The blow-up of solutions of a class of nonlinear viscoelastic Kirchhoff equation with suitable initial data and Dirichlet boundary conditions is discussed. By constructing a suitable auxiliary function to overcome the difficulty of gradient estimation and making use of differential inequality technique, we establish a finite time blow-up result when the initial data is at arbitrary energy level. Moreover, a lower bound of the lifespan is also derived by constructing a control function with both nonlocal term and memory kernel. Compared with the previous literature, our approach to estimate the lifespan does not require the initial energy to control some norms of the solution.

中文翻译：

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非线性粘弹性基尔霍夫方程解的爆破和寿命

讨论了具有合适初始数据和Dirichlet边界条件的一类非线性粘弹性Kirchhoff方程解的膨胀问题。通过构造合适的辅助函数来克服梯度估计的困难，并利用微分不等式技术，当初始数据处于任意能级时，我们建立了有限时间爆炸结果。此外，还通过构造具有非局部项和记忆核的控制函数来推导出寿命的下界。与之前的文献相比，我们估计寿命的方法不需要初始能量来控制解决方案的一些规范。