Blow-up properties of solutions to a class of p-Kirchhoff evolution equations

Abstract

This paper is devoted to an initial-boundary value problem for a class of $ p $-Kirchhoff type parabolic equations. Firstly, we consider this problem with a general nonlocal coefficient $ M(\|\nabla u\|_p^p) $ and a general nonlinearity $ k(t)f(u) $. A new finite time blow-up criterion is established, also, the upper and lower bounds for the blow-up time are derived. Secondly, we deal with the case that $ M(\|\nabla u\|_p^p) = a+b\|\nabla u\|_p^p $, $ k(t)\equiv1 $ and $ f(u) = |u|^{q-1}u $, which was considered by Li and Han [Math. Model. Anal. 2019; 24: 195-217] only for $ q > 2p-1 $. The threshold results for the existence of global and finite time blow-up solutions to this problem are obtained for the case $ 1 < q\leq 2p-1 $, which, together with the results given by Li and Han, shows that $ q = 2p-1 $ is critical for the existence of finite time blow-up solutions to this problem. These results partially generalize and extend some recent ones in previous literature.

Keywords

• p-kirchhoff equation
• general nonlinearity
• blow-up
• critical exponent