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Blow up for a diffusion-advection equation

Published online by Cambridge University Press:  14 November 2011

Nicholas D. Alikakos ,
Peter W. Bates  and
Christopher P. Grant
Nicholas D. Alikakos
Affiliation:
Department of Mathematics, University of Tennessee, Knoxville, TN 37996, U.S.A
Peter W. Bates
Affiliation:
Applied Mathematics, National Science Foundation, Washington, D.C. 20550, U.S.A
Christopher P. Grant
Affiliation:
Department of Mathematics, Brigham Young University, Provo, UT 84602, U.S.A
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Synopsis

These results describe the asymptotic behaviour of solutions to a certain non-linear diffusionadvection equation on the unit interval. The “no flux” boundary conditions prescribed result in mass being conserved by solutions and the existence of a mass-parametrised family of equilibria. A natural question is whether or not solutions stabilise to equilibria and if not, whether they blow up in finite time. Here it is shown that for non-linearities which characterise “fast association” there is a criticalmass such that initial data which have supercritical mass must lead to blow up in finite time. It is also shown that there exist initial data with arbitrarily small mass which also lead to blow up in finite time.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 113 , Issue 3-4 , 1989 , pp. 181 - 190
Copyright
Copyright © Royal Society of Edinburgh 1989

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