Communications in Mathematical Sciences

Volume 17 (2019)

Number 3

Special solutions to a fourth-order nonlinear parabolic equation in non-divergence form

Pages: 817 – 833



Xiangsheng Xu (Department of Mathematics and Statistics, Mississippi State University, Mississippi State, M.S., U.S.A.)


In this paper we study a crystal surface model first proposed by H. Al Hajj Shehadeh, R.V. Kohn, and J. Weare (Physica D, 240:1771–1784, 2011). By seeking a solution of a particular function form, we are led to a boundary value problem for a fourth-order nonlinear elliptic equation. The mathematical challenge of the problem is due to the fact that the degeneracy in the equation is directly imposed by one of the two boundary conditions. An existence theorem is established in which a meaningful mathematical interpretation of one of the boundary conditions remains open. Our proof seems to suggest that this is unavoidable. We also obtain self-similar solutions to the crystal surface model which are positive and unbounded. This is in sharp contrast with the linear biharmonic heat equation.


existence, nonlinear fourth-order elliptic equations, degeneracy, crystal surface models

2010 Mathematics Subject Classification

35D30, 35J40, 35J66, 35K41, 35K65

Received 28 August 2018

Accepted 2 February 2019

Published 30 August 2019