General decay for the wave model with nonlinear dissipation and a localized memory

Nacer Mahi; Mohamed  Medjden; Abbes Benaissa

doi:10.5269/bspm.76424

Nacer Mahi University of Science and Technology Houari Boumedienne
Mohamed Medjden University of Science and Technology Houari Boumedienne
Abbes Benaissa Djillali Liabes University, Algeria

Abstract

In this paper, we study the asymptotic behavior as well as the global existence of the solution of a dissipative wave equation. The exponential decay results of the energy are established via suitable Lyapunov functionals in a bounded domain $\Omega$ of $\mathbb{R}^n$.

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References

Aassila, M., A note on the boundary stabilization of a compacity coupled system of wave equations. Applied Mathematics Letters 12 (1999), 19-24.

Cavalcanti, M.M., Domingos Cavalcanti, V.N., Tebou, L.T., stabilization of the wave equation with localized compensating frictional and Kelvin-Voigt dissipating mechanism. Electron. J. Differential Equations (2017), Paper No. 83, 18 pp.

Cavalcanti, M.M., Domingos Cavalcanti, V.N., Prates Filho, J.S. and Soriano, J.A., Existence and uniform decay rates for viscoelastic problems with nonlinear boundary damping. Differential and Integral Equations 14, (2001)-1, 85-116.

Cavalcanti, M.M., Domingos Cavalcanti, V.N. and Soriano, J.A. Exponential decay for the solution of semilinear viscoelastic wave equations with localized damping Electronic Journal of Differential Equations, Vol. 2002(2002), No. 44, pp. 1-14.

Cavalcanti, M.M., Khemmoudj, A., M.Medjden, M., Uniform stabilization of the damped Cauchy-Ventcel problem with variable coefficients and dynamic boundary conditions, J.Math.Anal.Appl. 338 (2007)-2, 900-930.

Khemmoudj, A. and Mokhtari, Y., General decay of the solution to a nonlinear viscoelastic modified Von-Karman system with delay, Discrete Contin. Dyn. Syst. 39 (2019)-7, 3839-3866.

Lasiecka, I. and Tataru, D., Uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping, Differential and Integral Equations, 6 (1993)-3, 507-533.

Lebeau, G., Equation des ondes amorties, Algebraic and geometric methods in mathematical physics, 73-109, (1996) Kluwer Acad. Publ., Dordrecht.

Sabbagh, Z., Khammoudj, A. and Abdelli, M., Well-posedness and stability for a viscoelastic Petrovsky equation with a localized nonlinear damping,SeMA 81 (2024)-2, 307-328.

Tebou, L.T, Stabilization of the wave equation with a localized nonlinear strong damping. Angew. Math. Phys. 71 (2020)-1, Paper No. 22, 29 pp.

Wang, H. K. and Chen, G., Asymptotic behaviour of solutions of the one dimensional wave equation with a nonlinear boundary stabilizer, SIAM J. Control and Opt., 27 (1989)-4, 758-775.

Zuazua, E., Uniform stabilization of the wave equation by nonlinear boundary feedback. SIAM J. Control Optim. 28 (1990)-2, 466-477.

Zuazua, E., Exponential decay for the semilinear wave equation with localized damping in unbounded domains. J. Math. Pures. Appl. 70 (1991)-4, 513-529.