Acta mathematica scientia,Series A ›› 2021, Vol. 41 ›› Issue (5): 1333-1346.

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Blow-Up Properties of Solutions to a Class of Parabolic Type Kirchhoff Equations

Hui Yang*(),Yuzhu Han()   

  1. School of Mathematics, Jilin University, Changchun 130012
  • Received:2020-04-17 Online:2021-10-26 Published:2021-10-08
  • Contact: Hui Yang;
  • Supported by:
    the NSFC(11401252);the Education Department of Jilin Province(JJKH20190018KJ)


In this paper, blow-up properties of solutions to an initial-boundary value problem for a parabolic type Kirchhoff equation are studied. The main results contain two parts. In the first part, we consider this problem with a general diffusion coefficient $M(\|\nabla u\|_2^2)$ and general nonlinearity $f(u)$. A new finite time blow-up criterion is established, and the upper and lower bounds for the blow-up time are also derived. In the second part, we deal with the case that $M(\|\nabla u\|_2^2)=a+b\|\nabla u\|_2^2$ and $f(u)=|u|^{q-1}u$, which was considered in[Computers and Mathematics with Applications, 2018, 75:3283-3297] with $q>3$, where global existence and finite time blow-up of solutions were obtained for subcritical, critical and supercritical initial energy. Their results are complemented in this paper in the sense that $q=3$ will be shown to be critical for the existence of finite time blow-up solutions to this problem.

Key words: Kirchhoff equation, General nonlinearity, Blow-up, Blow-up time, Critical exponent

CLC Number: 

  • O29