Research article

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

  • Received: 01 September 2021 Revised: 24 February 2022 Accepted: 27 February 2022 Published: 16 May 2022
  • 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.

    Citation: Hui Yang, Futao Ma, Wenjie Gao, Yuzhu Han. Blow-up properties of solutions to a class of $ p $-Kirchhoff evolution equations[J]. Electronic Research Archive, 2022, 30(7): 2663-2680. doi: 10.3934/era.2022136

    Related Papers:

  • 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.



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