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 \(J(u_0)\leq d\), where \(J(u_0)\) 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.
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