Abstract: | The title of this paper states precisely what the subject is. The first part of the paper concerns the radially-symmetric
problem in the exterior of the unit ball. It is shown that in time the solution of the problem converges to one of two specific
self-similar solutions of the porous media equation, dependent upon the dimensionality of the problem. Moreover, the free
boundary of the solution converges to that of the self-similar solution. The critical space dimension is two, for which there
is no distinction between the self-similar solutions, and the form of the convergence is exceptional. The technique used is
a comparison principle involving a variable that is a weighted integral of the solution. The second part of the paper is devoted
to the problem in an arbitrary spatial domain with no conditions of symmetry. A special invariance principle and the results
obtained for the radially-symmetric case are used to determine the large-time behaviour of solutions and their free boundaries.
This behaviour is decidedly different from when the boundary data are fixed and not homogeneous. |