The value of the critical exponent for reaction-diffusion equations in cones |
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Authors: | Howard A Levine Peter Meier |
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Institution: | (1) Department of Mathematics, Iowa State University, Ames |
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Abstract: | Let D R
N
be a cone with vertex at the origin i.e., D = (0, )x where S
N–1 and x D if and only if x = (r, ) with r=¦x¦, . We consider the initial boundary value problem: u
t
= u+u
p
in D×(0, T), u=0 on Dx(0, T) with u(x, 0)=u
0(x) 0. Let 1 denote the smallest Dirichlet eigenvalue for the Laplace-Beltrami operator on and let
+ denote the positive root of ( +N–2) =
1. Let p
* = 1 + 2/(N + +). If 1 < p < p
*, no positive global solution exists. If p>p
*, positive global solutions do exist. Extensions are given to the same problem for u
t= +¦x¦
u
p
.This research was supported in part by the Air Force Office of Scientific Research under Grant # AFOSR 88-0031 and in part by NSF Grant DMS-8 822 788. The United States Government is authorized to reproduce and distribute reprints for governmental purposes not withstanding any copyright notation therein. |
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Keywords: | |
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