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Modulation Theory for the Blowup of Vector-Valued Nonlinear Heat Equations

Authors
Journal
Journal of Differential Equations
0022-0396
Publisher
Elsevier
Publication Date
Volume
116
Issue
1
Identifiers
DOI: 10.1006/jdeq.1995.1031

Abstract

Abstract This paper is concerned with the blowup of solutions of the nonlinear vector-valued heat equation U t − ΔU = |U| p − 1U, U(0) = U 0 , where U( x, t) = ( u 1( x, t), ..., u m ( x, t)) is a vector-valued function from R n × (0, T) to R m and 1 < p < (3 n + 8)/(3 n − 4). Working with the equation in similarity variables, and using modulation theory and ideas from center manifold theory, we obtain the asymptotic behavior of U in a backward space-time parabola near any blowup point.

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