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**Global existence, finite time blow-up and vacuum isolating phenomena for semilinear parabolic equation with conical degeneration.**
*(English)*
Zbl 1406.35164

Summary: This paper is devoted to studying a semilinear parabolic equation with conical degeneration. First, we extend previous results on the vacuum isolating of solution with initial energy \(J(u_0) < d\), where \(d\) is the mountain pass level. Concretely, we obtain the explicit vacuum region, the global existence region and the blow-up region. Moreover, as far as the blow-up solution is concerned, we estimate the upper bound of the blow-up time and blow-up rate. Second, for all \(p > 1\), we get a new sufficient condition, which demonstrates the finite time blow-up for arbitrary initial energy, and the upper bound estimate of blow-up time is obtained.

### MSC:

35K61 | Nonlinear initial, boundary and initial-boundary value problems for nonlinear parabolic equations |

35B44 | Blow-up in context of PDEs |

35K10 | Second-order parabolic equations |

35K55 | Nonlinear parabolic equations |

35D30 | Weak solutions to PDEs |

### Keywords:

cone Sobolev space; vacuum isolating phenomena; global existence; finite time blow-up; blow-up time; blow-up rate### References:

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