the language of a simple eco-grammar system where the acting teams of agents are formed according to two of the above conditions.
Comparing Truly Concurrent Semantics for Contextual Place/Transition Nets
Authors: Busi Nadia , Pinna G. Michele
Published: December 2024

The paper is centered around the study and comparison of truly concurrent semantics for P/T nets with inhibitor and read arcs (called henceforth contextual P/T nets). We start proposing a causal semantics for P/T nets, that we prove to be equivalent to history preserving bisimulation defined on nonsequential processes. Then we develop a conservative extension of the causal semantics to contextual P/T nets and we prove this one to be finer than step semantics. Finally, a comparison of causal semantics with the process based semantics for contextual P/T systems proposed in \cite{buspin99} is carried out.
Normalized Decision Functions and Measures for Inconsistent Decision Tables Analysis
Authors: Ślezak Dominik
Published: December 2024

We consider the family of normalized decision functions acting over conditional frequency distributions computed from data tables. We draw the connection between such functions and approaches to generating inexact decision rules for the new case classification. We also introduce the family of normalized decision measures corresponding to particular decision functions. They enable us to express efficiency of particular strategies of reasoning with respect to a given data. We show the properties of approximate decision rules and decision reducts based on normalized decision functions and measures. As a result, we obtain an intuitive and flexible tool for extracting approximate classification models from data.