Asymptotic Analysis - Volume 127, issue 1-2

Authors: Kopylova, Elena

Article Type: Research Article

Abstract: We improve previous results on dispersive decay for 1D Klein–Gordon equation. We develop a novel approach, which allows us to establish the decay in more strong norms and weaken the assumption on the potential.

Keywords: Klein–Gordon equation, dispersive estimates, scattering

DOI: 10.3233/ASY-201670

Citation: Asymptotic Analysis, vol. 127, no. 1-2, pp. 1-13, 2022

Authors: Arruda, Suellen Cristina Q. | Figueiredo, Giovany M. | Nascimento, Rubia G.

Article Type: Research Article

Abstract: In this paper we study the asymptotic behaviour of a family of elliptic systems, as far as the existence of solutions is concerned. We give a special attention to the asymptotic behaviour of W and V as ε goes to zero in the system − ε 2 Δ u + W ( x ) u = Q u ( u , v ) in  R N , − ε 2 Δ v + V ( x ) v = …Q v ( u , v ) in  R N , u , v ∈ H 1 ( R N ) , u ( x ) , v ( x ) > 0 for each  x ∈ R N , where ε > 0 , W and V are positive potentials of C 2 class and Q is a p -homogeneous function with subcritical growth. We establish the existence of a positive solution by considering two classes of potentials W and V . Our arguments are based on penalization techniques, variational methods and the Moser iteration scheme. Show more

Keywords: Elliptic system, asymptotic analysis, Palais–Smale condition, positive solution

DOI: 10.3233/ASY-201671

Citation: Asymptotic Analysis, vol. 127, no. 1-2, pp. 15-34, 2022

Authors: Piersanti, Paolo

Article Type: Research Article

Abstract: In this paper we show that the solution of an obstacle problem for linearly elastic shallow shells enjoys higher differentiability properties in the interior of the domain where it is defined.

Keywords: Shallow shell, obstacle problems, elliptic variational inequalities, improved regularity, fourth order problems

DOI: 10.3233/ASY-211672

Citation: Asymptotic Analysis, vol. 127, no. 1-2, pp. 35-55, 2022

Authors: Ding, Yutao | Jiang, Ning

Article Type: Research Article

Abstract: We study the zero viscosity and thermal diffusivity limit of an initial boundary problem for the linearized Navier–Stokes–Fourier equations of a compressible viscous and heat conducting fluid in the half plane. We consider the case that the viscosity and thermal diffusivity converge to zero at the same order. The approximate solution of the linearized Navier–Stokes–Fourier equations with inner and boundary expansion terms is analyzed formally first by multiscale analysis. Then the pointwise estimates of the error terms of the approximate solution are obtained by energy methods. Thus establish the uniform stability for the linearized Navier–Stokes–Fourier equations in the zero viscosity …and heat conductivity limit. This work is based on (Comm. Pure Appl. Math. 52 (1999 ), 479–541) and generalize their results from isentropic case to the general compressible fluid with thermal diffusive effect. Besides the viscous layer as in (Comm. Pure Appl. Math. 52 (1999 ), 479–541), the thermal layer appears and couples with the viscous layer linearly. Show more

Keywords: Compressible Navier–Stokes, Prandtl equations, boundary layer, thermal diffusivity, zero viscosity

DOI: 10.3233/ASY-211673

Citation: Asymptotic Analysis, vol. 127, no. 1-2, pp. 57-96, 2022

Authors: Belaud, Y. | Shishkov, A.

Article Type: Research Article

Abstract: We study the property of extinction in a finite time for nonnegative solutions of 1 q ∂ ∂ t ( u q ) − ∇ ( | ∇ u | p − 2 ∇ u ) + a ( x ) u λ = 0 for the Dirichlet Boundary Conditions when q > λ > 0 , p ⩾ 1 + q , p ⩾ 2 , a ( x ) ⩾ 0 …and Ω a bounded domain of R N (N ⩾ 1 ). We prove some necessary and sufficient conditions. The threshold is for power functions when p > 1 + q while finite time extinction occurs for very flat potentials a ( x ) when p = 1 + q . Show more

Keywords: Extinction time, semi-classical limit, nonlinear eigenvalue problem

DOI: 10.3233/ASY-211674

Citation: Asymptotic Analysis, vol. 127, no. 1-2, pp. 97-119, 2022

Authors: Ferrari, Gianluca | Squassina, Marco

Article Type: Research Article

Abstract: We obtain some nonlocal characterizations for a class of variable exponent Sobolev spaces arising in nonlinear elasticity theory and in the theory of electrorheological fluids. We also get a singular limit formula extending Nguyen results to the anisotropic case.

Keywords: Anisotropic Sobolev spaces, singular limit formulas, nonlocal characterizations

DOI: 10.3233/ASY-211675

Citation: Asymptotic Analysis, vol. 127, no. 1-2, pp. 121-142, 2022

Authors: Badieti Matala, Padouette Boubati | Moukoko, Daniel | Evrard Isseret Goyaud, Mayeul

Article Type: Research Article

Abstract: In this article, we study a hyperbolic equation of Cahn–Hilliard with a proliferation term and Dirichlet boundary conditions. In particular, we prove the existence and uniqueness of the solution, and also the existence of the global attractor.

Keywords: Cahn–Hilliard equation, proliferation term, dissipativity, global attractor, comparison principle

DOI: 10.3233/ASY-211676

Citation: Asymptotic Analysis, vol. 127, no. 1-2, pp. 143-165, 2022

Authors: Caruso, Noè Angelo | Michelangeli, Alessandro | Novati, Paolo

Article Type: Research Article

Abstract: In the framework of abstract linear inverse problems in infinite-dimensional Hilbert space we discuss generic convergence behaviours of approximate solutions determined by means of general projection methods, namely outside the standard assumptions of Petrov–Galerkin truncation schemes. This includes a discussion of the mechanisms why the error or the residual generically fail to vanish in norm, and the identification of practically plausible sufficient conditions for such indicators to be small in some weaker sense. The presentation is based on theoretical results together with a series of model examples and numerical tests.

Keywords: Linear inverse problems, infinite-dimensional Hilbert space, ill-posed problems, orthonormal basis discretisation, bounded linear operators, Krylov subspaces, Krylov solution, GMRES, conjugate gradient, LSQR

DOI: 10.3233/ASY-211678

Citation: Asymptotic Analysis, vol. 127, no. 1-2, pp. 167-189, 2022

Authors: Hireche, Faouzi | Ghomari, Kaoutar

Article Type: Research Article

Abstract: This article is devoted to an analysis of semiclassical Schrödinger operators for two-frequency resonance of the type 1 : p where p is even. The Birkhoff–Gustavson normal form is applied to describe the discrete spectrum in the case where the potential is smooth and admits a nondegenerate global minimum at the origin 0.

Keywords: Birkhoff–Gustavson normal form, Weyl quantization, Schrödinger operator, Harmonic oscillator, Resonance, Bargmann transform

DOI: 10.3233/ASY-211691

Citation: Asymptotic Analysis, vol. 127, no. 1-2, pp. 191-200, 2022