Asymptotic Analysis - Volume 26, issue 1

Authors: Vancostenoble, J.

Article Type: Research Article

Abstract: We study the problem of asymptotic behavior as t→+∞ of solutions of the weakly damped wave equation utt −Δu=−a(x)β(ut ,∇u) in Ω, u=0 on ∂Ω, where Ω is a bounded, open, connected set in RN , N≥1, with smooth boundary and where λ=(λ1 ,…,λN+1 )→β(λ1 ,…,λN+1 ) satisfies the basic assumption ∀λ, λ1 β(λ)≥0. We prove the weak asymptotic stabilization in H1 0 (Ω)×L2 (Ω) of all global solutions under very weak assumptions on β (in so far as we need neither hypothesis of monotonicity on β nor condition restricting its asymptotic growth at infinity). In particular, we generalize an earlier result …of M. Slemrod. Moreover the method also applies to other equations and above all to other feedbacks (boundary or pointwise feedbacks) and even to hybrid systems. Show more

Keywords: wave equation, distributed control, nonlinear nonmonotone feedback, weak asymptotic stabilization

Citation: Asymptotic Analysis, vol. 26, no. 1, pp. 1-20, 2001

Authors: Haraux, A. | Jendoubi, M.A.

Article Type: Research Article

Citation: Asymptotic Analysis, vol. 26, no. 1, pp. 21-36, 2001

Authors: Kogut, P. | Leugering, G.

Article Type: Research Article

Abstract: We consider the optimal control problems for the linear elliptic equations in perforated domains. Each components of mathematical description of such optimal control problem depends on small parameter ε. Problems of this type appear in sensitivity analysis, perturbation theory, and homogenization of heterogenous material. We study the problem of passing to the limit within the framework of variational S‐convergence. We derive conditions under which the so‐called fictitious limiting optimal control problem can be made explicit.

Keywords: homogenization, S‐convergence, optimal control, variable domains

Citation: Asymptotic Analysis, vol. 26, no. 1, pp. 37-72, 2001

Authors: Marušić, Sanja

Article Type: Research Article

Abstract: We consider an injection of incompressible viscous quasi‐Newtonian fluid in a curved pipe with a smooth central curve γ. Starting from the Navier–Stokes system with shear‐dependent viscosity, used to describe the polymeric flow or the blood flow, the 1‐dimensional model was obtained via singular perturbation as ε, the ratio between cross section area and the length of the pipe, tends to 0. The method of two‐scale convergence was applied. The limit depends only on the tangential injection along γ and the velocity, as well as the pressure drop have the tangential direction.

Citation: Asymptotic Analysis, vol. 26, no. 1, pp. 73-89, 2001