Electronic Journal of Differential Equations, Vol. 2022 (2022), No. 38, pp. 1-30.

Title: Global solutions and blow-up for a Kirchhoff-type problem on 
 a geodesic ball of the Poincare ball model
       
Authors: Hang Ding (Southwest Univ., Chongqing, China)
         Jun Zhou  (Southwest Univ., Chongqing, China)

Abstract:   
This article concerns a Kirchhoff-type parabolic problem on a geodesic ball of hyperbolic 
space. Firstly, we obtain  conditions for finite time blow-up, and for the existence of global 
solutions for J(u_0)&le; d, where J(u<sub>0</sub>) denotes the initial energy and d denotes
the depth of the potential well. 
Secondly, we estimate the upper and lower bounds of the blow-up time. 
In addition, we  derive the growth rate of the blow-up solution and the
decay rate of the global solution. 
Thirdly, we establish a new finite time blow-up condition
which is independent of d and prove that the solution can blow up in finite time with
arbitrary high initial energy, by using this blow-up condition. 
Finally, we present some equivalent conditions for the solution existing globally or blowing up 
in finite time.

Submitted July 18, 2021 Published May 13, 2022.
Math Subject Classifications: 35K55, 35B40, 35B44.
Key Words: Parabolic problem of Kirchhoff type; hyperbolic space;
           poincare ball model; global solution; blow-up.