Decay and Blow up of Solutions for a Delayed Wave Equation with Variable-Exponents

Year 2020, Volume: 3 Issue: 1, 91 - 96, 15.12.2020

Erhan Pişkin , Hazal Yüksekkaya

Abstract

This work deals with a nonlinear wave equation with delay term and variable exponents. Firstly, we prove the blow up of solutions in a finite time for negative initial energy. After, we obtain the decay results by applying an integral inequality due to Komornik. These results improve and extend earlier results in the literature. Generally, time delays arise in many applications. For instance, it appears in physical, chemical, biological, thermal and economic phenomena. Moreover, delay is source of instability. A small delay can destabilize a system which is uniformly asymptotically stable. Recently, 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, Decay, Delay term

References

• 1 S. Antontsev, Wave equation with $p(x; t)$-Laplacian and damping term: blow-up of solutions, C. R. Mecanique, 339(12) (2011), 751-755.
• 2 S. Antontsev, Wave equation with $p(x; t)$-Laplacian and damping term: existence and blow-up, Differential Equations Appl., 3(4) (2011), 503-525.
• 3 L. Diening, P. Hasto, P. Harjulehto, M.M. Ruzicka, Lebesgue and Sobolev Spaces with Variable Exponents, Springer-Verlag, 2011.
• 4 X.L. Fan, J.S. Shen, D. Zhao, Sobolev embedding theorems for spaces $Wk;p(x) ()$ , J. Math. Anal. Appl., 263 (2001), 749-760.
• 5 M. Kafini, S.A. Messaoudi, A blow-up result in a nonlinear wave equation with delay, Mediterr. J. Math., 13 (2016), 237-247.
• 6 O. Kovacik, J. Rakosnik, On spaces $Lp(x) ()$ ; and $Wk;p(x) ()$ , Czech. Math. J., 41(116) (1991), 592-618.
• 7 V. Komornik, Exact Controllability and Stabilization. The Multiplier Method, Masson and Wiley, 1994.
• 8 D. Lars, P. Harjulehto, P. Hasto and M. Ruzicka, Lebesque and Sobolev spaces with variable exponents, Springer, 2011.
• 9 S.A. Messaoudi, A.A. Talahmeh, B low up in solutions of a quasilinear wave equation with variable-exponent nonlinearities, Math. Meth. Appl. Sci., 40 (2017), 6976-6986.
• 10 S.A. Messaoudi, Jamal H. Al-Smail and A. A. Talahmeh, Decay for solutions of a nonlinear damped wave equation with variable-exponent nonlinearities, Comput. Math. Appl., 76 (2018), 1863-1875.
• 11 S.A. Messaoudi, 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.
• 12 S. Nicaise, 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.
• 13 E. Pişkin, Sobolev Spaces, Seçkin Publishing,(2017). (in Turkish).
• 14 M. Ruzicka, Electrorheological Fluids: Modeling and Mathematical Theory, Lecture Notes in Mathematics, Springer (2000).