Nonexistence of Solutions of a Delayed Wave Equation with Variable-Exponents

Abstract

In this paper, we deal with a nonlinear Timoshenko equation with delay term and variable exponents. Under suitable conditions, we prove the blow-up of solutions in a finite time. Our results are more general than the earlier results. Time delays arise in many applications, for instance, it appears in physical, chemical, biological, thermal and economic phenomena. Also, delay is source of instability, a small delay can destabilize a system which is uniformly asymptotically stable. Several physical phenomena such as flows of electro-rheological fluids or fluids with temperature-dependent viscosity, nonlinear viscoelasticity, filtration processes through a porous media and image processing are modelled by equations with variable exponents.

Keywords

• Blow up
• Timoshenko equation
• Delay term

References

1. 1 S. Antontsev, Wave equation with $p(x; t)$-Laplacian and damping term: blow-up of solutions, C. R. Mecanique, 339 (2011), 751-755.
2. 2 S. Antontsev, Wave equation with $p(x; t)$-Laplacian and damping term: existence and blow-up, Differential Equations Appl., 3 (2011), 503-525.
3. 3 Y. Chen, S. Levine, M. Rao, Variable Exponent, Linear Growth Functionals in Image Restoration, SIAM Journal on Applied Mathematics, 66 (2006), 1383-1406.
4. 4 L. Diening, P. Hasto, P. Harjulehto, M.M. Ruzicka, Lebesgue and Sobolev Spaces with Variable Exponents, Springer-Verlag, 2011.
5. 5 X.L. Fan, J.S. Shen, D. Zhao, Sobolev embedding theorems for spaces $Wk;p(x) ()$, J. Math. Anal. Appl., 263 (2001), 749-760.
6. 6 M. Kafini and S. A. Messaoudi, A blow-up result in a nonlinear wave equation with delay, Mediterr. J. Math., 13 (2016), 237-247.
7. 7 O. Kovacik , J. Rakosnik, On spaces $Lp(x) ()$ and $Wk;p(x) ()$, Czech. Math. J., 41(116) (1991), 592-618.
8. 8 S. A. Messaoudi and M. Kafini, On the decay and global nonexistence of solutions to a damped wave equation with variable-exponent nonlinearity and delay, Ann. Pol. Math., 122.1 (2019), doi: 10.4064/ap180524-31-10.

9. 9 S. Nicaise and C. Pignotti, Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks, SIAM J. Control Optim., 45 (2006), 1561-1585.
10. 10 E. Pişkin, Sobolev Spaces, Seçkin Publishing, 2017 (in Turkish).
11. 11 S. P. Timoshenko, On the correction for shear of the differential equation for transverse vibrations of prismatic bars, Philos. Mag. J. Sci., 41(6) (1921), 744-746.