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Year 2019, , 70 - 78, 28.06.2019
https://doi.org/10.32323/ujma.484532

Abstract

References

  • [1] W. Liu, Uniform decay of solutions for a quasilinear system of viscoelastic equations, Nonlinear Anal., 71 (2009) 2257-2267.
  • [2] B.S. Houari, Exponential growth of positive initial-energy solutions of a system of nonlinear viscoelastic wave equations with damping and source terms, Z. Angew. Math. Phys., 62 (2011) 115-133.
  • [3] X. Han , M. Wang, Global existence and blow-up solutions for a system of nonlinear viscoelastic wave equations with damping and source, Nonlinear Anal., 71 (2009) 5427-5450.
  • [4] S.A. Messaoudi , B.S. Houari, Global nonexistence of positive initial-energy solutions of a system of nonlinear viscoelastic wave equations with damping and source terms, J. Math. Anal. Appl., 365 (2010) 277-287.
  • [5] E. Pişkin, A lower bound for the blow up time of a system of viscoelastic wave equations with nonlinear damping and source terms, J. Nonlinear Funct. Anal., 2017 (2017) 1-9.
  • [6] E. Pişkin, Global nonexistence of solutions for a system of viscoelastic wave equations with weak damping terms, Malaya Journal of Matematik, 3(2) (2015) 168-174.
  • [7] B.S. Houari, S.A. Messaoudi, A. Guesmia, General decay of solutions of a nonlinear system of viscoelastic wavw equations, Nonlinear Differ. Equ. Appl., 18 (2011) 659-684.
  • [8] Y. Zhao , Q. Wang, Blow-up of arbitrarily positive initial energy solutions for a viscoelastic wave system with nonlinear damping and source terms, Boundary Value Problems, 35 (2018) 1-13.
  • [9] J. Hao, S. Niu, H. Men, Global nonexistence of solutions for nonlinear coupled viscoelastic wave equations with damping and source terms, Boundary Value Problems, 250 (2014) 1-11.
  • [10] L. Fei, G. Hongjun, Global nonexistence of positive initial energy solutions for coupled nonlinear wave equations with damping and source terms, Abst. Appl. Anal., 2011 (2011) 1-14.
  • [11] J. Hao, L. Cai, Global existence and blow up of solutions for nonlinear coupled wave equations with viscoelastic terms, Math. Meth. Appl. Sci., 39 (2016) 1977-1989.

Exponential Growth of Solutions for Nonlinear Coupled Viscoelastic Wave Equations

Year 2019, , 70 - 78, 28.06.2019
https://doi.org/10.32323/ujma.484532

Abstract

In this work, we consider an initial-boundary value problem related to the nonlinear coupled viscoelastic equations \[ \left\{ \begin{array}{c} \left\vert u_{t}\right\vert ^{j}u_{tt}-\Delta u_{tt}-div\left( \left\vert \nabla u\right\vert ^{\alpha -2}\nabla u\right) -\Delta u+\int\limits_{0}^{t}g\left( t-s\right) \Delta uds+\left\vert u_{t}\right\vert ^{m-1}u_{t}=f_{1}\left( u,v\right) ,\text{ } \\ \left\vert v_{t}\right\vert ^{j}v_{tt}-\Delta v_{tt}-div\left( \left\vert \nabla v\right\vert ^{\beta -2}\nabla v\right) -\Delta v+\int\limits_{0}^{t}h\left( t-s\right) \Delta vds+\left\vert v_{t}\right\vert ^{r-1}v_{t}=f_{2}\left( u,v\right) .\text{ } \end{array} \right. \] We will show the exponential growth of solutions with positive initial energy.

