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Blow-up similarity solutions of the fourth-order unstable thin film equation

Published online by Cambridge University Press:  01 April 2007

J. D. EVANS ,
V. A. GALAKTIONOV  and
J. R. KING
J. D. EVANS
Affiliation:
Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, UK email: [email protected]
V. A. GALAKTIONOV
Affiliation:
Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, UK email: [email protected]
J. R. KING
Affiliation:
Theoretical Mechanics Section, University of Nottingham, Nottingham NG7 2RD, UK email: [email protected]
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Abstract

We study blow-up behaviour of solutions of the fourth-order thin film equation which contains a backward (unstable) diffusion term. Our main goal is a detailed study of the case of the first critical exponent where N ≥ 1 is the space dimension. We show that the free-boundary problem with zero contact angle and zero-flux conditions admits continuous sets (branches) of blow-up self-similar solutions. For the Cauchy problem in RN × R+, we detect compactly supported blow-up patterns, which have infinitely many oscillations near interfaces and exhibit a “maximal” regularity there. As a key principle, we use the fact that, for small positive n, such solutions are close to the similarity solutions of the semilinear unstable limit Cahn-Hilliard equation which are better understood and have been studied earlier [19]. We also discuss some general aspects of formation of self-similar blow-up singularities for other values of p.

Type
Papers
Information
European Journal of Applied Mathematics , Volume 18 , Issue 2 , April 2007 , pp. 195 - 231
Copyright
Copyright © Cambridge University Press 2007

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