Asymptotic Analysis - Volume 38, issue 2

Authors: Boutat, M. | D'Angelo, Y. | Hilout, S. | Lods, V.

Article Type: Research Article

Abstract: The aim of this paper is to study the evolution of the surface of a crystal structure, constituted by a linearly elastic substrate and a thin film. After appropriate scalings, a formal asymptotical expansion of the displacement, under some assumptions, yields the following nonlinear PDE \begin{equation}\frac{\curpartial h}{\curpartial t}=-\frac{\curpartial ^{2}}{\curpartial x^{2}}\big((1-\theta h)h''-\frac{\theta }{2}h'^{2}\big),\end{equation} where θ is a coefficient related to the crystal, and h(t,x) describes the spatial evolution of the film surface. We give here some results about the finite‐time blow‐up and prove the existence and uniqueness of a solution in L2 (0,t* ;Hper 4 (0,1))∩L∞ (0,t* ;Hper …2 (0,1)). We also present some numerical computations confirming the blow‐up scenario. Show more

Keywords: nonlinear partial differential equations, finite time blow‐up, initial boundary value problem, local solution

Citation: Asymptotic Analysis, vol. 38, no. 2, pp. 93-128, 2004

Authors: Bourgeat, A. | Pankratov, L. | Panfilov, M.

Article Type: Research Article

Abstract: We study by homogenization the various macroscopic models associated to the diffusion through a fissured media, i.e., a set of low conductivity blocks crossed by a net of highly conducting fissures. According to the fissure thickness, δε, range we obtain different models. For this, first we homogenize by seeking the limit when ε, the small parameter associated to the blocks size, goes to zero and then we study the limit when δ goes to zero. In each situation we prove the convergence to the corresponding macroscopic model.

Citation: Asymptotic Analysis, vol. 38, no. 2, pp. 129-141, 2004

Authors: Kozlov, Vladimir | Maz'ya, Vladimir

Article Type: Research Article

Abstract: It is well known that distributional solutions of an elliptic equation with constant coefficients behave asymptotically near an interior point as sums of polynomials and linear combinations of derivatives of a fundamental solution. We consider a class of quasilinear elliptic systems and give mild conditions ensuring the same asymptotic behaviour. The sharpness of our conditions is illustrated by examples. The results are obtained as corollaries of a general theorem on the asymptotics of solutions to nonlinear ordinary differential equations in Banach spaces.

Citation: Asymptotic Analysis, vol. 38, no. 2, pp. 143-165, 2004

Authors: Perla Menzala, G. | Muñoz Rivera, Jaime E. | Konotop, V.V.

Article Type: Research Article

Abstract: In this paper we study a nonlinear lattice with memory and show that the problem is globally well posed. Furthermore we find uniform rates of decay of the total energy. Our main result shows that the memory effect is strong enough to produce a uniform rate of decay. That is, if the relaxation function decays exponentially then, the corresponding solution also decays exponentially. When the relaxation kernel decays polynomially then, the solution also decays polynomially as time goes to infinity.

Citation: Asymptotic Analysis, vol. 38, no. 2, pp. 167-185, 2004