Asymptotic Analysis - Volume 7, issue 1

Article Type: Other

DOI: 10.3233/ASY-1993-7101

Citation: Asymptotic Analysis, vol. 7, no. 1, pp. 1-2, 1993

Authors: Haraux, A.

Article Type: Research Article

Abstract: Let Ω be a bounded connected open subset of RN with a Lipschitz continuous boundary and f:R→R a locally Lipschitz continuous function such that f(0)=0, f is strictly convex on [0,+∞) and f′d (0)<−λ1 (Ω) where λ1 (Ω) is the smallest eigenvalue of (−Δ) in H1 0 (Ω), and f(s)→+∞ as s→+∞. For any h∈L∞ (R×Ω) with h(t,x)≥0, a.e. on R×Ω, the semilinear parabolic problem ut −Δu+f(u)=h(t,x) in R×Ω; u=0 on R×∂Ω has one and only one nonnegative global solution ω∈CB (R;H1 0 (Ω)∩L∞ (Ω)) such that w(t,·) does not tend to 0 as t→−∞. In addition, if h …is T-periodic, then ω is T-periodic. If h:R→L∞ (Ω) is continuous and almost-periodic, then ω:R→H0 1 (Ω)∩L∞ (Ω) is almost-periodic. Finally assuming either that u0 ≥0 is not identically 0 or that h(t,x)>0 on a subset of positive measure of R+ ×Ω, the unique solution u of ut −Δu+f(u)=h(t,x) in R+ ×Ω; u=0 on R+ ×∂Ω such that u(0)=u0 satisfies ‖u(t,·)−ω(t,·)‖∞ ≤C(u0 )exp (−γt), for all t≥0, with γ>0 independent of u0 and h. Show more

DOI: 10.3233/ASY-1993-7102

Citation: Asymptotic Analysis, vol. 7, no. 1, pp. 3-13, 1993

Authors: Levitin, Michael R.

Article Type: Research Article

Abstract: In this paper we consider the equations of small (acoustic) vibrations of a viscous compressible barotropic fluid in a bounded smooth domain under different boundary conditions. The structure of the spectrum is investigated; the estimates of the norm of the resolvent operator of the forced vibration problems are presented; the infinite asymptotic expansions of eigenfrequencies and eigenfunctions in series in a vanishing viscosity coefficient are obtained; the leading terms of these expansions are determined analytically.

DOI: 10.3233/ASY-1993-7103

Citation: Asymptotic Analysis, vol. 7, no. 1, pp. 15-35, 1993

Authors: Gamba, Irene Martínez

Article Type: Research Article

Abstract: We study how the equations of the potential of a semiconductor device, as described by Markowich (1986) and Selberherr (1984), behave when the width and the permittivity of the oxide region go to zero. According to their ratio, the problems converge to different asymptotic limits. This provides the correct boundary value approximation to the full problem avoiding costly computations (see, as an example, the discussion in Chapter 5 of (Selberherr, 1984), about the boundary conditions).

DOI: 10.3233/ASY-1993-7104

Citation: Asymptotic Analysis, vol. 7, no. 1, pp. 37-48, 1993

Authors: Karasev, M.V. | Pereskokov, A.V.

Article Type: Research Article

Abstract: In this paper we consider an ordinary differential equation with nonlinearity which becomes constant if the solution amplitude increases. We use a modification of the Kuzmak–Whitham method, solutions of boundary layer type and a model Painleve equation. We constructed a global asymptotic solution including neighbourhoods of turning points and calculated the phase shifts at turning points taking into account the nonlinear character of corrections. An asymptotics of eigenvalues is found.

DOI: 10.3233/ASY-1993-7105

Citation: Asymptotic Analysis, vol. 7, no. 1, pp. 49-66, 1993

Authors: Kratz, Werner

Article Type: Research Article

DOI: 10.3233/ASY-1993-7106

Citation: Asymptotic Analysis, vol. 7, no. 1, pp. 67-80, 1993