## Proceedings of the Centre for Mathematics and its Applications

### Waiting-time behaviour for a fourth-order nonlinear diffusion equation

N. F. Smyth

#### Abstract

The fourth order nonlinear diffusion equation $u_t + (u^nu_{xxx})_x = 0 (n \gt 0)$ governs a number of important physical processes, such as the flow of a surface tension dominated thin liquid film and the diffusion of dopant in semiconductors. This equation will be analysed using a perturbation scheme in the limit of small $n (ie 0 \lt n \ll 1)$. In this limit, the solution is determined by a system of nonlinear hyperbolic equations. An analysis of the solution shows that if the initial condition is of compact support, the solution does not move outside of its initial domain. Shocks, corresponding to jumps in $u_x$, can form in the solution. An examination of the shock jump condition shows that a shock cannot propagate outside of the domain of the initial condition. It is concluded that all solutions of $u_t + (u^nu_{xxx})_x = 0 (n \gt 0)$ for $0 \lt n \ll 1$ are waiting-time solutions.

#### Article information

Dates
First available in Project Euclid: 18 November 2014