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Symmetry
Volume 18
Issue 9
10.3390/sym18091522
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This is an early access version, the complete PDF, HTML, and XML versions will be available soon.
Article

Cross-Tempered Fractional Damping in Coupled Viscoelastic Wave Equations: Global Existence and Long-Time Behavior

by
Iqra Kanwal
1,*,
Jianghao Hao
1,
Ahmed Bchatnia
2,*,
Muhammad Fahim Aslam
3 and
Muhammad Afnan
4
1
School of Mathematics and Statistics, Shanxi University, Taiyuan 030006, China
2
Department of Mathematics, College of Science, Qassim University, P.O. Box 6644, Buraydah 52571, Saudi Arabia
3
Department of Mathematics, University of Kotli Azad Jammu and Kashmir (UOKAJK), Kotli 11100, Pakistan
4
School of Electrical Engineering and Computer Science (SEECS), National University of Sciences and Technology (NUST), Islamabad 44000, Pakistan
*
Authors to whom correspondence should be addressed.
Symmetry 2026, 18(9), 1522; https://doi.org/10.3390/sym18091522
Submission received: 19 August 2026 / Revised: 28 August 2026 / Accepted: 9 September 2026 / Published: 11 September 2026
(This article belongs to the Topic Modeling, Stability, and Control of Dynamic Systems and Their Applications)
Versions Notes

Abstract

This paper studies a coupled system of viscoelastic wave equations with frictional damping, cross-tempered fractional damping, viscoelastic memory, and logarithmic source nonlinearities. The fractional damping acts across the two components, so that the fractional feedback in each equation is generated by the velocity of the other component. To handle the memory and fractional terms, we introduce suitable history and diffusive variables and reformulate the problem as an evolution system in an extended energy space. Under appropriate assumptions on the relaxation kernels, fractional parameters, and nonlinear exponent, we establish the local well-posedness of mild and strong solutions using semigroup theory. We then use a potential-well argument to prove global existence for initial data in the stable set. Under an additional decay condition on the relaxation kernels, an appropriate Lyapunov functional is constructed to establish the exponential decay of the energy. Finally, numerical simulations based on a finite-difference scheme and a physics-informed neural network (PINN) are used to illustrate the predicted decay behavior.
Keywords: coupled viscoelastic wave equations; cross-tempered fractional damping; tempered Caputo derivative; logarithmic nonlinearity; potential-well method; global existence; exponential stability; partial differential equations coupled viscoelastic wave equations; cross-tempered fractional damping; tempered Caputo derivative; logarithmic nonlinearity; potential-well method; global existence; exponential stability; partial differential equations

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MDPI and ACS Style

Kanwal, I.; Hao, J.; Bchatnia, A.; Aslam, M.F.; Afnan, M. Cross-Tempered Fractional Damping in Coupled Viscoelastic Wave Equations: Global Existence and Long-Time Behavior. Symmetry 2026, 18, 1522. https://doi.org/10.3390/sym18091522

AMA Style

Kanwal I, Hao J, Bchatnia A, Aslam MF, Afnan M. Cross-Tempered Fractional Damping in Coupled Viscoelastic Wave Equations: Global Existence and Long-Time Behavior. Symmetry. 2026; 18(9):1522. https://doi.org/10.3390/sym18091522

Chicago/Turabian Style

Kanwal, Iqra, Jianghao Hao, Ahmed Bchatnia, Muhammad Fahim Aslam, and Muhammad Afnan. 2026. "Cross-Tempered Fractional Damping in Coupled Viscoelastic Wave Equations: Global Existence and Long-Time Behavior" Symmetry 18, no. 9: 1522. https://doi.org/10.3390/sym18091522

APA Style

Kanwal, I., Hao, J., Bchatnia, A., Aslam, M. F., & Afnan, M. (2026). Cross-Tempered Fractional Damping in Coupled Viscoelastic Wave Equations: Global Existence and Long-Time Behavior. Symmetry, 18(9), 1522. https://doi.org/10.3390/sym18091522

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