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In this paper, we consider a second-order abstract viscoelastic equation in Hilbert spaces with infinite memory, time delay, and a kernel function h ∶ R + → R + satisfying, for all t ≥ 0, h ′ (t) ≤ − (t)G(h(t)) where and G are functions satisfying some specific properties. For this much larger class of kernel functions and under a suitable conditions, we prove well-posedness of solution by using semi-group theory. Then, we establish an explicit and general decay results of the energy solution by introducing a suitable Lyapunov functional and some properties of the convex functions. Finally, some applications are given. This work improves the previous results with finite memory to infinite memory and without time delay term to those with delay.

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The paper is concerned with a second-order abstract viscoelastic equation with time delay and a relaxation function satisfying $ h^{\prime}(t)\leq -\zeta(t) G(h(t))$. Under a suitable conditions, we establish an explicit and general decay rate results of the energy by introducing a suitable Lyaponov functional and some proprieties of the convex functions. Finally, some applications are given. This work generalizes the previous results without time delay term to those with delay.

No abstract