Resumen de Asymptotic behaviour of the porous media equation in an exterior domain

Fernando Quirós Gracián , Juan Luis Vázquez

• 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.