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 (1997), 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.