Energy decay of solutions for the wave equation with a time varying delay term in the weakly nonlinear internal feedbacks

Mohamed Ferhata,*

Energy decay of solutions for the wave equation with a time varying delay term in the weakly nonlinear internal feedbacks

Authors :

Mohamed Ferhat^a,*

Author Address :

^aDepartment of Mathematics, Usto University – P. O. Box 89, Oran 31000, ALGERIA.

*Corresponding author.

Abstract :

We consider the nonlinear wave equation in a bounded domain with a time varying delay term in the weakly nonlinear internal feedback
$$
\left(|u_{t}|^{\gamma-2}u_{t}\right)_{t}-\Delta_{x}u-\int_{0}^{t}g(t-s)\Delta
u(s)ds+ \mu_{1} \psi(u_{t}(x, t))+ \mu_{2} \psi(u_{t}(x, t-\tau(t)))=0,
$$
we study the asymptotic behavior of solutions in using the Lyapunov functional , we extend and improve the previous result due to [30] .