Stabilization of semilinear wave equations with time-dependent variable coefficients and memory

Authors

  • Sheng-Jie Li Shanxi Univ., Taiyuan, Shanxi, China
  • Shugen Chai Shanxi Univ., Taiyuan, Shanxi, China

DOI:

https://doi.org/10.58997/ejde.2023.36

Keywords:

Semilinear wave equation, time-dependent variable coefficient, memory, Riemannian geometry method

Abstract

In this article, we study the stabilization of semilinear wave equations with time-dependent variable coefficients and memory in the nonlinear boundary feedback. We obtain the energy decay rate of the solution by an equivalent energy approach in the framework of Riemannian geometry.

For more information see https://ejde.math.txstate.edu/Volumes/2023/36/abstr.html

References

M. Aassila, M. M. Cavalcanti; On nonlinear hyperbolic problems with nonlinear boundary feedback, Bulletin of the Belgian Mathematical Society-Simon Stevin, 7 (2000), no. 4, 521-540. https://doi.org/10.36045/bbms/1103055613

G. Chen; A note on the boundary stabilization of the wave equation, SIAM Journal on Control and optimization, 19 (1981), no. 1, 106-113. https://doi.org/10.1137/0319008

M. M. Cavalcanti, V. Cavalcanti, J. A. Soriano; Exponential decay for the solution of semi- linear viscoelastic wave equation with localized damping, Electronic Journal of Differential Equations, 2002 (2002) no. 44, 1-14. https://doi.org/10.57262/die/1356123377

X. M. Cao, P.-F. Yao; General decay rate estimates for viscoelastic wave equation with variable coefficients, Journal of System Science and Complexity, 27 (2014), no. 5, 836-852. https://doi.org/10.1007/s11424-014-1056-x

S. G. Chai, Y. X. Guo; Boundary stabilization of wave equations with variable coefficients and memory, Differential and Integral Equations, 17 (2004), no. 5-6, 669-680. https://doi.org/10.57262/die/1356060354

S. G. Chai, K. S. Liu; Boundary stabilization of the transmission of wave equations with variable coefficients, Chinese Annals of Mathematics, series A, 5 (2005), no. 5, 605-612.

A. Guesmia; A new approach of stabilization of nondissipative distributed systems, SIAM Journal on Control and Optimization, 42 (2003), no. 1, 24-52. https://doi.org/10.1137/S0363012901394978

B.Z. Guo, Z.C. Shao; On exponential stability of a semilinear wave equation with variable coefficients under the nonlinear boundary feedback, Nonlinear Analysis-Theory, Methods and Applications, 71 (2009), no. 12, 5961-5978. https://doi.org/10.1016/j.na.2009.05.018

Y. X. Guo, P.-F. Yao; Stabilization of Euler-Bernoulli plate equation with variable coefficients by nonlinear boundary feedback, Journal of Mathematical Analysis and Applications, 317 (2006), no. 1, 50-70. https://doi.org/10.1016/j.jmaa.2005.12.006

T. G. Ha; Global solutions and blow-up for the wave equation with variable Coefficients: I. Interior supercritical source, Applied Mathematics and Optimization, 84 (2021), 767-803. https://doi.org/10.1007/s00245-021-09785-5

J.-M. Jeong, J. Y. Park, Y. H. Kang; Energy decay rates for the semilinear wave equation with memory boundary condition and acoustic boundary conditions, Computers and Mathematics with Applications, 76 (2018), no. 31 661-671. https://doi.org/10.1016/j.camwa.2018.05.006

V. Komornik, E. Zuazua; A direct method for the boundary stabilization of wave equation, Journal de mathématiques pures et appliques, 69 (1990), no. 1, 33-54.

J. Lagnese; Note on boundary stabilization of wave equations, SIAM Journal on Control and Optimization, 26 (1988), 1250-1256. https://doi.org/10.1137/0326068

J. L. Lions; Contrlabilité exacte des systèmes distribués, Comptes Rendus de l Académie des Sciences-Series I-Mathematics, 13 (1986), 471-475.

