M ar 1 99 8 Similarity , attraction and initial conditions in an example of nonlinear diffusion

Similarity solutions play an important role in many fields of science. The recent book of Barenblatt [2] 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 [10], 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.