## 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

## 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 $u_t = (|u_x|^{p-2} u_x)_x + u_{yy} + u^{p-1}$. For both equations, we prove that there exists a solution that blows up in the segment $B(u) = [-L,L] \times \{ 0 \} \subset \mathbb {R}^2$.## References

- Xu-Yan Chen and Hiroshi Matano,
*Convergence, asymptotic periodicity, and finite-point blow-up in one-dimensional semilinear heat equations*, J. Differential Equations**78**(1989), no. 1, 160–190. MR**986159**, DOI 10.1016/0022-0396(89)90081-8 - Carmen Cortázar, Manuel del Pino, and Manuel Elgueta,
*On the blow-up set for $u_t=\Delta u^m+u^m$, $m>1$*, Indiana Univ. Math. J.**47**(1998), no. 2, 541–561. MR**1647932**, DOI 10.1512/iumj.1998.47.1399 - Carmen Cortázar, Manuel Del Pino, and Manuel Elgueta,
*Uniqueness and stability of regional blow-up in a porous-medium equation*, Ann. Inst. H. Poincaré C Anal. Non Linéaire**19**(2002), no. 6, 927–960 (English, with English and French summaries). MR**1939091**, DOI 10.1016/S0294-1449(02)00107-5 - Carmen Cortázar, Manuel Elgueta, and Patricio Felmer,
*Symmetry in an elliptic problem and the blow-up set of a quasilinear heat equation*, Comm. Partial Differential Equations**21**(1996), no. 3-4, 507–520. MR**1387457**, DOI 10.1080/03605309608821194 - Victor A. Galaktionov and Juan L. Vázquez,
*The problem of blow-up in nonlinear parabolic equations*, Discrete Contin. Dyn. Syst.**8**(2002), no. 2, 399–433. Current developments in partial differential equations (Temuco, 1999). MR**1897690**, DOI 10.3934/dcds.2002.8.399 - 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**, DOI 10.1002/cpa.3160450303 - Carl E. Mueller and Fred B. Weissler,
*Single point blow-up for a general semilinear heat equation*, Indiana Univ. Math. J.**34**(1985), no. 4, 881–913. MR**808833**, DOI 10.1512/iumj.1985.34.34049 - Alexander A. Samarskii, Victor A. Galaktionov, Sergei P. Kurdyumov, and Alexander P. Mikhailov,
*Blow-up in quasilinear parabolic equations*, De Gruyter Expositions in Mathematics, vol. 19, Walter de Gruyter & Co., Berlin, 1995. Translated from the 1987 Russian original by Michael Grinfeld and revised by the authors. MR**1330922**, DOI 10.1515/9783110889864.535 - Fred B. Weissler,
*Single point blow-up for a semilinear initial value problem*, J. Differential Equations**55**(1984), no. 2, 204–224. MR**764124**, DOI 10.1016/0022-0396(84)90081-0 - Hatem Zaag,
*Determination of the curvature of the blow-up set and refined singular behavior for a semilinear heat equation*, Duke Math. J.**133**(2006), no. 3, 499–525. MR**2228461**, DOI 10.1215/S0012-7094-06-13333-1 - Hatem Zaag,
*On the regularity of the blow-up set for semilinear heat equations*, Ann. Inst. H. Poincaré C Anal. Non Linéaire**19**(2002), no. 5, 505–542 (English, with English and French summaries). MR**1922468**, DOI 10.1016/S0294-1449(01)00088-9

## 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
- MR Author ID: 601009
- ORCID: 0000-0001-7622-2759
- Email: jrossi@dm.uba.ar
- 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: https://doi.org/10.1090/S0002-9939-07-09028-4
- MathSciNet review: 2358500