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Blow up at well defined time for a coupled system of one spatial variable Emden-Fowler type in viscoelasticities with strong nonlinear sources

1 Department of mathematics, Faculty of sciences, University 20 Août 1955- Skikda, 21000, Algeria
2 Department of Mathematics, College of Sciences and Arts, Qassim University, Ar-Rass, Saudi Arabia
3 Laboratoire de Mathématiques Appliquées et de Modélisation, Université 8 Mai 1945 Guelma. B. P. 401 Guelma 24000 Algérie
4 Faculty of Science and Engineering, Yamato University, 2-5-1, Katayama-cho, Suita-shi, Osaka, 564-0082, Japan
5 Department of Mathematics and Computer Science, Faculty of Science, Port Said University, Egypt

Special Issues: Blow-up phenomena for nonlinear evolution equations arising from the modeling of life sciences

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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