Local well-posedness and blow-up of solution for a higher-order wave equation with viscoelastic term and variable-exponent

Document Type : Research Paper


1 Department of Mathematics, Laboratory of Applied Mathematics and History and Didactics of Mathematics (LAMAHIS), University of 20 August 1955, Skikda, Algeria

2 Department of Sciences and Technology, Laboratory of Applied Mathematics and History and Didactics of Mathematics (LAMAHIS), University of 20 August 1955, Skikda, Algeria


We investigate in this paper a value problem related to the following nonlinear higher-order wave equation $$
    \eta_{tt}+\left(  -\Delta\right)  ^{m}\eta-%
    %TCIMACRO{\dint \limits_{0}^{t}}%
    g\left(  t-s\right)  \left(  -\Delta\right)  ^{m}\eta\left(  s\right)
    ds+\eta_{t}=\left\vert \eta\right\vert ^{p\left(  x\right)  -2}\eta.
Firstly, we prove the existence and uniqueness of the local solution under suitable conditions for the relaxation function $g$ and viable-exponent $p\left(  .\right)  $, using a method, which is a mixture of the Faedo-Galarkin and Banach fixed point theorem, and prove also the solution blows up in finite time. Finally, we give a two-dimensional numerical example to illustrate the blow-up result.


[1] D. Andrade, L.H. Fatori and J.E.M. Rivera, Nonlinear transmission problem with a dissipative boundary condition of memory type, Electron J. Differ. Equ. 2006 (2006), no. 53, 1–16.
[2] S. Antontsev, J. Ferreira and E. Pi¸skin, Existence and blow up of petrovsky equation solutions with strong damping and variable exponents, Electron. J. Differ. Equ. 2021 (2021), 1–18.
[3] P. Bernner and W. von Whal, Global classical solutions of nonlinear wave equations, Math. Z. 176 (1981), 87–221.
[4] M.M. Cavalcanti, V.N. Domingos Cavalcanti, T.F. Ma and J.A. Soriano, Global existence and asymptotic stability for viscoelastic problem, Differ. Integral Equ. 15 (2002), 731–748.
[5] D. Lars, P. Harjulehto, P. Hasto and M. Ruzicka, Lebesgue and Sobolev spaces with variable exponents, Lecture Notes in Mathematics, 2017.
[6] S.A. Messaoudi and N.-e. Tatar, Exponential and polynomial decay for a quasilinear viscoelastic equation, Nolinear Anal. 68 (2008), 785-793.
[7] S.A. Messaoudi and A. Talahmeh, A blow-up result for a nonlinear wave equation with variable-exponent nonlinearities, Appl. Anal. 96 (2017), 71509–1515.
[8] S.A. Messaoudi and A. Talahmeh, Blowup in solutions of a quasilinear wave equation with variable-exponent nonlinearities, Math. Meth. Appl. Sci. 40 (2017), no. 18, 6976–6986.
[9] S.A. Messaoudi, A.A. Talahmeh and J.H. Al-Shail, Nonlinear damped wave equation: Existence and blow-up , Comput. Math. Appl. 74 (2017), no. 12, 3024–3041.
[10] J.E. Munoz Rivera, E.C. Lapa and R. Baretto, Decay rates for viscolastic plates with memory, J. Elastic. 40 (1996), 61–87.
[11] A. Ouaoua and M. Maouni, Blow-up, exponential growth of solution for a nonlinear parabolic equation with p(x)-Laplacian, Int. J. Anal. Appl. 17 (2019), no. 4, 620–629.
[12] A. Ouaoua and M. Maouni, Exponential growth of positive initial energy solutions for coupled nonlinear KleinGordon equations with degenerate damping and source terms, Bol. Soc. Paran. Mat. 40 (2022), 1–9.
[13] A. Ouaoua, A. Khaldi and M. Maouni, Existence and stabiliy results of a nonlinear Timoshenko equation with damping and source terms, Theor. Appl. Mech. 48 (2021), no. 1, 53–66.
[14] A. Ouaoua, M. Maouni and A. Khaldi, Exponential decay of solutions with Lp -norm for a class to semilinear wave equation with damping and source terms, Open J. Math. Anal. 4 (2020), no. 2, 123–131.
[15] S.H. Park and J.R. Kang, Blow-up of solutions for a viscoelastic wave equation with variable exponents, Math. Methods Appl. Sci. 42 (2019), no. 6, 2083–2097.
[16] H. Pecher, Die existenz regulaer Losungen fur Cauchy-und anfangs-randwertproble-me michtlinear wellengleichungen, Math. Z. 140 (1974), 263–679.
[17] E. Pi¸skin, Global nonexistence of solutions for a nonlinear Klein-Gordon equation with variable exponents, Appl. Math. E-Notes. 19 (2019), 315–323.
[18] A. Quarteroni and A. Valli, Numerical Approximation of Partial Differential Equations, SCM Series, vol. 23 ,Springer Verlag, Heidelberg, 1994.
[19] A. Quarteroni, R. Sacconand, and F. Saleri, Numerical Mathematics, second ed., TAM Series, vol. 37 , SpringerVerlag, New-York, 2007.
[20] F. Tahamatani and M. Shahrouzi, General existence and blow up of solutions to a Petrovsky equation with memory and nonlinear source, Bound. Value Probl. 2012 (2012), no. 50, 1–15.
[21] Y. Wang, A global nonexistence theorem for viscoelastic equation with arbitrary positive initial energy, Appl. Math. Lett. 22 (2009), 1394–1400.
[22] S.T. Wu and L.Y. Tsai, Existence and nonexistence of global solutions for a nonlinear wave equation, Taiwanese J. Math. 13 (2009), no. 6B, 2069–2091.
[23] Y. Ye, Global existence and blow-up of solutions for higher-order viscoelastic wave equation with a nonlinear source term, Nonlinear Anal. 112 (2015), 129–146.
[24] K. Zennir, Exponential growth of solutions with Lp- norm of a nonlinear viscoelastic hyperbolic equation, J. Nonlinear Sci. Appl. 6 (2013), 252–262.
[25] Y. Zhou, Global existence and nonexistence for a nonlinear wave equation with damping and source terms, Math. Nachr. 278 (2005), no. 11, 1341–1358.
Volume 14, Issue 4
April 2023
Pages 111-124
  • Receive Date: 23 December 2022
  • Revise Date: 15 February 2023
  • Accept Date: 15 February 2023
  • First Publish Date: 19 February 2023