Export file:

Format

• RIS(for EndNote,Reference Manager,ProCite)
• BibTex
• Text

Content

• Citation Only
• Citation and Abstract

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

Abstract    Full Text(HTML)    Figure/Table    Related pages

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 $\ln T^{\ast} = \frac{2}{\rho+1} \Big( \sum_{i=1}^2 \int_{r_1}^{r_2} \vert u_{i0} \vert^2 \, dx \Big) \Big( \sum_{i=1}^2 \int_{r_1}^{r_2} \left( 2u_{i0}u_{i1} - \vert u_{i0} \vert^2 \right) \, dx\Big)^{-1}.$
Figure/Table
Supplementary
Article Metrics

Citation: Fahima Hebhoub, Khaled Zennir, Tosiya Miyasita, Mohamed Biomy. Blow up at well defined time for a coupled system of one spatial variable Emden-Fowler type in viscoelasticities with strong nonlinear sources. AIMS Mathematics, 2021, 6(1): 442-455. doi: 10.3934/math.2021027

References

• 1. A. Benaissa, D. Ouchenane, Kh. Zennir, Blow up of positive initial-energy solutions to systems of nonlinear wave equations with degenerate damping and source terms, Nonl. Stud., 19 (2012), 523-535.
• 2. S. Chandrasekhar, An introduction to the study of stellar structure, New York: Dover Publications, Inc., 1957.
• 3. R. Emden, Gaskugeln: Anwendungen der mechanischen wärmetheorie auf kosmologie und meteorologische probleme, Berlin: B. G. Teubner, 1907.
• 4. R. H. Fowler, The form near infinity of real, continuous solutions of a certain differential equation of the second order, Quart. J. Math., 45 (1914), 289-350.
• 5. R. H. Fowler, The solution of Emden's and similar differential equations, Mon. Not. Roy. Astron. Soc., 91 (1930), 63-91.
• 6. R. H. Fowler, Some results on the form near infinity of real continuous solutions of a certain type of second order differential equation, P. London Math. Soc., 13 (1914), 341-371.
• 7. R. H. Fowler, Further studies of Emden's and similar differential equations, Quart. J. Math., 2 (1931), 259-288.
• 8. M. R. Li, Nonexistence of global solutions of Emden-Fowler type semilinear wave equations with non-positive energy, Electron. J. Diff. Equ., 93 (2016), 1-10.
• 9. M. R. Li, Existence and uniqueness of local weak solutions for the Emden-Fowler wave equation in one dimension, Electron. J. Diff. Equ., 145 (2015), 1-10.
• 10. S. A. Messaoudi, B. Said-Houari, Global nonexistence of positive initial-energy solutions of a systemof nonlinear viscoelastic wave equations with damping and source terms, J. Math. Anal. Appl., 365 (2010), 277-287.
• 11. D. Ouchenane, Kh. Zennir, M. Bayoud, Global nonexistence of solutions for a system of nonlinear viscoelastic wave equations with degenerate damping and source terms, Ukrainian Math. J., 65 (2013), 723-739.
• 12. M. A. Rammaha, S. Sakuntasathien, Global existence and blow up of solutions to systems of nonlinear wave equations with degenerate damping and source terms, Nonlinear Anal. Theor., 72 (2010), 2658-2683.
• 13. M. Reed, Abstract non-linear wave equations, Berlin-New York: Springer-Verlag, 1976.
• 14. A. Ritter, Untersuchungen über die Höhe der Atmosphäre und die Constitution gasformiger Weltkörper, Wiedemann Annalen der Physik, 249 (1881), 360-377.
• 15. W. Thomson, On the convective equilibrium of temperature in the atmosphere, In: Memoirs of the literary and philosophical society of Manchester, Vol. 2, Manchester: The Society, 1865, 125-131.
• 16. Kh. Zennir, A. Guesmia, Existence of solutions to nonlinear κth-order coupled Klein-Gordon equations with nonlinear sources and memory terms, Appl. Math. E-Notes, 15 (2015), 121-136.
• 17. Kh. Zennir, Growth of solutions with positive initial energy to system of degeneratly damped wave equations with memory, Lobachevskii J. Math., 35 (2014), 147-156.