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Self-similar boundary blow-up for higher-order quasilinear parabolic equations

Published online by Cambridge University Press:  12 July 2007

V. A. Galaktionov  and
A. E. Shishkov
V. A. Galaktionov
Affiliation:
Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, UK and Keldysh Institute of Applied Mathematics, Miusskaya Sq. 4, 125047 Moscow, Russia ([email protected])
A. E. Shishkov
Affiliation:
Institute of Applied Mathematics and Mechanics, National Academy of Sciences of Ukraine, R. Luxemburg str. 74, 83114 Donetsk, Ukraine ([email protected])
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Abstract

We study evolution properties of boundary blow-up for 2mth-order quasilinear parabolic equations in the case where, for homogeneous power nonlinearities, the typical asymptotic behaviour is described by exact or approximate self-similar solutions. Existence and asymptotic stability of such similarity solutions are established by energy estimates and contractivity properties of the rescaled flows.

Further asymptotic results are proved for more general equations by using energy estimates related to Saint-Venant's principle. The established estimates of propagation of singularities generated by boundary blow-up regimes are shown to be sharp by comparing with various self-similar patterns.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 135 , Issue 6 , December 2005 , pp. 1195 - 1227
Copyright
Copyright © Royal Society of Edinburgh 2005

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