Analysis of Some Problems Having Matched Asymptotic Expansion Solutions

Several examples are presented which illustrate some capabilities and some limitations of the method of matched asymptotic expansions for solving evolution equations. The results are listed according to spectral properties of the linear problem resulting near a known steady state of the system. When the linear problem is stable, it is shown that the solution can be written as a (finite) sum of terms, each responding on a different time scale. When the linear problem is unstable, it is shown that the method can be used to determine initial data which excite only decaying modes, and, in the case of bifurcation of new steady states, to construct the new states as well as the transients to them.