Taiwanese Journal of Mathematics

Bounds for the Lifespan of Solutions to Fourth-order Hyperbolic Equations with Initial Data at Arbitrary Energy Level

Bin Guo and Xiaolei Li

Advance publication

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Abstract

This paper deals with lower and upper bounds for the lifespan of solutions to a fourth-order nonlinear hyperbolic equation with strong damping: \[ u_{tt} + \Delta^{2} u - \Delta u - \omega \Delta u_t + \alpha(t) u_t = |u|^{p-2} u. \] First of all, the authors construct a new control function and apply the Sobolev embedding inequality to establish some qualitative relationships between initial energy value and the norm of the gradient of the solution for supercritical case ($2(N-2)/(N-4) \lt p \lt 2N/(N-4)$, $N \geq 5$). And then, the concavity argument is used to prove that the solution blows up in finite time for initial data at low energy level, at the same time, an estimate of the upper bound of blow-up time is also obtained.

Subsequently, for initial data at high energy level, the authors prove the monotonicity of the $L^{2}$ norm of the solution under suitable assumption of initial data, furthermore, we utilize the concavity argument and energy methods to prove that the solution also blows up in finite time for initial data at high energy level.

At last, for the supercritical case, a new control functional with a small dissipative term and an inverse Hölder inequality with correction constants are employed to overcome the difficulties caused by the failure of the embedding inequality ($H^{2}(\Omega) \cap H^{1}_{0}(\Omega) \hookrightarrow L^{2p-2}$ for $2(N-2)/(N-4) \lt p \lt 2N/(N-4)$) and then an explicit lower bound for blow-up time is obtained. Such results extend and improve those of [S. T. Wu, J. Dyn. Control Syst. 24 (2018), no. 2, 287--295].

Article information

Source
Taiwanese J. Math., Advance publication (2019), 17 pages.

Dates
First available in Project Euclid: 17 January 2019

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1547715693

Digital Object Identifier
doi:10.11650/tjm/190103

Subjects
Primary: 35L05: Wave equation 35L20: Initial-boundary value problems for second-order hyperbolic equations 35L71: Semilinear second-order hyperbolic equations

Keywords
concavity argument energy estimate mountain pass level

Citation

Guo, Bin; Li, Xiaolei. Bounds for the Lifespan of Solutions to Fourth-order Hyperbolic Equations with Initial Data at Arbitrary Energy Level. Taiwanese J. Math., advance publication, 17 January 2019. doi:10.11650/tjm/190103. https://projecteuclid.org/euclid.twjm/1547715693


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References

  • R. A. Adams, Sobolev Spaces, Pure and Applied Mathematics 65, Academic Press, New York, 1975.
  • L. J. An, Loss of hyperbolicity in elastic-plastic material at finite strains, SIAM J. Appl. Math. 53 (1993), no. 3, 621–654.
  • L. J. An and A. Peirce, The effect of microstructure on elastic-plastic models, SIAM J. Appl. Math. 54 (1994), no. 3, 708–730.
  • ––––, A weakly nonlinear analysis of elasto-plastic-microstructure models, SIAM J. Appl. Math. 55 (1995), no. 1, 136–155.
  • K. Baghaei, Lower bounds for the blow-up time in a superlinear hyperbolic equation with linear damping term, Comput. Math. Appl. 73 (2017), no. 4, 560–564.
  • J. A. Esquivel-Avila, Dynamics around the ground state of a nonlinear evolution equation, Nonlinear Anal. 63 (2005), no. 5-7, e331–e343.
  • B. Guo and W. Gao, Finite-time blow-up and extinction rates of solutions to an initial Neumann problem involving the $p(x,t)$-Laplace operator and a non-local term, Discrete Contin. Dyn. Syst. 36 (2016), no. 2, 715–730.
  • ––––, Blow-up of solutions to quasilinear parabolic equations with singular absorption and a positive initial energy, Mediterr. J. Math. 13 (2016), no. 5, 2853–2861.
  • B. Guo, Y. Li and W. Gao, Singular phenomena of solutions for nonlinear diffusion equations involving $p(x)$-Laplace operator and nonlinear sources, Z. Angew. Math. Phys. 66 (2015), no. 3, 989–1005.
  • B. Guo and F. Liu, A lower bound for the blow-up time to a viscoelastic hyperbolic equation with nonlinear sources, Appl. Math. Lett. 60 (2016), 115–119.
  • Q. Lin, Y. H. Wu and S. Lai, On global solution of an initial boundary value problem for a class of damped nonlinear equations, Nonlinear Anal. 69 (2008), no. 12, 4340–4351.
  • Y. Liu and R. Xu, Fourth order wave equations with nonlinear strain and source terms, J. Math. Anal. Appl. 331 (2007), no. 1, 585–607.
  • ––––, A class of fourth order wave equations with dissipative and nonlinear strain terms, J. Differential Equations 244 (2008), no. 1, 200–228.
  • G. A. Philippin, Lower bounds for blow-up time in a class of nonlinear wave equations, Z. Angew. Math. Phys. 66 (2015), no. 1, 129–134.
  • G. A. Philippin and S. Vernier Piro, Lower bound for the lifespan of solutions for a class of fourth order wave equations, Appl. Math. Lett. 50 (2015), 141–145.
  • D. H. Sattinger, On global solution of nonlinear hyperbolic equations, Arch. Rational Mech. Anal. 30 (1968), 148–172.
  • L. Sun, B. Guo and W. Gao, A lower bound for the blow-up time to a damped semilinear wave equation, Appl. Math. Lett. 37 (2014), 22–25.
  • S.-T. Wu, Lower and upper bounds for the blow-up time of a class of damped fourth-order nonlinear evolution equations, J. Dyn. Control Syst. 24 (2018), no. 2, 287–295.
  • J. Zhou, Lower bounds for blow-up time of two nonlinear wave equations, Appl. Math. Lett. 45 (2015), 64–68.