Single-point blow-up for a doubly degenerate parabolic equation with nonlinear source.

*(English)*Zbl 1223.35089The authors consider positive solutions of the Cauchy problem for the doubly degenerate equation

$${u}_{t}-\text{div}\left(\right|\nabla {u}^{m}{|}^{\sigma}\nabla {u}^{m})={u}^{\beta},\phantom{\rule{1.em}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}x\in {\mathbb{R}}^{N},\phantom{\rule{4pt}{0ex}}t>0,$$

where $\sigma >0$, $m>1$, $\beta >m(1+\sigma )$, $N\ge 1$. The authors study the set of blow-up points and the behavior of $u$ at the blow-up point. They prove single-point blow-up for a large class of radial decreasing solutions. The upper and lower estimates of the blow-up solution near the single blow-up point are also obtained.

Reviewer: Yuanyuan Ke (Beijing)

##### MSC:

35B44 | Blow-up (PDE) |

35K65 | Parabolic equations of degenerate type |

35B09 | Positive solutions of PDE |

35K59 | Quasilinear parabolic equations |