Asymptotic Analysis - Volume 46, issue 2

Authors: Rezaoui, Med-Salem

Article Type: Research Article

Abstract: Sibuya was concerned in [Lecture Notes in Math., Vol. 810, 1980, pp. 135–140] with studying analyticity of formal solutions for nonlinear differential systems xp+1 \[$\frac{\mathrm{d}u}{\mathrm{d}x}$ =E(x,y,u), where E is an analytic function in its arguments, near the origin. We are aiming in this work, to generalize the previous Sibuya's paper to nonlinear differential systems A(x)\[$\frac{\mathrm{d}u}{\mathrm{d}x}$ =E(x,y,u), where A(x) is an N×N matrix, whose entries are holomorphic near the origin. Sibuya s'est intéressé dans [Lecture Notes in Math., Vol. 810, 1980, pp. 135–140] à l'analyticité des solutions formelles des systèmes différentiels non linéaires xp+1 \[$\frac{\mathrm{d}u}{\mathrm{d}x}$ =E(x,y,u), où …E est une fonction analytique en ses arguments, au voisinage de l'origine. Nous nous intéressons dans ce travail, à généraliser le résultat de Sibuya aux systèmes différentiels non linéaires A(x)\[$\frac{\mathrm{d}u}{\mathrm{d}x}$ =E(x,y,u), où A(x) est une matrice carrée N×N, à coefficients holomorphes au voisinage de l'origine. Show more

Keywords: differential operator, formal solutions, convergent solutions, Pfaffian system

Citation: Asymptotic Analysis, vol. 46, no. 2, pp. 93-122, 2006

Authors: González-Burgos, Manuel | Pérez-García, Rosario

Article Type: Research Article

Abstract: In this paper, we present new controllability results for some nonlinear coupled parabolic systems considered in a bounded domain Ω of \[$\mathbb{R}$ N (with N≥1 being arbitrary) when the control force acts on a unique equation of the system through an arbitrarily small open set ω⊂Ω. As a model example, we consider a nonlinear phase field system with certain superlinear nonlinearities and prove the null controllability, the exact controllability to the trajectories and the approximate controllability of the model. The crucial point in this paper is the new strategy developed to deal with the null controllability of linear …coupled parabolic systems by a unique control force. Global Carleman estimates and the parabolic regularizing effect of the problem are used. Show more

Keywords: controllability, nonlinear coupled systems of parabolic type

Citation: Asymptotic Analysis, vol. 46, no. 2, pp. 123-162, 2006

Authors: Khrennikov, A.Yu. | Shelkovich, V.M.

Article Type: Research Article

Abstract: The notion of the quasi-asymptotics adapted to the case of p-adic distributions (generalized functions) is introduced. p-Adic analogs of Tauberian theorems for distributions are proved. We show that some properties of Vladimirov's pseudo-differential operator Dα are connected with a prototype of the Tauberian type theorem (with respect to distributional asymptotic). We also prove the p-adic version of the Shannon–Kotelnikov theorem.

Keywords: p-adic distributions, distributional asymptotics, quasi-asymptotics, Vladimirov's pseudo-differential operator, p-adic Tauberian type theorems, p-adic Shannon–Kotelnikov theorem

Citation: Asymptotic Analysis, vol. 46, no. 2, pp. 163-187, 2006