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Published online by Cambridge University Press: 24 October 2008
Atkinson and Peletier (2,3) have considered similarity solutions of the differential equation
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
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
is a necessary condition for the solution to have compact support. In (3) they prove existence of similarity solutions when
and show that in this case the similarity solution has the property that f(η) > 0 for all η > 0.