Well-posedness and Exponential Decay of Energy for the Solution of a Wave Equation with Nonlinear Source and Localized Damping Termes
Download PDF
Authors: M. KOUIDRI, M. ABDELLI, M. BAHLIL AND A. B. AISSA
DOI: 10.46793/KgJMat2601.007K
Abstract:
We consider the wave equation with a locally damping and a nonlinear source term in a bounded domain. ytt − Δy + a(x)g(yt) = |y|p−2y, where p > 2. The damping is nonlinear and is effective only in a neighborhood of a suitable subset of the boundary. We show, for certain initial data and suitable conditions on g,a and p that this solution is global we use the Faedo-Galerkin method. Also we established the exponential decay of the energy when the nonlinear damping grows linearly by introducing a suitable Lyapunov functional.
Keywords:
Wave equation, localized nonlinear damping, well-posedness, Faedo-Galerkin, multiplier method, exponential stabilization.
References:
[1] R. A. Adams, Sobolev Spaces, Pure and Appl. Math. 65, Academic Press, 1978.
[2] K. Ammari, F. Hassine and L. Robbiano, Stabilization for the wave equation with singular Kelvin-Voigt damping, Arch. Ration. Mech. Anal. 236 (2020), 577–601. https://doi.org/10.1007/s00205-019-01476-4
[3] H. Brezis, Analyse Fonctionnelle. Theorie et Applications, Masson, Paris, 1983.
[4] S. Berrimi and S. A. Messaoudi, Existence and decay of solutions of a viscoelastic equation with a nonlinear source, Nonlinear Anal. 64 (2006), 2314–2331. https://doi.org/10.1016/j.na.2005.08.015.
[5] M. M. Cavalcanti, V. N. Domingos Cavalcanti and J. A. Soriano, Exponential decay for the solution of semilinear viscoelastic wave equations with localized damping, Electron. J. Differential Equations 44 (2002), 1–14.
[6] A. Haraux, Two Remarks on Dissipative Hyperbolic Problems, Research Notes in Mathematics 122, Pitman, Boston, MA, 1985, 161–179.
[7] V. Komornik, Well-posedness and decay estimates for a Petrovsky system by a semigroup approach, Acta Sci. Math. (Szeged) 60, (1995), 451–466.
[8] J. L. Lions, Quelques Methodes de Resolution des Problemes aux Limites non Lineaires, Dunod, Paris, 1969 (in French).
[9] P. Martinez, A new method to obtain decay rate estimates for dissipative systems with localized damping, Rev. Mat. Complut. 12(1) (1999), 251–283.
[10] M. Nakao, Decay of solutions of the wave equation with local degenerate dissipation, Israel J. Math. 95 (1996), 25–42. https://doi.org/10.1007/BF02761033
[11] M. Nakao, Decay of solution of the wave equation with a local nonlinear dissipation, Math. Ann. 305(3) (1996), 403–417. https://doi.org/10.1007/BF01444231
[12] L. Tebou, Stabilization of the wave equation with localized nonlinear damping, J. Differential Equations 145 (1998), 502–524. https://doi.org/10.1006/jdeq.1998.3416
[13] L. Tebou, Well-posedness and stability of a hinged plate equation with a localized nonlinear structural damping, Nonlinear Anal. 71 (2009), 2288–2297. https://doi.org/10.1016/j.na.2009.05.026
[14] L. Tebou, Stabilization of the wave equation with a localized nonlinear strong damping, Z. Angew. Math. Phys. (ZAMP) 2020 (2020), 7–22. https://doi.org/10.1007/s00033-019-1240-x
[15] E. Zuazua, Exponential decay for the semilinear wave equation with locally distributed damping, Comm. Partial Differential Equations 15 (1990), 205–235. https://doi.org/10.1080/03605309908820684
Login:
