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Saint-Venant's principle in blow-up for higher-order quasilinear parabolic equations

Published online by Cambridge University Press:  12 July 2007

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 (vag@maths.bath.ac.uk)
A. E. Shishkov
Affiliation:
Institute of Applied Mathematics and Mechanics of NAS of Ukraine, R. Luxemburg str. 74, 83114 Donetsk, Ukraine (shishkov@iamm.ac.donetsk.ua)
Corresponding

Abstract

We prove localization estimates for general 2mth-order quasilinear parabolic equations with boundary data blowing up in finite time, as tT. The analysis is based on energy estimates obtained from a system of functional inequalities expressing a version of Saint-Venant's principle from the theory of elasticity. We consider a special class of parabolic operators including those having fixed orders of algebraic homogenuity p > 0. This class includes the second-order heat equation and linear 2mth-order parabolic equations (p = 1), as well as many other higher-order quasilinear ones with p ≠ 1. Such homogeneous equations can be invariant under a group of scaling transformations, but the corresponding least-localized regional blow-up regimes are not group invariant and exhibit typical exponential singularities ~ e(Tt)−γ → ∞ as tT, with the optimal constant γ = 1/[m(p + 1) − 1] > 0. For some particular equations, we study the asymptotic blow-up behaviour described by perturbed first-order Hamilton–Jacobi equations, which shows that general estimates of exponential type are sharp.

Type
Research Article
Copyright
Copyright © Royal Society of Edinburgh 2003

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