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The asymptotic behaviour of the solutions of the Kassoy problem with a modified source term

Published online by Cambridge University Press:  14 November 2011

C. Budd  and
Y. Qi
C. Budd
Affiliation:
Oxford University, Computing Laboratory, 8-11 Keble Road, Oxford, OX1 3QD, U.K.
Y. Qi
Affiliation:
Oxford University, Computing Laboratory, 8-11 Keble Road, Oxford, OX1 3QD, U.K.
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Synopsis

We study the asymptotic behaviour as x →∞ of the solutions of the ordinary differential equation problem

This equation generalises the ordinary differential equation obtained by studying the blow-up of the similarity solutions of the semilinear parabolic partial differential equation vt=vxx = ev. We show that if λ≦1, all solutions of (*) tend to —∞ as rapidly as the function —exp (x2/4) (E- solutions). However, if λ>1, then there also exists a solution which tends to –∞, like 2λlog(x) (L-solutions). Thus, the case λ = 1, for which (*) reduces tothe Kassoy equation, is the borderline between two quite different forms of asymptotic behaviour of the function u(x).

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 113 , Issue 3-4 , 1989 , pp. 347 - 356
Copyright
Copyright © Royal Society of Edinburgh 1989

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