Gradient flow approach to an exponential thin film equation: global existence and latent singularity

¹ Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong.
² Department of Mathematics, Duke University, Durham, NC 27708, USA.
³ Department of Mathematics and Department of Physics, Duke University, Durham, NC 27708, USA.
⁴ Department of Mathematical Sciences, Lakehead University, Thunder Bay, ON P7B5E1, Canada.
⁵ Department of Mathematics and Statistics, McGill University, Montreal, QC H3A0B9, Canada.

^* Corresponding author: gy2012yg@gmail.com

Received: 18 October 2017
Accepted: 12 June 2018

Abstract

In this work, we study a fourth order exponential equation, u_t = Δe^−Δu derived from thin film growth on crystal surface in multiple space dimensions. We use the gradient flow method in metric space to characterize the latent singularity in global strong solution, which is intrinsic due to high degeneration. We define a suitable functional, which reveals where the singularity happens, and then prove the variational inequality solution under very weak assumptions for initial data. Moreover, the existence of global strong solution is established with regular initial data.