For one spatial variable, a new kind of coupled system for nonlinear wave equations of Emden-Fowler type is considered with boundary value and initial values. Under certain conditions on the initial data and the exponent $\rho$, we show that the viscoelastic terms lead our problem to be dissipative and that the global solutions cannot exist in $L^2$ beyond the given finite time i.e., \[ \int_{r_1}^{r_2} \Big( \vert u_1 \vert^2 + \vert u_2 \vert^2 \Big) \, dx \to +\infty \quad \hbox{ as } t\to T^{\ast}, \] where \[ \ln T^{\ast} = \frac{2}{\rho+1} \Big( \sum_{i=1}^2 \int_{r_1}^{r_2} \vert u_{i0} \vert^2 \, dx \Big) \Big( \sum_{i=1}^2 \int_{r_1}^{r_2} \left( 2u_{i0}u_{i1} - \vert u_{i0} \vert^2 \right) \, dx\Big)^{-1}. \]

中文翻译：

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具有强非线性源的粘弹性中一个空间变量Emden-Fowler型耦合系统在明确的时间发生爆炸

对于一个空间变量，考虑了一种新型的耦合系统，用于Emden-Fowler型非线性波动方程的边界值和初始值。在初始数据和指数$ \ rho $的特定条件下，我们证明了粘弹性项导致我们的问题具有耗散性，并且在给定的有限时间内，即$ [^ \ int_ {r_1} ^ {r_2} \ Big（\ vert u_1 \ vert ^ 2 + \ vert u_2 \ vert ^ 2 \ Big）\，dx \ to + \ infty \ quad \ hbox {as} t \ to T ^ { \ ast}，\]其中\ [\ ln T ^ {\ ast} = \ frac {2} {\ rho + 1} \ Big（\ sum_ {i = 1} ^ 2 \ int_ {r_1} ^ {r_2} \ vert u_ {i0} \ vert ^ 2 \，dx \ Big）\ Big（\ sum_ {i = 1} ^ 2 \ int_ {r_1} ^ {r_2} \ left（2u_ {i0} u_ {i1}-\ vert u_ {i0} \ vert ^ 2 \ right）\，dx \ Big）^ {-1}。\]