GENERAL DECAY FOR A SEMILINEAR WAVE EQUATION WITH BOUNDARY FRICTIONAL AND MEMORY CONDITIONS

- Journal title : Bulletin of the Korean Mathematical Society
- Volume 51, Issue 3, 2014, pp.681-689
- Publisher : The Korean Mathematical Society
- DOI : 10.4134/BKMS.2014.51.3.681

Title & Authors

GENERAL DECAY FOR A SEMILINEAR WAVE EQUATION WITH BOUNDARY FRICTIONAL AND MEMORY CONDITIONS

Park, Sun Hye;

Park, Sun Hye;

Abstract

In this paper, we investigate the influence of boundary dissipations on decay property of the solutions for a semilinear wave equation with damping and memory condition on the boundary using the multiplier technique.

Keywords

wave equation;boundary damping;memory condition;general decay rate;Lyapunov functional;

Language

English

Cited by

1.

2.

References

1.

M. Aassila, M. M. Cavalcanti, and J. A. Soriano, Asymptotic stability and energy decay rates for solutions of the wave equation with memory in a star-shaped domain, SIAM J. Control Optim. 38 (2000), no. 5, 1581-1602.

2.

F. Alabau-Boussouira, Convexity and weighted integral inequalities for energy decay rates of nonlinear dissipative hyperbolic systems, Appl. Math. Optim. 51 (2005), no. 1, 61-105.

3.

M. M. Cavalcanti, V. N. Domingos Cavalcanti, and P. Martinez, General decay rate estimates for viscoelastic dissipative systems, Nonlinear Anal. 68 (2008), no. 1, 177-193.

4.

M. M. Cavalcanti and A. Guesmia, General decay rates of solutions to a nonlinear wave equation with boundary conditions of memory type, Differential Integral Equations 18 (2005), no. 5, 583-600.

5.

A. Guesmia, A new approach of stabilization of nondissipative distributed systems, SIAM J. Control Optim. 42 (2003), no. 1, 24-52.

6.

A. Guesmia and S. A. Messaoudi, General energy decay estimates of Timoshenko systems with frictional versus viscoelastic damping, Math. Methods Appl. Sci. 32 (2009), no. 16, 2102-2122.

7.

V. Komornik, Exact Controllability and Stabilization: The Multiplier Method, John Wiley and Sons, Masson, 1994.

8.

V. Komornik and E. Zuazua, A direct method for the boundary stabilization of the wave equation, J. Math. Pures Appl. (9) 69 (1990), no. 1, 33-54.

9.

I. Lasiecka and D. Tataru, Uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping, Differential Integral Equations 6 (1993), no. 3, 507-533.

10.

P. Martinez, A new method to obtain decay rate estimates for dissipative systems, ESAIM Control Optim. Calc. Var. 4 (1999), 419-444.

11.

S. A. Messaoudi, General decay of solutions of a viscoelastic equation, J. Math. Anal. Appl. 341 (2008), no. 2, 1457-1467.

12.

S. A. Messaoudi and A. Soufyane, General decay of solutions of a wave equation with a boundary control of memory type, Nonlinear Anal. R.W.A. 11 (2010), no. 4, 2896-2904.

13.

J. Y. Park and S. H. Park, On solutions for a hyperbolic system with differential inclusion and memory source term on the boundary, Nonlinear Anal. 57 (2004), no. 3, 459-472.

14.

S. H. Park, J. Y. Park, and J. M. Jeong, Boundary stabilization of hyperbolic hemivariational inequalities, Acta Appl. Math. 104 (2008), no. 2, 139-150.

15.

R. Triggiani, Wave equation on a bounded domain with boundary dissipation: An operator approach, J. Math. Anal. Appl. 137 (1989), no. 2, 438-461.