Abbes Benaissa, Abderrahmane Beniani, Khaled Zennir

DOI Number
First page
Last page


A system of viscoelastic wave equations of Kirchhoff type is considered. For a wider class of relaxation functions, we use spaces weighted by the density function to establish a very general decay rate of the solution.


Lyapunov function; Viscoelastic; Kirchhoff type; Density; Decay rate; Weighted spaces; Coupled system


Lyapunov function, viscoelastic, Kirchho type, density, decay rate, weighted spaces, coupled system.

Full Text:



M. Abdelli and A. Benaissa, Energy decay of solutions of degenerate Kirchhoff equation with a weak nonlinear dissipation, Nonlinear Anal, 69 (2008) ,1999-2008.

F. Alabau-Boussouira, F. and P. Cannarsa, A general method for proving sharp energy decay rates for memory-dissipative evolution equations, C. R. Math. Acad. Sci. Paris, Ser. I 347, (2009), 867-872.

V. I. Arnold, Mathematical Methods of Classical Mechanics, Springer-Verlag, New York, 1989.

A. Benaissa, S. Mokeddem, Global existence and energy decay of solutions to the Cauchy problem for a wave equation with a weakly nonlinear dissipation, Abstr. Appl. Anal, 11(2004) 935-955.

K. J. Brown, ; N. M. Stavrakakis, Global bifurcation results for semilinear elliptic equations on all of Rn, Duke Math. J. 85 (1996), 77-94.

M.M. Cavalcanti, H.P. Oquendo, Frictional versus viscoelastic damping in a semilinear wave equation, SIAM J. Control Optim. 42(4)(2003)1310–1324.

C. M. Dafermos, H.P. Oquendo, Asymptotic stability in viscoelasticity, Arch. Ration. Mech. Anal. 37(1970), 297-308.

I. Lasiecka, S. A. Messaoudi and M. I. Mustafa, Note on intrinsic decay rates for abstract wave equations with memory, J. Math. Phys. 031504 (2013).

M. Kafini, uniforme decay of solutions to Cauchy viscoelastic problems with density, Elecron. J. Differential Equations Vol.2011 (2011)No. 93, pp. 1-9.

M. Kafini and S. A. Messaoudi, On the uniform decay in viscoelastic problem in Rn, Appl. Math. Comput 215 (2009) 1161-1169.

M. Kafini, S. A. Messaoudi and Nasser-eddine Tatar, Decay rate of solutions for a Cauchy viscoelastic evolution equation, Indag. Math. 22 (2011) 103-115.

N. I. karachalios, N.I; N.M. Stavrakakis, Existence of global attractor for semilinear dissipative wave equations on Rn, J. Differential Equations 157 (1999) 183-205.

G. Kirchhoff, Vorlesungen uber Mechanik,3rd ed., Teubner, Leipzig, (1983).

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

S. A. Messaoudi and Nasser-eddine Tatar, Uniform stabilization of solutions of a nonlinear system of viscoelastic equations, App. Anal. 87 (2008) 247-263.

Muhammad I. Mustafa and S. A. Messaoudi, General stability result for viscoelastic wave equations, J. Math. Phys. 53, 053702 (2012).

J. E. Munoz Rivera, Global solution on a quasilinear wave equation with memory, Boll. Unione Mat. Ital. B (7) 8 (1994), no. 2, 289-303.

Papadopulos, P.G. Stavrakakies, Global existence and blow-up results for an equations of Kirchhoff type on Rn, Topol. Methods Nolinear Anal. 17, (2001), 91-109.

M. L. Santos, Decay rates for solutions of a system of wave equations with memory, Elec. J. Diff. Equ. 38 (2002), 1-17.

R. Torrejon and J. M. Yong, On a quasilinear wave equation with memory, Nonlinear Anal. 16 (1991), no. 1, 61-78.

Kh. Zennir, General decay of solutions for damped wave equation of Kirchhoff type with density in Rn, Ann Univ Ferrara. DOI 10.1007/s11565-015-0223-x (2015).

Y. Zhou, A blow-up result for a nonlinear wave equation with damping and vanishing initial energy in Rn, Appl. Math. Lett. 18 (2005), 281-286.


  • There are currently no refbacks.

© University of Niš | Created on November, 2013
ISSN 0352-9665 (Print)
ISSN 2406-047X (Online)