Proceedings of the Royal Society of Edinburgh: Section A Mathematics

Research Article

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

Luis Alvareza1 and Jesus Ildefonso Diaza2*

a1 Depto. de Informatica y Sistemas, Univ. de Las Palmas, 35017, Las Palmas, Spain

a2 Depto. de Matematica Aplicada, Univ. Complutense de Madrid, 28040 Madrid, Spain

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.

(Received March 10 1992)

Footnotes

* Partially sponsored by the DGICYT (Spain) under project number PB90/0620.