Optimal decay of an abstract nonlinear viscoelastic equation in Hilbert spaces with delay term in the nonlinear internal damping

Article type: Research Article

Authors: Chellaoua, Houria^a | Boukhatem, Yamna^a | Feng, Baowei^{b; *}

Affiliations: [a] Laboratory of Pure and Applied Mathematics, University of Laghouat, P.O. BOX 37G, Laghouat (03000), Algeria. E-mails: h.chellaoua@lagh-univ.dz, y.boukhatem@lagh-univ.dz | [b] Department of Economic Mathematics, Southwestern University of Finance and Economics, Chengdu 611130, China. E-mail: bwfeng@swufe.edu.cn

Correspondence: [*] Corresponding author. E-mail: bwfeng@swufe.edu.cn.

Abstract: In this paper, we consider a second-order abstract viscoelastic equation in Hilbert spaces with delay term in the nonlinear internal damping and a nonlinear source term. Under some suitable assumptions on the weight of the delayed feedback, the weight of the non-delayed feedback and the behavior of the relaxation function, we establish two explicit and general decay rate results of the energy by introducing a suitable Lyaponov functional and some properties of the convex functions. Moreover, we give some applications and examples. This work generalizes the previous results without time delay term to those with delay in the nonlinear damping.

Keywords: Abstract evolution equation, general decay, nonlinear damping, nonlinear source term, time delay

DOI: 10.3233/ASY-201664

Journal: Asymptotic Analysis, vol. 126, no. 1-2, pp. 65-94, 2022