Asymptotic Analysis - Volume 69, issue 1-2

Authors: Baklouti, Hamadi

Article Type: Research Article

Abstract: We develop a microlocal study for scalar classical analytic pseudo-differential operators of principal type. Next, we reduce a system of pseudo-differential operators to the normal form. Then we obtain asymptotic expansion and the exact exponential decay rate for the width of resonances arising from microlocal tunneling effect.

Keywords: BKW, analytic pseudo-differential operator, microlocal analysis, asymptotic, resonances

DOI: 10.3233/ASY-2010-0993

Citation: Asymptotic Analysis, vol. 69, no. 1-2, pp. 1-29, 2010

Authors: Ben Hassen, Imen

Article Type: Research Article

Abstract: We estimate the rate of decay of the difference between a solution and its limiting equilibrium for the nonautonomous first-order problem u˙+ℳu=g(t),  t∈R+ , and for the second-order problem u¨+u˙+ℳu=g(t),  t∈R+ , where ℳ is a gradient operator and g is a function which tends to 0 when t tends to ∞.

Keywords: convergence, rate of decay, analytic nonlinearity, nonautonomous, evolution equation

DOI: 10.3233/ASY-2010-0994

Citation: Asymptotic Analysis, vol. 69, no. 1-2, pp. 31-44, 2010

Authors: Borthwick, David

Article Type: Research Article

Abstract: For certain compactly supported metric and/or potential perturbations of the Laplacian on Hn+1 , we establish an upper bound on the resonance counting function with an explicit constant that depends only on the dimension, the radius of the unperturbed region in Hn+1 , and the volume of the metric perturbation. This constant is shown to be sharp in the case of scattering by a spherical obstacle.

Keywords: resonances, scattering theory, hyperbolic space

DOI: 10.3233/ASY-2010-0995

Citation: Asymptotic Analysis, vol. 69, no. 1-2, pp. 45-85, 2010

Authors: Warnault, Guillaume

Article Type: Research Article

Abstract: This article is concerned with the non-existence of stable solutions for a fourth-order semilinear elliptic equation Δ2 u=f(u) in RN , where f is a smooth nonlinearity. We establish the non-existence of stable radial solutions which verify decay conditions at infinity. Our Liouville-type results do not depend on the specific nonlinearity f. Moreover, in low dimensions and under no radial symmetry assumption, we prove Liouville-type results when f is increasing.

Keywords: biharmonic operator, Liouville result

DOI: 10.3233/ASY-2010-0997

Citation: Asymptotic Analysis, vol. 69, no. 1-2, pp. 87-98, 2010

Authors: Yu, Yifeng

Article Type: Research Article

Abstract: It was proved in Arch. Ration. Mech. Anal. 162 (2002), 1–23 that singularities of solutions for Hamilton–Jacobi equations will propagate along generalized characteristics. In this paper, we consider the homogenization of generalized characteristics. It is natural to think that (*) generalized characteristics associated to viscosity solutions of oscillatory Hamilton–Jacobi equations will converge to generalized characteristics associated to solutions of the effective equation. We show that this is indeed correct if the spatial dimension is 1. For high dimensions, we need to add some extra assumptions of singularities near generalized characteristics. We provide a counterexample that (*) fails without those assumptions. …Some issues related to the weak KAM theory are also discussed. Show more

Keywords: homogenization, generalized characteristics, Hamilton–Jacobi equations

DOI: 10.3233/ASY-2010-1001

Citation: Asymptotic Analysis, vol. 69, no. 1-2, pp. 99-116, 2010

Authors: Thomann, Laurent

Article Type: Research Article

Abstract: We prove the equivalence between the smoothing effect for a Schrödinger operator and the decay of the associate spectral projectors. We give two applications to the Schrödinger operator in dimension one.

Keywords: Schrödinger equation, potential, smoothing effect

DOI: 10.3233/ASY-2010-0986

Citation: Asymptotic Analysis, vol. 69, no. 1-2, pp. 117-123, 2010