Blow up at well defined time for a coupled system of one spatial variable Emden-Fowler type in viscoelasticities with strong nonlinear sources
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...
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| Language: | English |
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AIMS Press
2021-10-01
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| Series: | AIMS Mathematics |
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| Online Access: | https://www.aimspress.com/article/10.3934/math.2021027/fulltext.html |
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| author | Fahima Hebhoub Khaled Zennir Tosiya Miyasita Mohamed Biomy |
| author_facet | Fahima Hebhoub Khaled Zennir Tosiya Miyasita Mohamed Biomy |
| author_sort | Fahima Hebhoub |
| collection | DOAJ |
| description | 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 \[ \ln T^{\ast} = \frac{2}{\rho+1} \Big( \sum_{i=1}^2 \int_{r_1}^{r_2} \vert u_{i0} \vert^2 \, dx \Big) \Big( \sum_{i=1}^2 \int_{r_1}^{r_2} \left( 2u_{i0}u_{i1} - \vert u_{i0} \vert^2 \right) \, dx\Big)^{-1}. \] |
| first_indexed | 2024-12-21T15:15:23Z |
| format | Article |
| id | doaj.art-38b8f854b11a4cde988c138e46d011c3 |
| institution | Directory Open Access Journal |
| issn | 2473-6988 |
| language | English |
| last_indexed | 2024-12-21T15:15:23Z |
| publishDate | 2021-10-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | AIMS Mathematics |
| spelling | doaj.art-38b8f854b11a4cde988c138e46d011c32022-12-21T18:59:10ZengAIMS PressAIMS Mathematics2473-69882021-10-016144245510.3934/math.2021027Blow up at well defined time for a coupled system of one spatial variable Emden-Fowler type in viscoelasticities with strong nonlinear sourcesFahima Hebhoub0Khaled Zennir1Tosiya Miyasita2Mohamed Biomy31 Department of mathematics, Faculty of sciences, University 20 Août 1955- Skikda, 21000, Algeria2 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 Algérie4 Faculty of Science and Engineering, Yamato University, 2-5-1, Katayama-cho, Suita-shi, Osaka, 564-0082, Japan2 Department of Mathematics, College of Sciences and Arts, Qassim University, Ar-Rass, Saudi Arabia5 Department of Mathematics and Computer Science, Faculty of Science, Port Said University, EgyptFor 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 \[ \ln T^{\ast} = \frac{2}{\rho+1} \Big( \sum_{i=1}^2 \int_{r_1}^{r_2} \vert u_{i0} \vert^2 \, dx \Big) \Big( \sum_{i=1}^2 \int_{r_1}^{r_2} \left( 2u_{i0}u_{i1} - \vert u_{i0} \vert^2 \right) \, dx\Big)^{-1}. \]https://www.aimspress.com/article/10.3934/math.2021027/fulltext.htmlviscoelasticemden-fowler wave equationcoupled systemblow-up |
| spellingShingle | 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 viscoelastic emden-fowler wave equation coupled system blow-up |
| title | Blow up at well defined time for a coupled system of one spatial variable Emden-Fowler type in viscoelasticities with strong nonlinear sources |
| title_full | Blow up at well defined time for a coupled system of one spatial variable Emden-Fowler type in viscoelasticities with strong nonlinear sources |
| title_fullStr | Blow up at well defined time for a coupled system of one spatial variable Emden-Fowler type in viscoelasticities with strong nonlinear sources |
| title_full_unstemmed | Blow up at well defined time for a coupled system of one spatial variable Emden-Fowler type in viscoelasticities with strong nonlinear sources |
| title_short | Blow up at well defined time for a coupled system of one spatial variable Emden-Fowler type in viscoelasticities with strong nonlinear sources |
| title_sort | blow up at well defined time for a coupled system of one spatial variable emden fowler type in viscoelasticities with strong nonlinear sources |
| topic | viscoelastic emden-fowler wave equation coupled system blow-up |
| url | https://www.aimspress.com/article/10.3934/math.2021027/fulltext.html |
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