In this paper, we study a nonlinear Petrovsky type equation with nonlinear weak damping, a superlinear source and time-dependent coefficients

$${u_{tt}} + {{\rm{\Delta }}^2}u + {k_1}\left(t \right){\left| {{u_t}} \right|^{m - 2}}{u_t} = {k_2}\left(t \right){\left| u \right|^{p - 2}}u,\,\,\,\,\,\,\,\,\,\,x \in {\rm{\Omega,}}\,\,\,{\rm{t > 0,}}$$

where Ω is a bounded domain in Rⁿ. Under certain conditions on k₁(t), k₂(t) and the initial-boundary data, the upper bound for blow-up time of the solution with negative initial energy function is given by means of an auxiliary functional and an energy estimate method if p > m. Also, a lower bound of blow-up time are obtained by using a Sobolev-type inequality and a first order differential inequality technique for n = 2, 3 and n > 4.

中文翻译：

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具时变系数的非线性Petrovsky型方程解的爆破

在本文中，我们研究具有非线性弱阻尼，超线性源和时变系数的非线性Petrovsky型方程

$$ {u_ {tt}} + {{\ rm {\ Delta}} ^ 2} u + {k_1} \ left（t \ right）{\ left | {{u_t}} \ right | ^ {m-2}} {u_t} = {k_2} \ left（t \ right）{\ left | u \ right | ^ {p-2}} u，\，\，\，\，\，\，\，\，\，\，x \ in {\ rm {\ Omega，}} \，\，\ ，{\ rm {t> 0，}} $$

其中Ω是R ^n中的有界域。在k ₁（t），k ₂（t）和初始边界数据的某些条件下，借助于辅助函数和能量估计，给出具有负初始能量函数的解的爆破时间上限p> m时的方法。此外，对于n = 2、3和n > 4 ，使用Sobolev型不等式和一阶微分不等式技术可以获得爆燃时间的下限。