Asymptotic Analysis - Volume 45, issue 1-2

Authors: Coulombel, Jean-François | Golse, François | Goudon, Thierry

Article Type: Research Article

Abstract: It is a well-known fact that, in small mean free path regimes, kinetic equations can lead to diffusion equations. Besides, kinetic equations can be approached by a closed system of moments equations. In this paper, we are interested in a special closure based on an entropy minimization principle, as introduced earlier by Levermore. We investigate the behavior of the resulting nonlinear hyperbolic system in the diffusive scaling. We first establish various fundamental facts on this system, then we show that the hyperbolic system admits global smooth solutions, and is consistent with the diffusion limit. Similar features are also discussed for …a simpler limited flux equation. Show more

Keywords: diffusion approximation, hyperbolic systems, relaxation, global smooth solutions, nonlinear parabolic equations

Citation: Asymptotic Analysis, vol. 45, no. 1-2, pp. 1-39, 2005

Authors: Chae, Dongho

Article Type: Research Article

Abstract: We study a semilinear elliptic system modeling the physical system strings and antistrings in cosmology under the boundary condition of the symmetric vacuum (the nontopological type). We construct solutions with the representation having precise informations on the asymptotic behaviors near infinity for arbitrary location of strings and antistrings satisfying \[$1\leqslant M-N<\tfrac{1}{4\pi G}$ , where M and N are the total string number and the total antistring number respectively, and G is the gravitational constant. The asymptotic properties, in particular, are completely different to the solutions under the boundary condition of the asymmetric vacuum (the topological type) constructed previously by …Y. Yang [Phys. Rev. Lett. 80 (1999), 26–29]. We also compute the total magnetic flux, total energy and the total Gaussian curvature of our solutions. Show more

Citation: Asymptotic Analysis, vol. 45, no. 1-2, pp. 41-54, 2005

Authors: Eidelman, S.D. | Kamin, S.

Article Type: Research Article

Citation: Asymptotic Analysis, vol. 45, no. 1-2, pp. 55-71, 2005

Authors: Korotyaev, Evgeny

Article Type: Research Article

Abstract: The spectral properties of the Schrödinger operator Tt y=−y″+qt y in \[$L^{2}(\mathbb{R})$ are studied, with a potential qt (x)=p1 (x), x<0, and qt (x)=p(x+t), x>0, where p1 ,p are periodic potentials and \[$t\in \mathbb{R}$ is a parameter of dislocation. Under some conditions there exist simultaneously gaps in the continuous spectrum of T0 and eigenvalues in these gaps. The main goal of this paper is to study the discrete spectrum and the resonances of Tt . The following results are obtained: (i) In any gap of Tt there exist 0,1 or 2 eigenvalues. Potentials with …0, 1 or 2 eigenvalues in the gap are constructed. (ii) The dislocation, i.e., the case p1 =p is studied. If t→0, then in any gap in the spectrum there exist both eigenvalues (≤2) and resonances (≤2) of Tt which belong to a gap on the second sheet and their asymptotics as t→0 are determined. (iii) The eigenvalues of the half-solid, i.e., p1 =constant, are also studied. (iv) We prove that for any even 1-periodic potential p and any sequences {dn }1 ∞ , where dn =1 or dn =0 there exists a unique even 1-periodic potential p1 with the same gaps and dn eigenvalues of T0 in the n-th gap for each n≥1. Show more

Citation: Asymptotic Analysis, vol. 45, no. 1-2, pp. 73-97, 2005

Authors: Takamura, Hiroyuki

Article Type: Research Article

Abstract: In this paper we study a global in time existence of classical solutions of semilinear systems of 3-dimensional wave equations. We see that one component of the global in time solution can be arbitrarily large if another component is small enough according to some balance of each amplitude. Also its sharpness is discussed. This is a specific nature of strongly coupled systems.

Keywords: semilinear wave equation, classical solution, global existence, blow-up

Citation: Asymptotic Analysis, vol. 45, no. 1-2, pp. 99-112, 2005

Authors: Santos, M.L. | Rocha, M.P.C. | Ferreira, J.

Article Type: Research Article

Abstract: We prove the exponential decay in the case n>2 as time goes to infinity of regular solutions for the nonlinear coupled system of beam equations with memory and weak damping utt +Δ2 u−Δu+∫t 0 g1 (t−s)Δu(s) ds+αut +h(u−v)=0  in \[$\widehat{Q}$ , vtt +Δ2 v−Δv+∫t 0 g2 (t−s)Δv(s) ds+αvt −h(u−v)=0  in \[$\widehat{Q}$ in a noncylindrical domains of \[$\mathbb{R}^{n+1}$ (n≥1) under suitable hypothesis on the scalar functions g1 , g2 and h where α is a positive constant. Observe that the coupled is nonlinear and we worked directly in the noncylindrical domain that presents some …technical difficulties turning the interesting problem. We establish existence and uniqueness of regular solutions for any n≥1. Show more

Keywords: noncylindrical domain, strong solution, exponential decay

Citation: Asymptotic Analysis, vol. 45, no. 1-2, pp. 113-132, 2005

Authors: Shubov, Marianna A.

Article Type: Research Article

Abstract: This research is devoted to the asymptotic and spectral analysis of a coupled Euler–Bernoulli and Timoshenko beam model. The model is governed by a system of two coupled hyperbolic partial differential equations and a four-parameter family of the boundary conditions modeling the action of self-straining actuators. The aforementioned system of equations of motion together with the boundary conditions is equivalent to a single operator evolution equation in the energy space. That equation defines a semigroup of bounded operators. The dynamics generator of the semigroup is our main object of interest. For each set of boundary parameters, the dynamics generator has …a compact inverse. If all four boundary parameters are not purely imaginary numbers, then the dynamics generator is a nonself-adjoint operator in the energy space. We present two main results in this research. As the first one, we calculate the spectral asymptotics of the dynamics generator. We find that the complex spectrum lies in a strip parallel to the real axis, and is asymptotically close to the axis. The latter fact means that the system is stable, but is not uniformly stable. As the second main result, we prove that the set of the root vectors of the dynamics generator forms a Riesz basis in the energy space. The results obtained in the present paper allow us to conclude that the dynamics generator is a Riesz spectral operator in the sense of N. Dunford. Show more

Keywords: matrix differential operator, spectrum, root vectors, completeness, minimality, Riesz basis property

Citation: Asymptotic Analysis, vol. 45, no. 1-2, pp. 133-169, 2005