Y.-X. Liu; Exact controllability of the wave equation with time-dependent and variable coefficients, Nonlinear Analysis-Real World Applications, 45 (2019), 226-238. https://doi.org/10.1016/j.nonrwa.2018.07.005

Y.-X. Liu; Polynomial decay rate of a variable coefficient wave equation with memory type acoustic boundary conditions, Journal of Geometric Analysis, 32 (2022), 254. https://doi.org/10.1007/s12220-022-00991-3

S.-J. Li, S. G. Chai; Stabilization of the viscoelastic wave equation with variable coefficients and a delay term in boundary feedback, submitted.

L.Q. Lu, S.J. Li, and S.G. Chai; On a viscoelastic equation with nonlinear boundary damping and source terms: Global existence and decay of the solution, Nonlinear Analysis-Real World Applications, 12 (2011), no. 1, 295-303. https://doi.org/10.1016/j.nonrwa.2010.06.016

K. S. Liu, Z. Y. Liu, B.P. Rao; Exponential stability of an abstract nondissipative linear system, SIAM Journal on Control and Optimization, 40 (2001), no. 1, 149-165. https://doi.org/10.1137/S0363012999364930

H. Li, Z.-H. Ning, F. Y. Yang; Stabilization of the critical semilinear wave equation with Dirichlet-Neumann boundary condition on bounded domain, Journal of Mathematical Analysis and Applications, 506 (2022), no. 1, 125610. https://doi.org/10.1016/j.jmaa.2021.125610

I. Lasiecka, R. Triggiani, P.-F. Yao; Inverse/observability estimates for second-order hyperbolic equations with variable coefficients, Journal of Mathematical Analysis and Applications, 235 (1999), no. 1, 13-57. https://doi.org/10.1006/jmaa.1999.6348

M. I. Mustafa, S. A. Messaoudi; General stability result for viscoelastic wave equations, Journal of Mathematical Physics, 53 (2012), no. 5, 867-872. https://doi.org/10.1063/1.4711830

Z.-H. Ning, F.Y. Yang; Stabilization of wave equations with variable coefficients and internal memory, Electronic Journal of Differential Equations, 2018 (2018), no. 160, 1-19.

J. Y. Park, T. G. Ha; Energy decay for nondissipative distributed systems with boundary damping and source term, Nonlinear Analysis-Theory, Methods and Applications, 70 (2009), no. 6, 2416-2434. https://doi.org/10.1016/j.na.2008.03.026

J. Q. Wu, S. J. Li, S.G. Chai; Uniform decay of the solution to a wave equation with memory conditions on the boundary, Nonlinear Analysis-An International Multidisciplinary Journal, 2010 (2010), no. 7, 2213-2220. https://doi.org/10.1016/j.na.2010.05.050

J. Q. Wu, S. J. Li, F. Feng; Energy decay of a variable-coefficient wave equation with memory type acoustic boundary conditions, Journal of Mathematical Analysis and Applications, 434 (2016), no. 1, 882-893. https://doi.org/10.1016/j.jmaa.2015.09.039

H. X. Wu, C. L. Shen, Y. L. Yu; An Introduction to Riemannian Geometry (in Chinese), Peking University Press, Beijing, 1989.

P.-F. Yao; On the observability inequality for exact controllability of wave equations with variable coefficients, SIAM Journal on Control and Optimization, 37 (1999), no. 5, 1568-1599. https://doi.org/10.1137/S0363012997331482

P.-F. Yao; Modeling and Control in Vibrational and Structural Dynamics-A Differential Geometric Approach, Chapman and Hall/CRC Applied Mathematics and Nonlinear Science Series. CRC Press, Boca Raton, FL, 2011.

E. Zuazua; Uniform stabilization of the wave equation by nonlinear boundary feedback, SIAM Journal on Control and Optimization, 28 (1990), no. 2, 466-477. https://doi.org/10.1137/0328025

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Published

2022-12-15

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How to Cite

Stabilization of semilinear wave equations with time-dependent variable coefficients and memory. (2022). Electronic Journal of Differential Equations, 2023(01-87), No. 36, 1-14. https://doi.org/10.58997/ejde.2023.36