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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

  • 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}}. $

    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

    Related Papers:

  • 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}}. $


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