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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Nontrivial compact blow-up sets of smaller dimension


Authors: Mayte Pérez-Llanos and Julio D. Rossi
Journal: Proc. Amer. Math. Soc. 136 (2008), 593-596
MSC (2000): Primary 35B40, 35K65, 35J25, 35J60
Published electronically: October 24, 2007
MathSciNet review: 2358500
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Abstract: 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 = (\vert u_x\vert^{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 \RR^2$.


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Additional Information

Mayte Pérez-Llanos
Affiliation: Departamento de Matemáticas, Universidad Carlos III de Madrid, 28911 Leganés, Spain
Email: mtperez@math.uc3m.es

Julio D. Rossi
Affiliation: Departamento de Matemáticas, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Argentina
Email: jrossi@dm.uba.ar

DOI: http://dx.doi.org/10.1090/S0002-9939-07-09028-4
PII: S 0002-9939(07)09028-4
Keywords: Blow-up sets, $p$-Laplacian, porous media.
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
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.