Asymptotic behaviour of the porous media equation in an exterior domain
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Fernando Quirós [1] ; Juan Luis Vázquez [1]
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[1] Universidad Autónoma de Madrid
Universidad Autónoma de Madrid
Madrid, España
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- Localización: Annali della Scuola Normale Superiore di Pisa. Classe di scienze, ISSN 0391-173X, Vol. 28, Nº 2, 1999, págs. 183-227
- Idioma: inglés
- Enlaces
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Resumen
We study the asymptotic behaviour of weak solutions u (x, t) to the porous media equation in exterior domains with nontrivial boundary data which are constant in time. We prove that, when the space dimension is greater than one, this behaviour is given in the interior of the positivity set by a function, P (x), which has the same value as u in the fixed boundary and such that its m -th power, P’ (x), is harmonic in the exterior domain. We also prove that near the free boundary the asymptotic behaviour is given by a radial, self-similar solution of the porous media equation which is singular at the origin for all times. There is a whole family of such singular self-similar solutions. The precise one giving the asymptotic behaviour is determined through a process of matched asymptotics.
We also show that the free boundary approaches a sphere as t ~ oo, and give the asymptotic growth rate for the radius.
