Asymptotic Analysis - Volume 65, issue 1-2

Authors: Kopylova, E.A.

Article Type: Research Article

Abstract: We obtain a dispersive long-time decay in weighted energy norms for solutions to the 3D wave equation with generic potential. The decay extends the results obtained by Jensen and Kato for the 3D Schrödinger equation.

Keywords: dispersion, wave equation, resolvent, spectral representation, weighted spaces, continuous spectrum, Born series, convolution, long-time asymptotics

DOI: 10.3233/ASY-2009-0945

Citation: Asymptotic Analysis, vol. 65, no. 1-2, pp. 1-16, 2009

Authors: Fujiié, Setsuro | Lasser, Caroline | Nédélec, Laurence

Article Type: Research Article

Abstract: We study the resonant set of a two-level Schrödinger operator with a linear conical intersection. This model operator can be decomposed into a direct sum of first order systems on the real half-line. For these ordinary differential systems we locally construct exact WKB solutions, which are connected to global solutions, amongst which are resonant states. The main results are a generalized Bohr–Sommerfeld quantization condition and an asymptotic description of the set of resonances as a distorted lattice.

Keywords: conical intersection, exact WKB solutions, resonances, semiclassical analysis, Schrödinger systems

DOI: 10.3233/ASY-2009-0946

Citation: Asymptotic Analysis, vol. 65, no. 1-2, pp. 17-58, 2009

Authors: Maddouri, Feten

Article Type: Research Article

Abstract: We study an explicit interior observability estimate for the Schrödinger equation with potential in a bounded domain of RN (N≥1), and with zero Dirichlet boundary condition. The method combines Carleman inequality and multiplier techniques. Moreover, by microlocal analysis tools, we prove a precised observability estimate in high frequencies.

Keywords: Schrödinger equation, observability estimate, Carleman inequality, microlocal analysis

DOI: 10.3233/ASY-2009-0947

Citation: Asymptotic Analysis, vol. 65, no. 1-2, pp. 59-78, 2009

Authors: Jiang, Jie | Zhang, Yanyan

Article Type: Research Article

Abstract: In this paper, we study the asymptotic behavior of solutions to a chemotaxis model with volume-filling effect subject to the homogeneous Neumann boundary conditions. We establish a non-smooth version of Łojasiewicz–Simon inequality, and we prove that as time goes to infinity the solution to our system converges to an equilibrium in W1,p (Ω)×W1,p (Ω), p>max (n, 2), Ω⊂Rn . We also obtain an estimate of the decay rate to equilibrium.

Keywords: chemotaxis model, volume-filling effect, Łojasiewicz–Simon inequality

DOI: 10.3233/ASY-2009-0948

Citation: Asymptotic Analysis, vol. 65, no. 1-2, pp. 79-102, 2009

Authors: Jimbo, Shuichi | Kimura, Masato | Notsu, Hirofumi

Article Type: Research Article

Abstract: We study an asymptotic behaviour of the principal eigenvalue for an elliptic operator with large advection which is given by a gradient of a potential function. It is shown that the principal eigenvalue decays exponentially under the velocity potential well condition as the parameter tends to infinity. We reveal that the depth of the potential well plays an important role in the estimate. Particularly, in one-dimensional case, we give a much more elaborate characterization for the eigenvalue. Some numerical examples are also shown.

Keywords: elliptic eigenvalue problems, asymptotic behaviour

DOI: 10.3233/ASY-2009-0951

Citation: Asymptotic Analysis, vol. 65, no. 1-2, pp. 103-123, 2009