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Blow up of Solutions for a Nonlinear Petrovsky Type Equation with Time-dependent Coefficients

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Abstract

In this paper, we study a nonlinear Petrovsky type equation with nonlinear weak damping, a superlinear source and time-dependent coefficients

$${u_{tt}} + {{\rm{\Delta }}^2}u + {k_1}\left(t \right){\left| {{u_t}} \right|^{m - 2}}{u_t} = {k_2}\left(t \right){\left| u \right|^{p - 2}}u,\,\,\,\,\,\,\,\,\,\,x \in {\rm{\Omega,}}\,\,\,{\rm{t > 0,}}$$

where Ω is a bounded domain in Rn. Under certain conditions on k1(t), k2(t) and the initial-boundary data, the upper bound for blow-up time of the solution with negative initial energy function is given by means of an auxiliary functional and an energy estimate method if p > m. Also, a lower bound of blow-up time are obtained by using a Sobolev-type inequality and a first order differential inequality technique for n = 2, 3 and n > 4.

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References

  1. Fang, Z.B., Wang, Y.X. Blow-up analysis for a semilinear parabolic equation with time-dependent coefficients under nonlinear boundary flux. Z. Angew. Math. Phys., 66: 2525–2541 (2015)

    Article  MathSciNet  Google Scholar 

  2. Guo, B., Liu, F. A lower bound for the blow-up time to a viscoelastic hyperbolic equation with nonlinear sources. Appl. Math. Lett., 60: 115–119 (2016)

    Article  MathSciNet  Google Scholar 

  3. Li, F.S., Du, G.W. General energy decay for a degenerate viscoelastic Petrovsky-type plate equation with boundary feedback. J. Appl. Anal. Comput., 8: 390–401 (2018)

    MathSciNet  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  5. 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., 15: (2019)

  6. Liu, Z.Q., Fang, Z.B. Blow-up phenomena for a nonlocal quasilinear parabolic equation with time-dependent coefficents under nonlinear boundary flux. Discrete Contin. Dyn. Syst. Ser. B, 21: 3619–3635 (2016)

    Article  MathSciNet  Google Scholar 

  7. Messaoudi, S.A. Blow up and global existence in a nonlinear viscoelastic wave equation. Math. Nachr., 260: 58–66 (2003)

    Article  MathSciNet  Google Scholar 

  8. Messaoudi, S.A. Global Existence and Nonexistence in a System of Petrovsky. J. Math. Anal. Appl., 265: 296–308 (2002)

    Article  MathSciNet  Google Scholar 

  9. Peng, X.M., Shang, Y.D., Zheng, X.X. Lower bounds for the blow-up time to a nonlinear viscoelastic wave equation with strong damping. Appl. Math. Lett., 76: 66–73 (2018)

    Article  MathSciNet  Google Scholar 

  10. Philippina, G.A., Vernier Pirob, S. Lower bound for the lifespan of solutions for a class of fourth order wave equations. Appl. Math. Lett., 50: 141–145 (2015)

    Article  MathSciNet  Google Scholar 

  11. Philippin, G.A. Blow-up phenomena for a class of fourth order parabolic problems. Proc. Amer. Math. Soc., 143: 2507–2513 (2015)

    Article  MathSciNet  Google Scholar 

  12. Song, H.T. Blow up of arbitrarily positive initial energy solutions for a viscoelastic wave equation. Nonlinear Anal. RWA., 26: 306–314 (2015)

    Article  MathSciNet  Google Scholar 

  13. Xia, A.Y., Fan, M.S., Li, S. Blow-up and life span estimates for a class of nonlinear degenerate parabolic system with time-dependent coefficients. Acta Math. Sci., 37B: 974–984 (2017)

    Article  MathSciNet  Google Scholar 

  14. Yang, L., Liang, F., Guo, Z.H. Lower bounds for blow-up time of a nonlinear viscoelastic wave equation. Bound. Value Probl, 219: (2015)

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Acknowledgments

The authors are very much indebted to the editor and anonymous referees for their helpful comments and suggestions which greatly improved the original manuscript of this paper.

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Authors and Affiliations

  1. School of Mathematical Sciences, Qufu Normal University, Qufu, 273155, China

    Xiao-xiao Zheng

  2. School of Mathematics and Information Science, Guangzhou University, Guangzhou, 510006, China

    Ya-dong Shang

  3. School of Statistics and Mathematics, Guangdong University of Finance and Economics, Guangzhou, 510320, China

    Xiao-ming Peng

Authors
  1. Xiao-xiao Zheng
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  2. Ya-dong Shang
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  3. Xiao-ming Peng
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Corresponding author

Correspondence to Xiao-xiao Zheng.

Additional information

This paper is supported by the Natural Science Foundation of Shandong Province (No. ZR2018BA016).

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Zheng, Xx., Shang, Yd. & Peng, Xm. Blow up of Solutions for a Nonlinear Petrovsky Type Equation with Time-dependent Coefficients. Acta Math. Appl. Sin. Engl. Ser. 36, 836–846 (2020). https://doi.org/10.1007/s10255-020-0984-6

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  • DOI: https://doi.org/10.1007/s10255-020-0984-6

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