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Removable singularities of semilinear parabolic equations

Author: Kentaro Hirata
Journal: Proc. Amer. Math. Soc. 142 (2014), 157-171
MSC (2010): Primary 35B65; Secondary 35K91, 35K05
Published electronically: September 5, 2013
MathSciNet review: 3119191
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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=\vert u\vert^{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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Additional Information

Kentaro Hirata
Affiliation: Faculty of Education and Human Studies, Akita University, Akita 010-8502, Japan
Address at time of publication: Department of Mathematics, Graduate School of Science, Hiroshima University, Higashi-Hiroshima, 739-8526, Japan

Keywords: Removable singularities, blow-up, semilinear parabolic equation, heat equation.
Received by editor(s): February 16, 2011
Received by editor(s) in revised form: February 22, 2012
Published electronically: September 5, 2013
Additional Notes: This work was partially supported by Grant-in-Aid for Young Scientists (B) (No. 22740081), Japan Society for the Promotion of Science.
Communicated by: Tatiana Toro
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.