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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Nontrivial compact blow-up sets of smaller dimension
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by Mayte Pérez-Llanos and Julio D. Rossi PDF
Proc. Amer. Math. Soc. 136 (2008), 593-596 Request permission


We provide examples of solutions to parabolic problems with nontrivial blow-up sets of dimension strictly smaller than the space dimension. To this end we just consider different diffusion operators in different variables, for example, $u_t= (u^m)_{xx} + u_{yy} + u^m$ or $u_t = (|u_x|^{p-2} u_x)_x + u_{yy} + u^{p-1}$. For both equations, we prove that there exists a solution that blows up in the segment $B(u) = [-L,L] \times \{ 0 \} \subset \mathbb {R}^2$.
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Additional Information
  • Mayte Pérez-Llanos
  • Affiliation: Departamento de Matemáticas, Universidad Carlos III de Madrid, 28911 Leganés, Spain
  • Email:
  • Julio D. Rossi
  • Affiliation: Departamento de Matemáticas, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Argentina
  • MR Author ID: 601009
  • ORCID: 0000-0001-7622-2759
  • Email:
  • Received by editor(s): November 8, 2006
  • Published electronically: October 24, 2007
  • Additional Notes: The first author is partially supported by DGICYT grant PB94-0153 (Spain).
    The second author is partially supported by ANPCyT PICT 5009, UBA X066 and CONICET (Argentina).
  • Communicated by: David S. Tartakoff
  • © Copyright 2007 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 136 (2008), 593-596
  • MSC (2000): Primary 35B40, 35K65, 35J25, 35J60
  • DOI:
  • MathSciNet review: 2358500