Nontrivial compact blow-up sets of smaller dimension

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 $B(u) = [-L,L] \times \{ 0 \} \subset \RR^2$.