Removable singularities of semilinear parabolic equations

Hirata, Kentaro

doi:10.1090/S0002-9939-2013-11739-9

Removable singularities of semilinear parabolic equations
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by Kentaro Hirata

Proc. Amer. Math. Soc. 142 (2014), 157-171

DOI: https://doi.org/10.1090/S0002-9939-2013-11739-9

Published electronically: September 5, 2013

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Abstract:

This paper extends the recent result due to Hsu (2010) about removable singularities of semilinear parabolic equations. Our result is applicable to solutions of equations of the form $-\Delta u+\partial _t u=|u|^{p-1}u$ with $0\le p<n/(n-2)$. The proof is based on the parabolic potential theory and an iteration argument. Also, we prove that if $0<p<(n+2)/n$, then integral solutions of semilinear parabolic equations with nonlinearity depending on space and time variables and $u^p$ are locally bounded. This implies that the blow-up for continuous solutions is complete.

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