General decay of solutions of a nonlinear system of viscoelastic wave equations

Belkacem Said-Houari (1) , Aissa Guesmia (2) , Salim A. Messaoudi (3)

1 Université Al Hosn
2 LMAM - Laboratoire de Mathématiques et Applications de Metz
3 KFUPM - King Fahd University of Petroleum and Minerals

This work is concerned with a system of two viscoelastic wave equations with nonlinear damping and source terms acting in both equations. Under some restrictions on the nonlinearity of the damping and the source terms, we prove that, for certain class of relaxation functions and for some restrictions on the initial data, the rate of decay of the total energy depends on those of the relaxation functions. This result improves many results in the literature, such as the ones in Messaoudi and Tatar (Appl. Anal. 87(3):247-263, 2008) and Liu (Nonlinear Anal. 71:2257-2267, 2009) in which only the exponential and polynomial decay rates are considered.

Domaines: Equations aux dérivées partielles [math.AP]

Belkacem Said-Houari, Aissa Guesmia, Salim A. Messaoudi. General decay of solutions of a nonlinear system of viscoelastic wave equations. Nonlinear Differential Equations and Applications, 2011, 18 (6), pp.659-684. ⟨10.1007/s00030-011-0112-7⟩. ⟨hal-01094953⟩