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A semilinear heat equation with singular initial data

Published online by Cambridge University Press:  14 November 2011

Minkyu Kwak
Minkyu Kwak
Affiliation:
Department of Mathematics, Chonnam National University, Kwangju, 500-757, Korea
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Abstract

We first prove existence and uniqueness of non-negative solutions of the equation

in in the range 1 < p < 1 + 2/N, when initial data u(x, 0) = a|x|−2(p−1), x ≠ 0, for a > 0. It is proved that the maximal and minimal solutions are self-similar with the form

where g = ga satisfies

After uniqueness is proved, the asymptotic behaviour of solutions of

is studied. In particular, we show that

The case for a = 0 is also considered and a sharp decay rate of the above equation is derived. In the final, we reveal existence of solutions of the first and third equations above, which change sign.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 128 , Issue 4 , 1998 , pp. 745 - 758
Copyright
Copyright © Royal Society of Edinburgh 1998

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