Asymptotic Analysis - Volume 32, issue 1

Authors: Macià, Fabricio | Zuazua, Enrique

Article Type: Research Article

Abstract: This paper is devoted to study the property of observability for wave equations guaranteeing that the total energy of solutions may be estimated by means of the energy concentrated on a subset of the domain or of the boundary. We prove that this property fails in three different situations. First, we consider the wave equation with piecewise smooth coefficients when the observation is made in the exterior boundary. We also present a wave equation with highly oscillating Hölder continuous coefficients for which observability fails from any open set that does not contain the origin. Finally, lack of observability is proved …for the constant coefficient wave equation when the observation is made from an interior hypersurface. All the counterexamples presented here are constructed using highly localized solutions known as Gaussian beams. Show more

Citation: Asymptotic Analysis, vol. 32, no. 1, pp. 1-26, 2002

Authors: Rybalko, Volodymyr

Article Type: Research Article

Abstract: We consider a spectral problem modeling natural vibrations of a complex medium consisting of an elastic medium and tiny rigid inclusions. We study the asymptotic behaviour of the eigenvalues and eigenvectors of this problem when the total number of inclusions and their density tend to infinity. We obtain a limiting problem, that is a spectral problem for a linear fractional operator pencil, describing the macroscopic behaviour of the system (global vibrations). We also show that there exist vibrations of the medium localized in small vicinities of the inclusions (local vibrations), which correspond to eigenvalues accumulating at the poles of the …operator pencil. Show more

Keywords: homogenization, spectral analysis, concentrated masses

Citation: Asymptotic Analysis, vol. 32, no. 1, pp. 27-62, 2002

Authors: Senba, Takasi | Suzuki, Takashi

Article Type: Research Article

Abstract: This paper is concerning with asymptotic behavior of infinite time blowup solutions to a parabolic‐elliptic system to chemotaxis. In this paper, we show that the solutions form a delta function singularity at each blowup point in infinite time and that the weight of each delta function singularity is equal to 8π and 4π if the blowup point is in the domain and on the boundary, respectively. We refer to the former and the latter phenomenon as chemotactic collapse and quantization, respectively. Next, we investigate the behavior of some norms of the solutions. Finally, we show that the location of the …collapse moves continuously if total mass of the solution is between 4π and 8π. Show more

Citation: Asymptotic Analysis, vol. 32, no. 1, pp. 63-89, 2002