The aim of this paper is to study bounds for blow-up time to the following viscoelastic hyperbolic equation of Kirchhoff type with initial-boundary value condition:

$$ |u_{t}|^{\rho }u_{tt}-M(\|\nabla u\|_{2}^{2})\Delta u+\int _{0}^{t}g(t- \tau )\Delta u(\tau )d\tau +|u_{t}|^{m(x)-2}u_{t}=|u|^{p(x)-2}u. $$

Compared with constant exponents, it is difficult to discuss the above problem due to the existence of a gap between the modular and the norm. The authors construct suitable function spaces to discuss the upper bound for blow-up time with positive initial energy by means of a differential inequality technique. In addition, lower bounds for blow-up time in different range of exponent are obtained. These improve and generalize some recent results.

中文翻译：

---

具有可变源的基尔霍夫型粘弹性双曲方程的爆破时间界

本文的目的是研究以下具有初边界值条件的基尔霍夫型粘弹性双曲方程的爆破时间界：

$$ | u_ {t} | ^ {\ rho} u_ {tt} -M（\ | \ nabla u \ | __ {2} ^ {2}）\ Delta u + \ int _ {0} ^ {t} g （t- \ tau）\ Delta u（\ tau）d \ tau + | u_ {t} | ^ {m（x）-2} u_ {t} = | u | ^ {p（x）-2} u 。$$

与常数指数相比，由于模块和规范之间存在间隙，因此很难讨论上述问题。作者构造了合适的函数空间，以微分不等式技术讨论具有正初始能量的爆炸时间的上限。此外，在不同的指数范围内获得了爆破时间的下限。这些改进和概括了一些最近的结果。