Geodesic
Asymptotic behaviour of the porous media equation in domains with holes
Cristina Brändle1 ; Fernando Quirós1 orcid ; Juan Luis Vázquez1
1Universidad Autónoma de Madrid, Spain
Interfaces and free boundaries, Tome 9 (2007) no. 2, pp. 211-232

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DOI

The paper deals with the asymptotic behaviour of solutions to the porous media equation, ut​=Δum, m>1, in an exterior domain, Ω, which excludes one or several holes, and with zero Dirichlet data on ∂Ω. When the space dimension is three or more this behaviour is given by a Barenblatt function away from the fixed boundary ∂Ω and near the free boundary. The asymptotic behaviour of the free boundary is given by the same Barenblatt function. On the other hand, if the solution is scaled according to its decay factor, away from the free boundary and close to the holes it behaves like a function whose m-th power is harmonic and vanishes on ∂Ω. The height of such a function is determined by matching with the Barenblatt solution representing the outer behaviour. The inner and the outer behaviour can be presented in a unified way through a suitable global approximation.
DOI : 10.4171/ifb/162
Classification : 35-XX, 65-XX, 76-XX, 92-XX
Mots-clés : Porous media equation, exterior domain, asymptotic behaviour, free boundary, matched asymptotics

Cristina Brändle  1   ; Fernando Quirós  1   ; Juan Luis Vázquez  1

1 Universidad Autónoma de Madrid, Spain 
Cristina Brändle; Fernando Quirós; Juan Luis Vázquez. Asymptotic behaviour of the porous media equation in domains with holes. Interfaces and free boundaries, Tome 9 (2007) no. 2, pp. 211-232. doi: 10.4171/ifb/162
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     title = {Asymptotic behaviour of the porous media equation in domains with holes},
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     pages = {211--232},
     year = {2007},
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     url = {http://geodesic.mathdoc.fr/articles/10.4171/ifb/162/}
}
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