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Blow-Up and Global Solutions of a Wave Equation with Initial-Boundary Conditions

Year 2015, Volume: 28 Issue: 2, 245 - 251, 22.06.2015

Abstract

In this paper, it is studied an initial and boundary value problem with interior source function and linear damping term. It is proved that the solution is global in time and blow-up in finite time under suitable condition.

References

  • V. Baryak and M. Can, “Global nonexistence of solutions of the quasilinear hyperbolic equation of the vibrations of a riser”, Mathematical & Computational Applications., 2(1): 45-52, (1997).
  • J. Hao and S. Li, Y. Zhang, “Blow up and global solution for a quasilinear riser problem”, Nonlinear Analysis., 67: 974-980, (2007).
  • J.-Q. Wu and S.-J. Li, “Global solution and blow up solution for a nonlinear damped beam with source term”, Applied Mathematics., 25(4): 447-453, (2010).
  • H. Feng, S. Li and X. Zhi, “Blow up solutions for a nonlinear wave equation with boundary damping and interior source”, Nonlinear Analysis., 75: 2273-2280, (2012).
  • Ü. Dinlemez and E. Aktaş, “Global and Blow up solutions for nonlinear hyperbolic equations with initial- boundary conditions”, International Journal of Differential Equations., 2014: 5, (2014).
  • A. O. Çelebi, Ş. Gür and V. K. Kalantarov, “Structural stability and decay estimate for marine riser equations”, Math. Comput. Modelling, 54(11-12): 3182-3188, (2011).
  • H. Takamura and K. Wakasa, “Almost global solutions of semilinear wave equtions with the critical exponent in high dimensions”, Nonlinear Anal., 109: 187- 229, (2014).
  • J. Hao and S. Li, “Global solution and blow up solutions for a nonlinear string with boundary input and output,” Nonlinear Analysis: Theory, Methods and Applications., 66(1): 131-137, (2007).
  • A. Benaissa, S.A. Messaoudi, “Blow up of solutions of a nonlinear wave equation,” J. Appl. Math., 2(2): 105– 108, (2002).
  • V.A. Galaktionov, S.I. Pohozaev, “Blow-up and critical exponents for nonlinear hyperbolic equations,” Nonlinear Anal., 53: 453–466, (2003).
  • H.A. Levine, P. Pucci, J. Serrin, “Some remarks on global nonexistence for nonautonomous abstract evolution equations,”Contemp. Math., 208: 253–263, (1997).
  • H.A. Levine, J. Serrin, “A global nonexistence theorem for quasilinear evolution equation with dissipation,” Arch. Ration. Mech. Anal., 137: 341–361, (1997).
  • S.A. Messaoudi, “Blow up in a nonlinearly damped wave equation,” Math. Nachr., 231: 1–7, (2001).
  • S.A. Messaoudi, “Blow up in solutions of a semilinear wave equation,” Int. J. Appl. Math., 1 (6): 621– 626, (1999).
  • Z.J. Yang, “Global existence, asymptotic behavior and blow up of solutions for a class of nonlinear wave equations with dissipative term,” J. Differential Equations., 187: 520–540, (2003).
Year 2015, Volume: 28 Issue: 2, 245 - 251, 22.06.2015

