This paper deals with a class of Petrovsky system with nonlinear damping $$\begin{aligned} w_{tt}+\Delta _{\mathbb{B}}^{2}w-k_{2} \Delta _{\mathbb{B}}w_{t}+aw_{t} \vert w_{t} \vert ^{m-2}=bw \vert w \vert ^{p-2} \end{aligned}$$ on a manifold with conical singularity, where $\Delta _{\mathbb{B}}$ is a Fuchsian-type Laplace operator with totally characteristic degeneracy on the boundary $x_{1}=0$ . We first prove the global existence of solutions under conditions without relation between m and p, and establish an exponential decay rate. Furthermore, we obtain a finite time blow-up result for local solutions with low initial energy $E(0)< d$ .

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具锥变性的非线性彼得罗夫斯基系统解的衰减和爆破

本文研究了一类具有非线性阻尼$$-begin {align} w_ {tt} + \ Delta _ {\ mathbb {B}} ^ {2} w-k_ {2} \ Delta _ {_ mathbb的Petrovsky系统{B}} w_ {t} + aw_ {t} \ vert w_ {t} \ vert ^ {m-2} = bw \ vert w \ vert ^ {p-2} \ end {aligned} $$在流形上具有圆锥形奇点，其中$ \ Delta _ {\\ mathbb {B}} $是在边界$ x_ {1} = 0 $上具有完全特征简并性的Fuchsian型Laplace算子。我们首先证明在m与p之间没有关系的条件下解的整体存在性，并建立指数衰减率。此外，我们获得了具有低初始能量$ E（0）<d $的局部解的有限时间爆炸结果。