The asymptotics of extinction in nonlinear diffusion reaction equations
R. E. Grundy
(Received 15 November 1990; revised 12 January 1991)
Abstract
In this paper we consider the asymptotics of extinction for the nonlinear diffusion reaction equation
¶u ¶ 2(um)
¾ = ¾¾ - up , m > 1, 0 < p < 1,
¶t ¶x2
with non-negative initial data possessing finite support. For t > 0, both solution and support vanish as t ® T
and x ® x0. With T as the extinction time
we construct the asymptotic solution as t = T - t ® 0
near the extinction point x0 using matched expansions. Taking x0 = 0,
we first form an outer expansion valid when h = xt -(m - p) / 2(1 - p) = O(1).
This is nonuniformly valid for large |h| and has to be replaced by an intermediate
expansion valid for |x| = O(t -1/l 0) where
l0 is an even integer greater than unity. If p + m ³ 2
this expansion is uniformly valid while for p + m < 2, there are regions near the edge of the support where
diffusion becomes important. The zero order solution in these inner regions is discussed numerically.
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Author
R. E. Grundy
Department of Mathematical Sciences, University of St. Andrews, Scotland.
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