Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

Request Permissions   Purchase Content 


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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

  • [1] P. Baras and L. Cohen, Complete blow-up after $ T_{{\rm max}}$ for the solution of a semilinear heat equation, J. Funct. Anal. 71 (1987), no. 1, 142-174. MR 879705 (88e:35105),
  • [2] Haïm Brézis and Laurent Véron, Removable singularities for some nonlinear elliptic equations, Arch. Rational Mech. Anal. 75 (1980/81), no. 1, 1-6. MR 592099 (83i:35071),
  • [3] Joseph L. Doob, Classical potential theory and its probabilistic counterpart, Classics in Mathematics, Springer-Verlag, Berlin, 2001. Reprint of the 1984 edition. MR 1814344 (2001j:31002)
  • [4] Yoshikazu Giga and Robert V. Kohn, Nondegeneracy of blowup for semilinear heat equations, Comm. Pure Appl. Math. 42 (1989), no. 6, 845-884. MR 1003437 (90k:35034),
  • [5] Shu-Yu Hsu, Removable singularities of semilinear parabolic equations, Adv. Differential Equations 15 (2010), no. 1-2, 137-158. MR 2588392 (2011a:35274)
  • [6] Kin Ming Hui, Another proof for the removable singularities of the heat equation, Proc. Amer. Math. Soc. 138 (2010), no. 7, 2397-2402. MR 2607869 (2011c:35213),
  • [7] P.-L. Lions, Isolated singularities in semilinear problems, J. Differential Equations 38 (1980), no. 3, 441-450. MR 605060 (82g:35040),
  • [8] Frank Merle, Solution of a nonlinear heat equation with arbitrarily given blow-up points, Comm. Pure Appl. Math. 45 (1992), no. 3, 263-300. MR 1151268 (92k:35160),
  • [9] Luc Oswald, Isolated positive singularities for a nonlinear heat equation, Houston J. Math. 14 (1988), no. 4, 543-572. MR 998457 (90k:35128)
  • [10] Pavol Quittner and Frédérique Simondon, A priori bounds and complete blow-up of positive solutions of indefinite superlinear parabolic problems, J. Math. Anal. Appl. 304 (2005), no. 2, 614-631. MR 2126555 (2005m:35151),
  • [11] Steven D. Taliaferro, Isolated singularities of nonlinear parabolic inequalities, Math. Ann. 338 (2007), no. 3, 555-586. MR 2317931 (2008h:35174),
  • [12] J. J. L. Velázquez, Higher-dimensional blow up for semilinear parabolic equations, Comm. Partial Differential Equations 17 (1992), no. 9-10, 1567-1596. MR 1187622 (93k:35044),
  • [13] Laurent Véron, Singular solutions of some nonlinear elliptic equations, Nonlinear Anal. 5 (1981), no. 3, 225-242. MR 607806 (82f:35076),
  • [14] N. A. Watson, Green functions, potentials, and the Dirichlet problem for the heat equation, Proc. London Math. Soc. (3) 33 (1976), no. 2, 251-298. MR 0425145 (54 #13102)
  • [15] Qi S. Zhang, The boundary behavior of heat kernels of Dirichlet Laplacians, J. Differential Equations 182 (2002), no. 2, 416-430. MR 1900329 (2003f:35006),

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 35B65, 35K91, 35K05

Retrieve articles in all journals with MSC (2010): 35B65, 35K91, 35K05

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.

American Mathematical Society