AIMS Mathematics, 2018, 3(2): 322-342. doi: 10.3934/Math.2018.2.322.

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Existence and nonexistence of global solutions to the Cauchy problem of thenonlinear hyperbolic equation with damping term

School of Mathematics and Information Science, Guangzhou University, Guangzhou, Guangdong, 510006, P. R. China

This paper concerns with the Cauchy problem for two classes of nonlinear hyperbolic equations with double damping terms. Firstly, by virtue of the Fourier transform method, we prove that the Cauchy problem of a class of high order nonlinear hyperbolic equation admits a global smooth solution $u(x, t)\in C^{\infty}((0, T]; H^{\infty}(\mathbb{R}))$$\bigcap C([0, T]; H^{3}(\mathbb{R}))$$\bigcap C^{1}([0, T]; H^{-1}(\mathbb{R}))$ as long as initial value $u_{0}\in W^{4, 1}(\mathbb{R})\bigcap H^{3}(\mathbb{R}), u_{1}\in L^{1}(\mathbb{R})\bigcap H^{-1}(\mathbb{R})$. Moreover, we give the sufficient conditions on the blow-up of the solution of a nonlinear damped hyperbolic equation with the initial value conditions in finite time and an example.
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Keywords nonlinear damped hyperbolic equation; Cauchy problem; Fourier transform; globalsmooth solution; blow-up

Citation: Jiali Yu, Yadong Shang, Huafei Di. Existence and nonexistence of global solutions to the Cauchy problem of thenonlinear hyperbolic equation with damping term. AIMS Mathematics, 2018, 3(2): 322-342. doi: 10.3934/Math.2018.2.322

References

  • 1. H. T. Banks and D. S. Gilliam, Global solvability for damped abstract nonlinear hyperbolic systems, Differ. Integral. Equ., 10 (1997), 309–332.
  • 2. Z. J. Yang and C. M. Song, Blow up of solutions for a class of quasi-linear evolution equations, Nonlinear Anal., 28 (1997), 2017–2032.
  • 3. Z. J. Yang and G. W. Chen, Global existence of solutions for quasi-linear wave equations with viscous damping, J. Math. Anal. Appl., 285 (2003), 604–618.    
  • 4. Y. C. Liu and R. Z. Xu, A class of fourth order wave equations with dissipative and nonlinear strain terms, J. Differ. Equ., 244 (2008), 200–228.    
  • 5. G.W. Chen and Z. J. Yang, Existence and nonexistence of global solutions for a class of nonlinear wave equations, Math. Meth. Appl. Sci., 23 (2000), 615–631.
  • 6. G. A. Philippin and S. V. Piro, Lower bound for the lifespan of solutions for a class of fourth order wave equations, Appl. Math. Lett., 50 (2015), 141–145.    
  • 7. R. Z. Xu, S. Wang, Y. B. Yang, et al. Initial boundary value problem for a class of fourth-order wave equation with viscous damping term, Appl. Anal., 92 (2013), 1403–1416.    
  • 8. A. Khelghati and K. Baghaei, Blow-up phenomena for a class of fourth-order nonlinear wave equations with a viscous damping term, Math. Meth. Appl. Sci., 41 (2018), 490–494.
  • 9. W. F. Zhao andW. J. Liu, A note on blow-up of solutions for a class of fourth-order wave equation with viscous damping term, Appl. Anal., 1 (2017), 1–9.    
  • 10. K. Baghaei, Lower bounds for the blow-up time in a superlinear hyperbolic equation with linear damping term, Comput. Math. Appl., 73 (2017), 560–564.    
  • 11. Z. J. Yang and G. W. Chen, Initial value problem for a nonlinear wave equation with damping term, Acta Math. Appl. Sinica., 23 (2000), 45–54.
  • 12. G. W. Chen and B. Lu, The initial-boundary value problems for a class of nonlinear wave equations with damping term, J. Math. Anal. Appl., 351 (2009), 1–15.    
  • 13. H. T. Banks and D. S. Gilliam, Well-posedness for a one dimensional nonlinear beam, In Computation and Control IV, Progress in Systems and Control Theory, Birkh¨auser, Boston, MA. 20 (1995), 1–21.    
  • 14. A. S. Ackleh, H. T. Banks and G. A. Pinter, A nonlinear beam equation, Appl. Math. Lett., 15 (2002), 381–387.
  • 15. G.W. Chen, Y. P.Wang and Z. C. Zhao, Blow-up of solution of an initial boundary value problem for a damped nonlinear hyperbolic equation, Appl. Math. Lett., 17 (2004), 491–497.    
  • 16. C. M. Song, Nonexistence of global solutions of nonlinear hyperbolic equation with material damping, Appl. Math. Mech., 27 (2006), 975–981.    
  • 17. G. W. Chen and F. Da, Blow-up of solution of Cauchy problem for three-dimensional damped nonlinear hyperbolic equation, Nonlinear Anal., 71 (2009), 358–372.    
  • 18. C. M. Song and Z. J. Yang, Existence and nonexistence of global solutions to the Cauchy problem for a nonlinear beam equation, Math. Meth. Appl. Sci., 33 (2010), 563–575.    
  • 19. G. W. Chen, R. L. Song and S. B. Wang, Local existence and global nonexistence theorems for a damped nonlinear hyperbolic equation, J. Math. Anal. Appl., 368 (2010), 19–31.    
  • 20. J. L. Yu, Y. D. Shang and H. F. Di, On decay and blow-up of solutions for a nonlinear beam equation with double damping terms, Bound.Value Probl., Accepted (2018).

 

This article has been cited by

  • 1. Jiali Yu, Yadong Shang, Huafei Di, On decay and blow-up of solutions for a nonlinear beam equation with double damping terms, Boundary Value Problems, 2018, 2018, 1, 10.1186/s13661-018-1067-y

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