Atkinson and Peletier (2,3) have considered similarity solutions of the differential equation
![](//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS030500410005636X/resource/name/S030500410005636X_eqnU1.gif?pub-status=live)
where the function k(s) is defined, real and continuous for s ≥ 0 and k(s) > 0 if s > 0 (in (2) k(0) = 0 is also assumed). In particular they look for similarity solutions of the form u(x, t) = f(η) where η = x(t+l)−½ with boundary conditions f(0) = A and
. They show that if k(s) satisfies the condition
![](//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS030500410005636X/resource/name/S030500410005636X_eqnU2.gif?pub-status=live)
then for any A > 0 there is a unique similarity solution which is non-negative and has compact support in [0, ∞). They also show in (2) that
![](//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS030500410005636X/resource/name/S030500410005636X_eqnU3.gif?pub-status=live)
is a necessary condition for the solution to have compact support. In (3) they prove existence of similarity solutions when
![](//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS030500410005636X/resource/name/S030500410005636X_eqnU4.gif?pub-status=live)
and show that in this case the similarity solution has the property that f(η) > 0 for all η > 0.