# Asymptotics near an extinction point for some semilinear heat equations

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Herrero, Miguel A. y Velázquez, J.J. L. (1993) Asymptotics near an extinction point for some semilinear heat equations. In Emerging applications in free boundary problems. Pitman Research Notes in Mathematics Series (280). Longman Scientific and Technical, Harlow, pp. 188-194. ISBN 0-582-08768-6

## Resumen

We consider the Cauchy problem u t -u xx +u p =0,x∈ℝ,t>0,u(x,0)=u 0 (x),x∈ℝ, where 0<p<1, u 0 (x) is continuous, nonnegative, and bounded, with a single maximum at x=0 and such that u 0 (-x)=u 0 (x) for any x, lim x→∞ u 0 (x)=0. It is well known that the solution has some features which are absent in the superlinear case p≥1. For instance, there exists T>0 such that u(x,t)¬≡0 if t<T and u(x,t)≡0 for t≥T. Moreover, there exists a continuous curve ζ(t) such that lim t→T ζ(t)=0 and Ω + (t)={x: u(x,t)>0}={x: -ζ(t)<x<ζ(t)}. In this communication we describe some asymptotic results. Namely, lim t→T (T-t) 1 1-p u(ξ(T-t) 1/2 |ln(T-t)| 1/2 ,t)=(1-p) 1 1-p 1 - (1-p) 4p ξ 2 + 1 1-p uniformly on sets |ξ|≤C(T-t) 1/2 |ln(T-t)| 1/2 , and lim t→T ζ(t) (T-t) 1/2 |ln(T-t)|=4p 1-p 1/2 ·

Tipo de documento: Sección de libro Proceedings of the Fifth International Colloquium on Free Boundary Problems: Theory and Applications held in Montreal, Quebec, June 13–22, 1990 Extinction point; sublinear equation Ciencias > Matemáticas > Ecuaciones diferenciales 22697 03 Sep 2013 14:23 22 Nov 2013 19:23

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