Nontrivial compact blow-up sets of smaller dimension

Pérez-Llanos, Mayte; Rossi, Julio

doi:10.1090/S0002-9939-07-09028-4

Nontrivial compact blow-up sets of smaller dimension
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by Mayte Pérez-Llanos and Julio D. Rossi

Proc. Amer. Math. Soc. 136 (2008), 593-596

DOI: https://doi.org/10.1090/S0002-9939-07-09028-4

Published electronically: October 24, 2007

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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 $u_t = (|u_x|^{p-2} u_x)_x + u_{yy} + u^{p-1}$. 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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