The Porous Medium Equation in an Exterior Domain
Victor A. Galaktionov, Juan Luis Vázquez · Birkhäuser Boston eBooks · 2004
As a second instance of application of the S-Therom in a problem with matched asymptotics, we consider the porous medium equation in an exterior domain with nontrivial boundary data. Assuming that the space dimension is greater than 1 and the boundary data are constant in time, we can describe the large-time behaviour by means of a two-region analysis. In the interior of the positivity set, it is given by a funcyion p(x), which has the same value as u in the fixed boundary and such that its m th power, p m (x), is harmonic in the exterior domain. Near the free boundary the asymptotic behaviour is given by a radial, self-similar solution of the PME which is singular at the origin for all times.There is a whole family of such singular self-similar solutions. The precise one given the asympotic behaviour is determined through matched asympotics. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.