Asymptotic Analysis - Volume 45, issue 3-4

Authors: Briani, Ariela

Article Type: Research Article

Abstract: We consider the sequence of optimal control problems having as state equation y′ (t)=an (t,y)+bn (t,u) (t∈(0,T], y(0)=x) and cost functional \[$J_{n}(y,u)=\mathop{\mathrm{ess}}\mathop{\mathrm{sup}}\nolimits_{t\in[0,T]}f_{n}(t,y(t),u(t))$ . We prove a Γ-convergence result and we study the entailed properties on the stability for the related Hamilton–Jacobi equations.

Citation: Asymptotic Analysis, vol. 45, no. 3-4, pp. 171-190, 2005

Authors: El Mehdi, Khalil | Hammami, Mokhless

Article Type: Research Article

Abstract: In this paper we consider the following biharmonic equation with critical exponent \[$(P_{\varepsilon})\dvt\Delta^{2}u=Ku^{\frac{n+4}{n-4}-\varepsilon}$ , u>0 in Ω and u=Δu=0 on \[$\curpartial \varOmega $ , where Ω is a smooth bounded domain in \[$\mathbb{R}^{n}$ , n≥5, ε is a small positive parameter, and K is a smooth positive function in \[$\overline{ \varOmega }$ . We construct solutions of (Pε ) which blow up and concentrate at strict local maximum of K either at the boundary or in the interior of Ω. We also construct solutions of (Pε ) concentrating at an interior strict local minimum point of …K. Finally, we prove a nonexistence result for the corresponding supercritical problem which is in sharp contrast to what happened for (Pε ). Show more

Keywords: fourth-order elliptic equations, critical Sobolev exponent, biharmonic operator

Citation: Asymptotic Analysis, vol. 45, no. 3-4, pp. 191-225, 2005

Authors: Gravejat, Philippe

Article Type: Research Article

Abstract: We investigate the asymptotic behaviour of the subsonic travelling waves of finite energy in the Gross–Pitaevskii equation in dimension larger than two. In particular, we give their first-order asymptotics in the case they are axisymmetric, and link it to their energy and momentum.

Keywords: nonlinear Schrödinger equation, travelling waves, asymptotic behaviour

Citation: Asymptotic Analysis, vol. 45, no. 3-4, pp. 227-299, 2005

Authors: Zheng, Songmu | Chipot, Michel

Article Type: Research Article

Abstract: In this paper the asymptotic behavior of the solutions, as time goes to infinity, to nonlinear parabolic equations with two classes of nonlocal terms is investigated. One of important features of our problems is that equilibria may be a continuum which is diffeomorphic to a piece of curve in R2 .

Keywords: nonlinear parabolic equations, nonlocal term, asymptotic behavior

Citation: Asymptotic Analysis, vol. 45, no. 3-4, pp. 301-312, 2005

Authors: Licht, C. | Michaille, G.

Article Type: Research Article

Abstract: We obtain a nonlocal functional as the variational limit of an integral functional associated with the strain energy of a pseudo-plastic material reinforced by a ε-periodic distribution of pseudo plastic fibers. We begin by studying separately the variational limit behavior, when ε goes to zero, of the structure made of the soft material without fibers, and of the structure constituted by the distribution of the small fibers. We show that the complete structure is, in the sense of the Γ-convergence, equivalent to a new homogeneous structure whose functional energy is the epigraphical sum of the limit integral functionals energies modeling …each two previous structures. Show more

Keywords: homogenization, Γ-convergence, convex functional of measures, pseudo-plasticity

Citation: Asymptotic Analysis, vol. 45, no. 3-4, pp. 313-339, 2005