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On the initial growth of interfaces in reaction–diffusion equations with strong absorption

Published online by Cambridge University Press:  14 November 2011

Luis Alvarez
Affiliation:
Depto. de Informatica y Sistemas, Univ. de Las Palmas, 35017, Las Palmas, Spain
Jesus Ildefonso Diaz
Affiliation:
Depto. de Matematica Aplicada, Univ. Complutense de Madrid, 28040 Madrid, Spain

Extract

We study the initial growth of the interfaces of non-negative local solutions of the equation ut = (um)xx−λuq when m ≧ 1 and 0<q <1. We show that if with C < C0, for some explicit C0 = C0(λ, m, q), then the free boundary Ϛ(t) = sup {x:u(x, t) > 0} is a ‘heating front’. More precisely Ϛ(t) ≧at(m−q)/2(1−q) for any t small enough and for some a>0. If on the contrary, with C<C0, then Ϛ(t) is a ‘cooling front’ and in fact Ϛ(t) ≧ −atm−q)/2(1−q) for any t small enough and for some a > 0. Applications to solutions of the associated Cauchy and Dirichlet problems are also given.

Type
Research Article
Copyright
Copyright © Royal Society of Edinburgh 1993

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