Optimal energy decay for a viscoelastic Kirchhoff equation with distributed delay acting on nonlinear frictional damping

Abstract

In this paper, we have analysed the influence of viscoelastic and frictional damping on the decay rate of solutions for a Kirchhoff-type viscoelastic wave equation with a distributed delay acting on nonlinear internal damping. Taking the relaxation function of a fairly large class and using the method of energy in which we introduce an adapted Lyapunov functional and by exploiting certain properties of convex functions, under certain assumptions on the constants of system, we obtain the optimal decay rate of energy in the sense that it is compatible with the decay rate of the relaxation function.

Keywords:

• Viscoelastic Kirchhoff equation
• frictional damping
• nonlinear distributed delay
• optimal decay rate

Acknowledgements

The authors would like to thank the anonymous referees for their valuable suggested comments. The first author would like to express his gratitude to DGRSDT for the financial support. The authors are grateful to the editors for their help.

Disclosure statement

No potential conflict of interest was reported by the author(s).