Similarity, attraction and initial conditions in an example of nonlinear diffusion

Abstract

Similarity solutions play an important role in many fields of science.
The recent book of Barenblatt (1996) discusses many
examples.
Often, outstanding unresolved issues are whether a similarity solution
is dynamically attractive, and if it is, to what particular solution
does the system evolve.
By recasting the dynamic problem in a form to which centre manifold
theory may be applied, based upon a transformation by
Wayne (1994), we may resolve these issues in many cases.
For definiteness we illustrate the principles by discussing the
application of centre manifold theory to a particular nonlinear
diffusion problem arising in filtration.
Theory constructs the similarity solution, confirms its relevance, and
determines the correct solution for any compact initial condition.
The techniques and results we discuss are applicable to a wide range
of similarity problems.