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

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_*;H_per⁴(0,1))∩L^∞(0,t_*;H_per²(0,1)). We also present some numerical computations confirming the blow‐up scenario.