Proceedings of the Centre for Mathematics and its Applications

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

N. F. Smyth

Full-text: Open access

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

Source
Mini-Conference on Free and Moving Boundary and Diffusion Problems. Amiya K. Pani and Robert S. Anderssen, eds. Proceedings of the Centre for Mathematics and its Applications, v. 30. (Canberra AUS: Centre for Mathematics and its Applications, Mathematical Sciences Institute, The Australian National University, 1992), 212-217

Dates
First available in Project Euclid: 18 November 2014

Permanent link to this document
https://projecteuclid.org/ euclid.pcma/1416323081

Mathematical Reviews number (MathSciNet)
MR1210760

Zentralblatt MATH identifier
0784.35047

Citation

Smyth, N. F. Waiting-time behaviour for a fourth-order nonlinear diffusion equation. Mini-Conference on Free and Moving Boundary and Diffusion Problems, 212--217, Centre for Mathematics and its Applications, Mathematical Sciences Institute, The Australian National University, Canberra AUS, 1992. https://projecteuclid.org/euclid.pcma/1416323081


Export citation