Global existence and general decay for a nonlinear viscoelastic equation with nonlinear localized damping and velocity-dependent material density

Abstract

In this article, we investigate a nonlinear viscoelastic equation with nonlinear localized damping and velocity-dependent material density. We prove the global existence of weak solutions and general decay of the energy by using the Faedo–Galerkin method [Z.Y. Zhang and X.J. Miao, Global existence and uniform decay for wave equation with dissipative term and boundary damping, Comput. Math. Appl. 59 (2010), pp. 1003–1018; J.Y. Park and J.R. Kang, Global existence and uniform decay for a nonlinear viscoelastic equation with damping, Acta Appl. Math. 110 (2010), pp. 1393–1406] and the perturbed energy method [Zhang and Miao (2010); X.S. Han, and M.X. Wang, Global existence and uniform decay for a nonlinear viscoelastic equation with damping, Nonlinear Anal. TMA. 70 (2009), pp. 3090–3098], respectively. Furthermore, for certain initial data and suitable conditions on the relaxation function, we show that the energy decays exponentially or polynomially depending the rate of the decay of the relaxation function. This result is an improvement over the earlier ones in the literature.

Keywords:

• global existence
• nonlinear viscoelastic equation
• general decay
• nonlinear localized damping
• perturbed energy method

AMS Subject Classifications::

• 35L35
• 35B40

Acknowledgements

The authors are highly grateful for the anonymous referee's careful reading and comments on this article. Z.-Y. Zhang would like to thank Dr. Professor Zhen-Hai Liu and Dr. Xiang-Yang Gan for their useful discussions concerning this article.