On the behaviour of blow-up interfaces for an inhomogeneous filtration equation |
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Authors: | GALAKTIONOV V A; KING J R |
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Institution: |
School of Mathematical Sciences, University of Bath Bath BA2 7AY, UK
Department of Theoretical Mechanics, University of Nottingham Nottingham NG7 2RD, UK
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Abstract: | We study the asymptotic behaviour of blow-up interfaces of thesolutions to the one-dimensional nonlinear filtration equationin inhomogeneous media
where m>1 isa constant and (x) = |x| (for |x| 1, with > 2) isa bounded, positive, smooth, and symmetric function. The initialdata are assumed to be smooth, bounded, compactly supported,symmetric, and monotone. It is known that due to the fast decayof the density (x) as |x|![->](http://imamat.oxfordjournals.org/math/rarr.gif) the support of the solution increasesunboundedly in a finite time T. We prove that as t T theinterface behaves like O((T t)b), where the exponentb > 0 (which depends on m and only) is given by a uniqueself-similar solution of the second kind satisfying the equation|x| ut = (um)xx. The corresponding rescaled profilesalso converge. We establish the stability of the self-similarsolution of the second kind for the exponential density (x)=e|x|for |x| 1. We give a formal asymptotic analysis of the blow-upbehaviour for the non-self-similar density (x) = e|x|2.Several exact self-similar solutions and their correspondingasymptotics are constructed. |
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