References

  • [1] W. Liu, Uniform decay of solutions for a quasilinear system of viscoelastic equations, Nonlinear Anal., 71 (2009) 2257-2267.
  • [2] B.S. Houari, Exponential growth of positive initial-energy solutions of a system of nonlinear viscoelastic wave equations with damping and source terms, Z. Angew. Math. Phys., 62 (2011) 115-133.
  • [3] X. Han , M. Wang, Global existence and blow-up solutions for a system of nonlinear viscoelastic wave equations with damping and source, Nonlinear Anal., 71 (2009) 5427-5450.
  • [4] S.A. Messaoudi , B.S. Houari, Global nonexistence of positive initial-energy solutions of a system of nonlinear viscoelastic wave equations with damping and source terms, J. Math. Anal. Appl., 365 (2010) 277-287.
  • [5] E. Pişkin, A lower bound for the blow up time of a system of viscoelastic wave equations with nonlinear damping and source terms, J. Nonlinear Funct. Anal., 2017 (2017) 1-9.
  • [6] E. Pişkin, Global nonexistence of solutions for a system of viscoelastic wave equations with weak damping terms, Malaya Journal of Matematik, 3(2) (2015) 168-174.
  • [7] B.S. Houari, S.A. Messaoudi, A. Guesmia, General decay of solutions of a nonlinear system of viscoelastic wavw equations, Nonlinear Differ. Equ. Appl., 18 (2011) 659-684.
  • [8] Y. Zhao , Q. Wang, Blow-up of arbitrarily positive initial energy solutions for a viscoelastic wave system with nonlinear damping and source terms, Boundary Value Problems, 35 (2018) 1-13.
  • [9] J. Hao, S. Niu, H. Men, Global nonexistence of solutions for nonlinear coupled viscoelastic wave equations with damping and source terms, Boundary Value Problems, 250 (2014) 1-11.
  • [10] L. Fei, G. Hongjun, Global nonexistence of positive initial energy solutions for coupled nonlinear wave equations with damping and source terms, Abst. Appl. Anal., 2011 (2011) 1-14.
  • [11] J. Hao, L. Cai, Global existence and blow up of solutions for nonlinear coupled wave equations with viscoelastic terms, Math. Meth. Appl. Sci., 39 (2016) 1977-1989.
There are 11 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Erhan Pişkin 0000-0001-6587-4479

Şeyhmus Altındağ This is me

Publication Date June 28, 2019
Submission Date November 17, 2018
Acceptance Date April 9, 2019
Published in Issue Year 2019

Cite

APA Pişkin, E., & Altındağ, Ş. (2019). Exponential Growth of Solutions for Nonlinear Coupled Viscoelastic Wave Equations. Universal Journal of Mathematics and Applications, 2(2), 70-78. https://doi.org/10.32323/ujma.484532
AMA Pişkin E, Altındağ Ş. Exponential Growth of Solutions for Nonlinear Coupled Viscoelastic Wave Equations. Univ. J. Math. Appl. June 2019;2(2):70-78. doi:10.32323/ujma.484532
Chicago Pişkin, Erhan, and Şeyhmus Altındağ. “Exponential Growth of Solutions for Nonlinear Coupled Viscoelastic Wave Equations”. Universal Journal of Mathematics and Applications 2, no. 2 (June 2019): 70-78. https://doi.org/10.32323/ujma.484532.
EndNote Pişkin E, Altındağ Ş (June 1, 2019) Exponential Growth of Solutions for Nonlinear Coupled Viscoelastic Wave Equations. Universal Journal of Mathematics and Applications 2 2 70–78.
IEEE E. Pişkin and Ş. Altındağ, “Exponential Growth of Solutions for Nonlinear Coupled Viscoelastic Wave Equations”, Univ. J. Math. Appl., vol. 2, no. 2, pp. 70–78, 2019, doi: 10.32323/ujma.484532.
ISNAD Pişkin, Erhan - Altındağ, Şeyhmus. “Exponential Growth of Solutions for Nonlinear Coupled Viscoelastic Wave Equations”. Universal Journal of Mathematics and Applications 2/2 (June 2019), 70-78. https://doi.org/10.32323/ujma.484532.
JAMA Pişkin E, Altındağ Ş. Exponential Growth of Solutions for Nonlinear Coupled Viscoelastic Wave Equations. Univ. J. Math. Appl. 2019;2:70–78.
MLA Pişkin, Erhan and Şeyhmus Altındağ. “Exponential Growth of Solutions for Nonlinear Coupled Viscoelastic Wave Equations”. Universal Journal of Mathematics and Applications, vol. 2, no. 2, 2019, pp. 70-78, doi:10.32323/ujma.484532.
Vancouver Pişkin E, Altındağ Ş. Exponential Growth of Solutions for Nonlinear Coupled Viscoelastic Wave Equations. Univ. J. Math. Appl. 2019;2(2):70-8.

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