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Discontinuity, Nonlinearity, and Complexity

Dimitry Volchenkov (editor), Dumitru Baleanu (editor)

Dimitry Volchenkov(editor)

Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA

Email: dr.volchenkov@gmail.com

Dumitru Baleanu (editor)

Cankaya University, Ankara, Turkey; Institute of Space Sciences, Magurele-Bucharest, Romania

Email: dumitru.baleanu@gmail.com


Global Existence and Finite Time Blow-Up in a New Class of Non-Linear Viscoelastic Wave Equation

Discontinuity, Nonlinearity, and Complexity 11(2) (2022) 275--284 | DOI:10.5890/DNC.2022.06.007

Tebba Zakia$^{1}$, Hakima Degaichia$^{2}$, Hadia Messaoudene$^{3}$

$^{1}$ Laboratory of Mathematics, Informatics and Systems, Larbi Tebessi, University, Tebessa, Algeria

$^{2}$ Department of Mathematics and Computer Science, Larbi Tebessi, University, Tebessa, Algeria

$^{3}$ Faculty of Economics Sciences and Management, Larbi Tebessi, University, Tebessa, Algeria

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Abstract

A new class of nonlinear viscoelastic wave equation is studied. Under appropriate conditions imposed on h, the global existence of solutions with any initial data is proved when $m\geq p$, and a finite time blow-up with negative initial energy is obtained when $p>m$.

References

  1. [1]  Haraux, A. and Zuazua, E. (1988), Decay estimates for some semilinear damped hyperbolic problems, Arch. Rational Mech. Anal., 150, 191-206.
  2. [2]  Kopackova, M. (1989), Remarks on bounded solutions of a semilinear dissipative hyperbolic equation, Comment. Math. Univ. Carolin., 30(4), 713-719.
  3. [3]  Ball, J. (1977), Remarks on blow up and nonexistence theorems for nonlinear evolutions equations, Quart. J. Math. Oxford., 28(4), 473-486.
  4. [4]  Kalantarov, V.K. and Ladyzhenskaya, O.A. (1978), The occurrence of collapse for quasilinear equations of parabolic and hyperbolic type, J. Soviet Math., 10(1), 53-70.
  5. [5]  Levine, H.A. (1974), Instability and nonexistence of global solutions of nonlinear wave equation of the form $Pu_{tt}=Au+F(u)$, Trans. Amer. Math. Soc., 192, 1-21.
  6. [6]  Levine, H.A. (1974), Some additional remarks on the nonexistence of global solutions to nonlinear wave equation, SIAM J. Math. Anal., 5(1), 138-146.
  7. [7]  Georgiev, V. and Todorova, G. (1994), Existence of solutions of the wave equation with nonlinear damping and source terms, J. Diff. Eqns., 109(2), 295-308.
  8. [8]  Levine, H.A. and Serrin, J. (1997), A global nonexistence theorem for quasilinear evolution equation with dissipation, Arch. Rational Mech. Anal., 137, 341-361.
  9. [9]  Levine, H.A. and Ro Park, S. (1998), Global existence and global nonexistence of solutions of the Cauchy problem for a nonlinearly damped wave equation, J. Math. Anal. Appl., 228(1), 181-205.
  10. [10]  Messaoudi, S.A. (2001), Blow up in a nonlinearly damped wave equation, Math. Nachr., 231, 1-7.
  11. [11]  Vitillaro, E. (1999), Global nonexistence theorems for a class of evolution equations with dissipation, Arch. Ration. Mech. Anal., 149(2), 155-182.
  12. [12]  Cavalcanti, M.M., Domingos Cavalcanti, V.N., and Soriano, J.A. (2002), Exponential decay for the solution of semilinear viscoelastic wave equations with localized damping, Elect. J. Diffe. Eqns., 44, 1-14.
  13. [13]  Messaoudi, S.A. (2003), Blow up and global existence in a nonlinear viscoelastic wave equation, Math. Nachr., 260(1), 58-66.
  14. [14]  Cavalcanti, M.M., Domingos Cavalcanti, V.N., and Ferreira, J. (2001), Existence and uniform decay for nonlinear viscoelastic equation with strong damping, Math. Meth. Appl. Sci., 24(14), 1043-1053.