Long-Time Behavior of a Wave Equation with Non-constant Delay

Abstract

This paper investigates the long-time behavior of a nonlinear wave equation with a time-varying delay and an external forcing term on a bounded domain. Under suitable assumptions on the nonlinear term, the external force, and the delay function, the existence and uniqueness of a global weak solution are recalled within the appropriate energy framework. An adapted Lyapunov functional incorporating the delay history is then constructed. By verifying the gradient property, the boundedness of the Lyapunov sublevel sets and stationary solutions, and the asymptotic smoothness of the associated dynamical system, we establish the existence of a compact global attractor. Furthermore, when the external force vanishes, an appropriate perturbed energy functional is employed to prove the exponential decay of the energy under a suitable relation between the damping coefficient, the delayed feedback coefficient, and the derivative of the delay function.

Keywords

Wave equation / Nonlinear system / Global attractor / Existence / Time-varying delay / Exponential decay / External forcing / 35L05 / 35B40 / 34K35 / 37L30

References

[1] Abd-Elmageed H, Roshid MM, Ahmed SAA, Abdalla M. Analysis of bifurcation, chaos, and multiplicative white noise intensity of the stochastic fractional nonlinear mCGL model in an optical communication system. J. Funct. Spaces, 2026, 202619279739.

[2] Babin, A.V., Vishik, M.I.: Attractors of Evolution Equations, Studies in Mathematics and its Applications, vol. 25. North-Holland, Amsterdam (1992)

[3] Boulaaras S, Choucha A. On a Hartree-type nonlinearity wave equation with distributed delay combined with a fractional condition. Netw. Heterogeneous Media, 2026, 211243260.

[4] Boulaaras SM, Guefaifia R, Mezouar N, Alghamdi AM. Global existence and decay for a system of two singular nonlinear viscoelastic equations with general source and localized frictional damping terms. J. Funct. Spaces, 2020, 20205085101

[5] Boulaaras SM, Choucha A, Zara A, Abdalla M, Cherif B-B. Global existence and decay estimates of energy of solutions for a new class of $p$-Laplacian heat equations with logarithmic nonlinearity. J. Funct. Spaces, 2021, 20215558818

[6] Boulaaras, S.: Nonexistence of Global Weak Solutions for Strongly Damped $p$-Laplacian Wave Equations with Logarithmic Source Terms. Appl. Anal., 1–37 (2026)

[7] Boulaaras S, Jan R, Choucha A, Zaraï A, Benzahi M. Blow-up and lifespan of solutions for an elastic membrane equation with distributed delay and logarithmic nonlinearity. Bound. Value Probl., 2024, 2024. 36

[8] Boulaaras S, Choucha A, Ouchenane D, Belkacem Cherif B. Blow-up of solutions of two singular nonlinear viscoelastic equations with general source and localized frictional damping terms. Adv. Differ. Equ., 2020, 2020. 310

[9] Chueshov, I., Lasiecka, I.: Long-Time Behavior of Second Order Evolution Equations with Nonlinear Damping, vol. 195, American Mathematical Society (2008)

[10] Chueshov, I., Lasiecka, I.: Von Kármán Evolution Equations: Well-Posedness and Long-Time Dynamics. Springer Monographs in Mathematics, Springer, New York (2010)

[11] Datko R, Lagnese J, Polis MP. An example on the effect of time delays in boundary feedback stabilization of wave equations. SIAM J. Control. Optim., 1986, 24: 152-156.

[12] Guesmia A. Well-posedness and exponential stability of an abstract evolution equation with infinite memory and time delay. IMA J. Math. Control Inform., 2013, 30: 507-526.

[13] Hale, J.K.: Asymptotic Behavior of Dissipative Systems, Mathematical Surveys and Monographs, vol. 25. American Mathematical Society, Providence, RI (1988)

[14] Jorge Silva MA, Ma TF. Long-time dynamics for a class of Kirchhoff models with memory. J. Math. Phys., 2013, 54. 021505

[15] Lions, J.-L.: Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires. Dunod-Gauthier-Villars, Paris (1969)

[16] Liu G, Zhang H. Well-posedness for a class of wave equations with past history and a delay. Z. Angew. Math. Phys., 2016, 67: 14.

[17] Liu G, Yue H, Zhang H. Long-time behavior for a wave equation with time delay. Taiwanese J. Math., 2017, 21: 107-129.

[18] Hai NM, Luc NH, Nghia BD. On the well-posedness of mild solutions to certain nonlinear wave equations. Filomat, 2025, 391967236742.

[19] Nicaise S, Pignotti C. Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks. SIAM J. Control. Optim., 2006, 45: 1561-1585.

[20] Nicaise S, Pignotti C. Interior feedback stabilization of wave equations with time delay. Electron. J. Differ. Equ., 2011, 201141120

[21] Nicaise S, Pignotti C. Stabilization of the wave equation with boundary or internal distributed delay. Differ. Integr. Equ., 2008, 21: 935-958

[22] Ragusa MA, Razani A. Weak solutions for a system of quasilinear elliptic equations. Contrib. Math., 2020, 1: 11-16

[23] Park SH. Long-time behavior for suspension bridge equations with time delay. Z. Angew. Math. Phys., 2018, 69: 1-12.

[24] Xu GQ, Yung SP, Li LK. Stabilization of wave systems with input delay in the boundary control. ESAIM Control Optim. Calc. Var., 2006, 12: 770-785.

Rights & permissions

School of Mathematical Sciences, University of Science and Technology of China and Springer-Verlag GmbH Germany, part of Springer Nature