Abstract

References

  • V. Baryak and M. Can, “Global nonexistence of solutions of the quasilinear hyperbolic equation of the vibrations of a riser”, Mathematical & Computational Applications., 2(1): 45-52, (1997).
  • J. Hao and S. Li, Y. Zhang, “Blow up and global solution for a quasilinear riser problem”, Nonlinear Analysis., 67: 974-980, (2007).
  • J.-Q. Wu and S.-J. Li, “Global solution and blow up solution for a nonlinear damped beam with source term”, Applied Mathematics., 25(4): 447-453, (2010).
  • H. Feng, S. Li and X. Zhi, “Blow up solutions for a nonlinear wave equation with boundary damping and interior source”, Nonlinear Analysis., 75: 2273-2280, (2012).
  • Ü. Dinlemez and E. Aktaş, “Global and Blow up solutions for nonlinear hyperbolic equations with initial- boundary conditions”, International Journal of Differential Equations., 2014: 5, (2014).
  • A. O. Çelebi, Ş. Gür and V. K. Kalantarov, “Structural stability and decay estimate for marine riser equations”, Math. Comput. Modelling, 54(11-12): 3182-3188, (2011).
  • H. Takamura and K. Wakasa, “Almost global solutions of semilinear wave equtions with the critical exponent in high dimensions”, Nonlinear Anal., 109: 187- 229, (2014).
  • J. Hao and S. Li, “Global solution and blow up solutions for a nonlinear string with boundary input and output,” Nonlinear Analysis: Theory, Methods and Applications., 66(1): 131-137, (2007).
  • A. Benaissa, S.A. Messaoudi, “Blow up of solutions of a nonlinear wave equation,” J. Appl. Math., 2(2): 105– 108, (2002).
  • V.A. Galaktionov, S.I. Pohozaev, “Blow-up and critical exponents for nonlinear hyperbolic equations,” Nonlinear Anal., 53: 453–466, (2003).
  • H.A. Levine, P. Pucci, J. Serrin, “Some remarks on global nonexistence for nonautonomous abstract evolution equations,”Contemp. Math., 208: 253–263, (1997).
  • H.A. Levine, J. Serrin, “A global nonexistence theorem for quasilinear evolution equation with dissipation,” Arch. Ration. Mech. Anal., 137: 341–361, (1997).
  • S.A. Messaoudi, “Blow up in a nonlinearly damped wave equation,” Math. Nachr., 231: 1–7, (2001).
  • S.A. Messaoudi, “Blow up in solutions of a semilinear wave equation,” Int. J. Appl. Math., 1 (6): 621– 626, (1999).
  • Z.J. Yang, “Global existence, asymptotic behavior and blow up of solutions for a class of nonlinear wave equations with dissipative term,” J. Differential Equations., 187: 520–540, (2003).
There are 15 citations in total.

Details

Primary Language English
Subjects Engineering
Journal Section Mathematics
Authors

Ülkü Dinlemez

Saghar Nabdel This is me

Publication Date June 22, 2015
Published in Issue Year 2015 Volume: 28 Issue: 2

Cite

APA Dinlemez, Ü., & Nabdel, S. (2015). Blow-Up and Global Solutions of a Wave Equation with Initial-Boundary Conditions. Gazi University Journal of Science, 28(2), 245-251.
AMA Dinlemez Ü, Nabdel S. Blow-Up and Global Solutions of a Wave Equation with Initial-Boundary Conditions. Gazi University Journal of Science. June 2015;28(2):245-251.
Chicago Dinlemez, Ülkü, and Saghar Nabdel. “Blow-Up and Global Solutions of a Wave Equation With Initial-Boundary Conditions”. Gazi University Journal of Science 28, no. 2 (June 2015): 245-51.
EndNote Dinlemez Ü, Nabdel S (June 1, 2015) Blow-Up and Global Solutions of a Wave Equation with Initial-Boundary Conditions. Gazi University Journal of Science 28 2 245–251.
IEEE Ü. Dinlemez and S. Nabdel, “Blow-Up and Global Solutions of a Wave Equation with Initial-Boundary Conditions”, Gazi University Journal of Science, vol. 28, no. 2, pp. 245–251, 2015.
ISNAD Dinlemez, Ülkü - Nabdel, Saghar. “Blow-Up and Global Solutions of a Wave Equation With Initial-Boundary Conditions”. Gazi University Journal of Science 28/2 (June 2015), 245-251.
JAMA Dinlemez Ü, Nabdel S. Blow-Up and Global Solutions of a Wave Equation with Initial-Boundary Conditions. Gazi University Journal of Science. 2015;28:245–251.
MLA Dinlemez, Ülkü and Saghar Nabdel. “Blow-Up and Global Solutions of a Wave Equation With Initial-Boundary Conditions”. Gazi University Journal of Science, vol. 28, no. 2, 2015, pp. 245-51.
Vancouver Dinlemez Ü, Nabdel S. Blow-Up and Global Solutions of a Wave Equation with Initial-Boundary Conditions. Gazi University Journal of Science. 2015;28(2):245-51.