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

Fahima Hebhoub; Khaled Zennir; Tosiya Miyasita; Mohamed Biomy; Fahima Hebhoub; Khaled Zennir; Tosiya Miyasita; Mohamed Biomy

doi:10.3934/math.2021027

AIMS Mathematics

2021, Volume 6, Issue 1: 442-455. doi: 10.3934/math.2021027

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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

Received: 20 August 2020 Accepted: 08 October 2020 Published: 16 October 2020
MSC : 35B44, 35D30, 35L05

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^ * } = \frac{2}{{\rho + 1}}(\sum\limits_{i = 1}^2 {\int_{{r_1}}^{{r_2}} {|{u_{i0}}{|^2}} } {\mkern 1mu} dx){(\sum\limits_{i = 1}^2 {\int_{{r_1}}^{{r_2}} {\left( {2{u_{i0}}{u_{i1}} - |{u_{i0}}{|^2}} \right)} } {\mkern 1mu} dx)^{ - 1}}. $
- viscoelastic,
- Emden-Fowler wave equation,
- coupled system,
- blow-up
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[J]. AIMS Mathematics, 2021, 6(1): 442-455. doi: 10.3934/math.2021027

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Abstract

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^ * } = \frac{2}{{\rho + 1}}(\sum\limits_{i = 1}^2 {\int_{{r_1}}^{{r_2}} {|{u_{i0}}{|^2}} } {\mkern 1mu} dx){(\sum\limits_{i = 1}^2 {\int_{{r_1}}^{{r_2}} {\left( {2{u_{i0}}{u_{i1}} - |{u_{i0}}{|^2}} \right)} } {\mkern 1mu} dx)^{ - 1}}. $

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. doi: 10.1093/mnras/91.1.63
[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. doi: 10.1007/s11253-013-0809-3
[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. doi: 10.1016/j.na.2009.11.013
[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. doi: 10.1002/andp.18812490616
[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.

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