Asymptotics near an extinction point for some semilinear heat equations



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Herrero, Miguel A. and 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


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 ·

Item Type:Book Section
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Proceedings of the Fifth International Colloquium on Free Boundary Problems: Theory and Applications held in Montreal, Quebec, June 13–22, 1990

Uncontrolled Keywords:Extinction point; sublinear equation
Subjects:Sciences > Mathematics > Differential equations
ID Code:22697
Deposited On:03 Sep 2013 14:23
Last Modified:12 Dec 2018 15:08

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