| Summary: | 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 \[ \ln T^{\ast} = \frac{2}{\rho+1} \Big( \sum_{i=1}^2 \int_{r_1}^{r_2} \vert u_{i0} \vert^2 \, dx \Big) \Big( \sum_{i=1}^2 \int_{r_1}^{r_2} \left( 2u_{i0}u_{i1} - \vert u_{i0} \vert^2 \right) \, dx\Big)^{-1}. \]
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