Asymptotic Analysis - Volume 8, issue 2

Authors: Kozlov, V. A. | Maz'ya, V. G. | Movchan, A. B.

Article Type: Research Article

Abstract: We consider a mixed boundary value problem for the Laplace operator in a union of a three-dimensional domain and thin cylinders. A rigorous procedure is developed to derive the total asymptotic expansion. The asymptotic formulae for the first eigenvalue and corresponding eigenfunction have been obtained.

DOI: 10.3233/ASY-1994-8201

Citation: Asymptotic Analysis, vol. 8, no. 2, pp. 105-143, 1994

Authors: Galaktionov, Victor A. | Vazquez, Juan L.

Article Type: Research Article

Abstract: We investigate the asymptotic behaviour as t→∞ of the non-negative weak solution to the Cauchy problem for the equation of superslow diffusion ut =(e−1/u )xx  for x∈R, t>0, with non-negative initial function u0 ∈L∞ (R)∩L1 (R), u0 $\not\equiv$ 0. We prove that asymptotic separation of variables takes place if we make the change of variables v=e−1/u and η=x/log t. The precise result says that as t→∞ tv(η log t,t)→½(a2 −η2 )+ , and the convergence is uniform in η∈R. The constant a>0 is exactly one half of the initial energy: a=½∫u0 (x)dx>0. This implies that u …evolves for large t towards a mesa-like profile of height 1/(log t) and width =2a log t. Show more

DOI: 10.3233/ASY-1994-8202

Citation: Asymptotic Analysis, vol. 8, no. 2, pp. 145-159, 1994

Authors: Quintela Estévez, P.

Article Type: Research Article

Abstract: An asymptotic method is presented to analyse perturbations of bifurcations of the solutions of nonlinear problems. The perturbations may result from imperfections, impurities or other inhomogeneities in the corresponding physical problem. Using a method of matched asymptotic expansions we obtain global representations of the solutions of the perturbed problem when the bifurcation solutions are known globally. Even if, in this paper, the asymptotic method is used to analyse the perturbed bifurcation in the von Kármán equations, the same analysis is also valid to study the perturbations in more general nonlinear problems.

DOI: 10.3233/ASY-1994-8203

Citation: Asymptotic Analysis, vol. 8, no. 2, pp. 161-184, 1994

Authors: Canalis-Durand, M.

Article Type: Research Article

Abstract: Nous nous intéressons à un problème d'équations différentielles singulièrement perturbées. Après avoir précisé la notion d'équation élémentaire sur C, nous étudions les solutions formelles de telles équations possédant certaines propriétés de régularité. Nous prouvons, par des majorations directes, le caractère Gevrey 1 de ces séries formelles. Nous obtenons alors par des techniques de sommation de séries divergentes, une nouvelle preuve de l'existence des solutions surstables d'une équation élémentaire. Ce texte est le développement de la note aux C. R. Acad. Sci. qui porte le titre: “Caractère Gevrey des solutions canard de l'équation de Van der Pol”.

DOI: 10.3233/ASY-1994-8204

Citation: Asymptotic Analysis, vol. 8, no. 2, pp. 185-216, 1994