Finite-time singularity formation for the one-dimensional surface growth model

Determine whether finite-time singularities can form from general initial data for the one-dimensional molecular-beam-epitaxy surface growth model u_t+u_{xxxx}+u_{xx}=-(u_x^2)_{xx} on \mathbb{R}.

Background

The paper studies the one-dimensional surface growth model associated with molecular beam epitaxy and analyzes the stability of its inverted-parabola steady solution. In discussing prior numerical observations, the paper notes that solutions may develop singular behavior near local minima, while the authors establish global stability only for sufficiently small perturbations of the inverted parabola.

The unresolved issue concerns arbitrary, rather than perturbatively small, initial data: it is not established whether solutions of the surface growth model can develop singularities in finite time. This question is distinct from the stability result proved in the paper, which applies only to small perturbations of a specific quadratic steady state.