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Blow-Up of Solutions to the Fourth-Order Equation with Variable-Exponent Nonlinear Weak Damping

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Abstract

In this note, we investigate again the blow-up phenomenon of weak solutions to the following initial-boundary value problem of the fourth-order equation with variable-exponent nonlinearity

$$\begin{aligned} u_{tt}+\Delta ^2 u-M(\Vert \nabla u\Vert _2^2)\Delta u-\Delta u_t+|u_t|^{m(x)-2}u_t=|u|^{p(x)-2}u. \end{aligned}$$

Our investigations reveal that the blow-up phenomenon will happen for arbitrarily high initial energy under \(m^-:=ess\inf _{x\in \Omega }m(x)>2\). It is worthy to point out that an upper bound of the blow-up time is also shown. These results answer the earlier unsolved question in our previous paper (Liao and Tan in Sci China Math 66, 285–302 (2023). https://doi.org/10.1007/s11425-021-1926-x).

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References

  1. Autuori, G., Pucci, P., Salvatori, M.C.: Asymptotic stability for nonlinear Kirchhoff systems. Nonlinear Anal. Real World Appl. 10, 889–909 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  2. Ball, J.M.: Initial-boundary value problems for an extensible beam. J. Math. Anal. Appl. 42, 61–90 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  3. Burgreen, D.: Free vibrations of a pin-ended column with constant distance between pin ends. J. Appl. Mech. 18, 135–139 (1951)

    Article  Google Scholar 

  4. Cavalcanti, M.M., Domingos Cavalcanti, V.N., Soriano, J.A.: Global existence and asymptotic stability for the nonlinear and generalized damped extensible plate equation. Commun. Contemp. Math. 6, 705–731 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  5. Chen, W.Y., Zhou, Y.: Global nonexistence for a semilinear Petrovsky equation. Nonlinear Anal. 70, 3203–3208 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  6. De Brito, E.H.: The damped elastic stretched string equation generalized: existence, uniqueness, regularity and stability. Appl. Anal. 13, 219–233 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  7. Dickey, R.W.: Free vibrations and dynamic buckling of the extensible beam. J. Math. Anal. Appl. 29, 443–454 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  8. Eisley, J.G.: Nonlinear vibrations of beams and rectangular plates. Z. Angew Math. Phys. 15, 167–175 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  9. Fan, X.L., Zhang, Q.H.: Existence of solutions for \(p(x)\)-Laplacian Dirichlet problem. Nonlinear Anal. 52, 1843–1852 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  10. Fan, X.L., Zhao, D.: On the spaces \(L^{p(x)}(\Omega )\) and \(W^{m, p(x)}(\Omega )\). J. Math. Anal. Appl. 263, 424–446 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  11. Guesmia, A.: Existence globale et stabilisation interne non linéaire dun système de Petrovsky. Bull. Belg. Math. Soc. 5, 583–594 (1998)

    MATH  Google Scholar 

  12. Guo, B., Li, X.L.: Bounds for the lifespan of solutions to fourth-order hyperbolic equations with initial data at arbitrary energy level. Taiwan. J. Math. 23, 1461–1477 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  13. Han, Y.Z., Li, Q.: Lifespan of solutions to a damped plate equation with logarithmic nonlinearity. Evol. Equ. Control Theory 11, 25–40 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  14. Kang, J.R.: Global nonexistence of solutions for von Karman equations with variable exponents. Appl. Math. Lett. 86, 249–255 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  15. Kouémou-Patcheu, S.: Global existence and exponential decay estimates for a damped quasilinear equation. Comm. Partial Differ. Equ. 22, 2007–2024 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  16. Li, F.S., Gao, Q.Y.: Blow-up of solution for a nonlinear Petrovsky type equation with memory. Appl. Math. Comput. 274, 383–392 (2016)

    MathSciNet  MATH  Google Scholar 

  17. Li, X.L., Liu, M.M.: Blow-up and asymptotic behavior of solutions for a class of fourth-order hyperbolic equations with mixed damping. Authorea (2022). https://doi.org/10.22541/au.165051739.94421600/v1

    Article  Google Scholar 

  18. Li, G., Sun, Y.N., Liu, W.J.: Global existence and blow-up of solutions for a strongly damped Petrovsky system with nonlinear damping. Appl. Anal. 91, 575–586 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  19. Liao, M.L., Tan, Z.: On behavior of solutions to a Petrovsky equation with damping and variable-exponent sources. Sci. China Math. (2022). https://doi.org/10.1007/s11425-021-1926-x

  20. Liu, L.S., Sun, F.L., Wu, Y.H.: Blow-up of solutions for a nonlinear Petrovsky type equation with initial data at arbitrary high energy level. Bound. Value Probl. 2019, 15 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  21. Messaoudi, S.A.: Global existence and nonexistence in a system of Petrovsky. J. Math. Anal. Appl. 265, 296–308 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  22. Sun, F.L., Liu, L.S., Wu, Y.H.: Blow-up of solutions for a nonlinear viscoelastic wave equation with initial data at arbitrary energy level. Appl. Anal. 98, 2308–2327 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  23. Tahamtani, F., Shahrouzi, M.: Existence and blow up of solutions to a Petrovsky equation with memory and nonlinear source term. Bound. Value Probl. 2012, 50 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  24. Woinowsky-Krieger, S.: The effect of an axial force on the vibration of hinged bars. J. Appl. Mech. 17, 35–36 (1950)

    Article  MathSciNet  MATH  Google Scholar 

  25. Wu, S.T., Tsai, L.Y.: On global solutions and blow-up of solutions for a nonlinearly damped Petrovsky system. Taiwan. J. Math. 13, 545–558 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  26. Wu, S.T., Tsai, L.Y.: Existence and nonexistence of global solutions for a nonlinear wave equation. Taiwan. J. Math. 13, 2069–2091 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  27. Yang, Z.J.: On an extensible beam equation with nonlinear damping and source terms. J. Differ. Equ. 254, 3903–3927 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  28. Zhou, J.: Global existence and blow-up of solutions for a Kirchhoff type plate equation with damping. Appl. Math. Comput. 265, 807–818 (2015)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors would like to express her sincere gratitude to both Professor Wenjie Gao and Bin Guo in Jilin University for their help and constant encouragement. This project is supported by Fundamental Research Funds for the Central Universities (3132022203) and by Basic Scientific Research Project of The Educational Department of Liaoning Province (LJKMZ20220368).

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Authors and Affiliations

  1. College of Science, Hohai University, Nanjing, 210098, China

    Menglan Liao

  2. School of Science, Dalian Maritime University, Dalian, 116026, China

    Qingwei Li

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  1. Menglan Liao
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Menglan Liao and Qingwei Li wrote the main manuscript text. All authors reviewed the manuscript.

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Correspondence to Qingwei Li.

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Liao, M., Li, Q. Blow-Up of Solutions to the Fourth-Order Equation with Variable-Exponent Nonlinear Weak Damping. Mediterr. J. Math. 20, 179 (2023). https://doi.org/10.1007/s00009-023-02391-5

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