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VOL. 30 | 1992 Waiting-time behaviour for a fourth-order nonlinear diffusion equation
N. F. Smyth

Editor(s) Robert S. Anderssen, Amiya K. Pani


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.


Published: 1 January 1992
First available in Project Euclid: 18 November 2014

zbMATH: 0784.35047
MathSciNet: MR1210760

Rights: Copyright © 1992, Centre for Mathematics and its Applications, Mathematical Sciences Institute, The Australian National University. This book is copyright. Apart from any fair dealing for the purpose of private study, research, criticism or review as permitted under the Copyright Act, no part may be reproduced by any process without permission. Inquiries should be made to the publisher.


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