Asymptotic Analysis - Volume 36, issue 1

Authors: Pivovarchik, Vjacheslav | Tretter, Christiane

Article Type: Research Article

Abstract: We investigate the spectrum of a damped quadratic operator pencil with 2×2 block operator matrix coefficients. Our main result is that the spectrum splits asymptotically into four branches of eigenvalues and we determine their asymptotic behaviour. An application to the vibrations of two interacting connected strings and beams is given where one of them is damped.

Citation: Asymptotic Analysis, vol. 36, no. 1, pp. 1-12, 2003

Authors: Reyes, Guillermo | Sánchez, Ariel

Article Type: Research Article

Abstract: We improve previous results on the qualitative properties of solutions to the Cauchy problem for the equation ρ(x)ut =(um )xx −c0 up , where 1<m<p and c0 >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.

Keywords: non‐homogeneous porous media, positivity, disappearance of interfaces

Citation: Asymptotic Analysis, vol. 36, no. 1, pp. 13-20, 2003

Authors: Fushan, Li

Article Type: Research Article

Abstract: 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.

Keywords: viscoelastic shell, asymptotic analysis, membrane shell

Citation: Asymptotic Analysis, vol. 36, no. 1, pp. 21-46, 2003

Authors: Ji, Ye

Article Type: Research Article

Abstract: We consider a dynamic linearly elastic generalized membrane shell with variable thickness and show that the solutions of the three‐dimensional equations, as the thickness of the shell goes to zero, converge to the solution of the two‐dimensional dynamic generalized membrane shell.

Keywords: generalized membrane shells, linear elasticity, dynamic problem, asymptotic analysis

Citation: Asymptotic Analysis, vol. 36, no. 1, pp. 47-62, 2003

Authors: Ikehata, Ryo

Article Type: Research Article

Abstract: Hyperbolic linear Cauchy problem εu″+Au+u′=0, u(0)=u0 , u′(0)=u1 , with “nonnegative” selfadjoint operator A in a real Hilbert space H is first considered. It is shown that the solution uε tends to some solution v as ε↓0 for the parabolic equation v′+Av=0 in a certain sense. Some applications are given. Finally, we present hyperbolic‐hyperbolic convergence results such as the solution for the damped wave equations goes to some solution for the free wave equations as the effect of the damping vanishes in a concrete context.

Keywords: dissipative wave equation, unbounded domain, singular limit, $L^{2}$‐convergence result

Citation: Asymptotic Analysis, vol. 36, no. 1, pp. 63-74, 2003

Authors: Peng, Yue‐Jun

Article Type: Research Article

Abstract: We study the zero‐electron‐mass limit, the zero‐relaxation‐time limit and the quasi‐neutral limit in steady‐state Euler–Poisson system for potential flow arising in mathematical modeling for plasmas and semiconductors. We show the existence and uniqueness of solutions when the electron‐mass is small enough. For the zero‐electron‐mass limit and the zero‐relaxation‐time limit, we prove the strong convergence of the sequence of solutions and give the corresponding error estimates. Whereas for the quasi‐neutral limit, we obtain the similar results only if the given data on the boundary are in equilibrium.

Keywords: zero‐electron‐mass limit, zero‐relaxation‐time limit, quasi‐neutral limit, Euler–Poisson equations, potential flow, boundary layer

Citation: Asymptotic Analysis, vol. 36, no. 1, pp. 75-92, 2003