On the Decay and Global Existence of Solutions to a Nonlinearly Damped Wave Equation with Variable Exponents and Delay
DOI:
https://doi.org/10.5890/JAND.2025.12.011Abstract
In this paper, we consider a nonlinear wave equation with variable exponents and time-varying delay. We prove a global existence result using the well depth method and by a lemma by Komornik, we establish the decay estimates for the solution under suitable assumptions on the variable exponents $m,p$ and the initial data. This work generalizes and extends several works in the literature.References
[1] Levine, H. (1974), Some additional remarks on the nonexistence of global solutions to nonlinear wave equations, SIAM Journal on Mathematical Analysis, 5(1) 138-146.
[2] Kopackova, M. (1989), Remarks on bounded solutions of a semilinear dissipative hyperbolic equation, Commentationes Mathematicae Universitatis Carolinae, 30(4), 713-719.
[3] Vitillaro, E. (1999), Global nonexistence theorems for a class of evolution equations with dissipation, Archive for Rational Mechanics and Analysis, 149(2), 155-182.
[4] Levine, H. and Serrin, J. (1997), Global nonexistence theorems for quasilinear evolution equations with dissipation, Archive for Rational Mechanics and Analysis, 137(4), 341-361.
[5] Messaoudi, S. (2001), Blow up in a nonlinearly damped wave equation, Mathematische Nachrichten, 231(1), 1-7.
[6] Nicaise, S. and Pignotti, C. (2006), Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks, SIAM Journal on Control and Optimization, 45(5), 1561-1585.
[7] Zuazua, E. (1990), Exponential decay for the semi-linear wave equation with locally distributed damping, Communications in Partial Differential Equations, 15, 205-235.
[8] Nicaise, S., Pignotti, C., and Valein J. (2011), Exponential stability of the wave equation with boundary time-varying delay, Discrete and Continuous Dynamical Systems - S, 4(3), 693-722.
[9] Nicaise, S. and Pignotti, C. (2008), Stabilization of the wave equation with boundary or internal distributed delay, Differential and Integral Equations, 21(9-10), 935-958.
[10] Kafini, M., Messaoudi, S. and Nicaise, S. (2016), A blow-up result in a nonlinear abstract evolution system with delay, Nonlinear Differential Equations and Applications NoDEA, 23(2), 1-14.
[11] Aboulaich, R., Meskine, D., Souissi, A. (2008), New diffusion models in image processing, Computers and Mathematics with Applications, 56(4), 874-882.
[12] Lian, S., Gao, W., Cao, C. and Yuan, H. (2008), Study of the solutions to a model porous medium equation with variable exponent of nonlinearity, Journal of Mathematical Analysis and Applications, 342(1), 27-38.
[13] Chen, Y., Levine, S. and Rao, M. (2006), Variable exponent, linear growth functionals in image restoration, SIAM Journal on Applied Mathematics, 66, 1383-1406.
[14] Antontsev, S. (2011), Wave equation with $p(x,t)$-Laplacian and damping term: blow-up of solutions. Comptes Rendus Mécanique, 339(12), 751-755.
[15] Antontsev, S. (2011), Wave equation with $p(x,t)$-Laplacian and damping term: existence and blow-up, Differential Equations & Applications, 3(4), 503-525.
[16] Guo, B. and Gao, W. (2014), Blow-up of solutions to quasilinear hyperbolic equations with $p(x, t)$-Laplacian and positive initial energy, Comptes Rendus Mécanique, 342(9), 513-519.
[17] Messaoudi, S. and Talahmeh, A. (2017), A blow-up result for a nonlinear wave equation with variable-exponent nonlinearities., Applicable Analysis, 96(9), 1509-1515.
[18] Korpusov, M. (2012), Non-existence of global solutions to generalized dissipative Klein-Gordon equations with positive energy, Electronic Journal of Differential Equations, 119:1-10.
[19] Antontsev, S. and Shmarev, S. (2015), Evolution PDEs With Nonstandard Growth Conditions, Existence, Uniqueness, Localization, Blow-Up, Atlantis Studies in Differential Equations, 4, Atlantis Press, Paris, p. xviii+409.
[20] Antontsev, S. (2011), Wave equation with $p(x, t)$-Laplacian and damping term: Existence and blow-up, Journal of Difference Equations and Applications, 3(4), 503-525.
[21] Galaktionov, V. and Pohozaev, S. (2003), Blow-up and critical exponents for nonlinear hyperbolic equations, Nonlinear Analysis Theory, Methods and Applications, 53(3), 453-466.
[22] Messaoudi, S., Talahmeh, A., and Al-Smail, J. (2017), Nonlinear damped wave equation: Existence and blow-up, Computers and Mathematics with Applications, 74(12), 3024-3041.
[23] Messaoudi, S. and Talahmeh, A. (2017), Blow up in solutions of a quasilinear wave equation with variable-exponent nonlinearities, Mathematical Methods in the Applied Sciences, 1-11.
[24] Komornik, V. (1994), Exact Controllability and Stabilization. The Multiplier Method, Masson-John Wiley, Paris.
Article Metrics
Usage tracking begins September 1, 2026.