Existence and finite‐time blow‐up for the solution to a thin‐film surface evolution problem

Article type: Research Article

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

Affiliations: Laboratoire d'Applications des Mathématiques, Université de Poitiers, Boulevard Marie et Pierre Curie, Téléport 2, BP 30179, 86962 Futuroscope Chasseneuil cedex, France | Laboratoire de Combustion et de Détonique, UPR 9028 CNRS, ENSMA, BP 109, 86960 Futuroscope, France | Département de Mathématiques Appliquées et Informatique, Faculté des Sciences et Techniques, BP 523, 23000 Béni‐Mellal, Maroc

Note: [] Corresponding author. E‐mail: [email protected].

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*;Hper4(0,1))∩L∞(0,t*;Hper2(0,1)). We also present some numerical computations confirming the blow‐up scenario.

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

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