Asymptotic Behavior for a Viscoelastic Kirchhoff-Type Equation with Delay and Source Terms

Asymptotic Stability for a Viscoelastic Equation With the Time-Varying Delay

“…In the absence of the source term b|u| p−2 u and the time delay is constant in problem (1.1), Wu [21] proved an energy decay by the similar method in [13], and generalized the results to the time-varying delay in [23]. There are many papers concerning with the stability of viscoelastic equations with time delay, the interested readers may refer to [2,7,18] and the reference therein. However, the relaxation function g are mainly limited to satisfying among the three conditions, which are (1.5), (1.6) and that g : R + → R + is a differentiable function satisfying g(0) > 0 and (1.2), and there exists a positive function G ∈ C 1 (R + ) and G is linear or strictly increasing and strictly convex C 2 function on (0, r], r < 1, with G(0) = G ′ (0) = 0, such that g ′ (t) ≤ −G(g(t)) for t > 0.…”

Asymptotic stability for a class of viscoelastic equations with general relaxation functions and the time delay

“…In the absence of the source term b|u| p−2 u in problem (1.1), Wu [21] proved an energy decay by the similar method in [11], and generalized the results to the time-varying delay in [22]. There are many papers concerning with the stability of viscoelastic equations with time delay, the interested readers may refer to [3,8,16] and the reference therein. However, the relaxation function g are mainly limited to satisfying among the three conditions, which are (1.5), (1.6) and that g : R + → R + is a differentiable function satisfying g(0) > 0 and (1.2), and there exists a positive function G ∈ C 1 (R + ) and G is linear or strictly increasing and strictly convex C 2 function on (0, r], r < 1, with G(0) = G ′ (0) = 0, such that g ′ (t) ≤ −G(g(t)) for t > 0.…”