%A Hui Yang,Yuzhu Han %T Blow-Up Properties of Solutions to a Class of Parabolic Type Kirchhoff Equations %0 Journal Article %D 2021 %J Acta mathematica scientia,Series A %R %P 1333-1346 %V 41 %N 5 %U {http://121.43.60.238/sxwlxbA/CN/abstract/article_16492.shtml} %8 2021-10-26 %X

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.