General decay result for nonlinear viscoelastic equations

Optimal Decay Rate Estimates of a Nonlinear Viscoelastic Kirchhoff Plate

“…In this paper, we continue to study (1)-(3), in which we consider σ � 1 for simplicity, with minimal conditions on the L 1 (0, ∞) relaxation function g (see (12)). We establish explicit and general energy decay results of systems (1)-(3) by using the idea of Mustafa [22,23] and some properties of convex functions developed in [18,39]. We point out that the decay results established here are optimal exponential and polynomial rates for 1 ≤ p < 2 when G(t) � t p , which improved the previous known results for 1 ≤ p < (3/2).…”

“…We refer the reader to Cavalcanti et al [15,16], Lasiecka et al [17,18], Mustafa [19], Mustafa and Messaoudi [20], and Xiao and Liang [21]. Very recently, in [22,23], Mustafa considered two classes of wave equations and proved general and explicit decay results of energy by using a new general assumption on g: g ′ (t) ≤ − ξ(t)H(g(t)). For viscoelastic plate equation, Rivera et al [24] studied the following equation:…”

General decay rate for a class of weakly dissipative second‐order systems with memory

“…Remark As is in Mustafa, we present the following: (1)From assumption (A.1) we deduce that and assumption (A.2) implies that there exists t 0 > 0 such that The nonincreasing property of g implies that Continuity of G on [0, r ] yields, for some constants a , b > 0 Consequently, for any t ∈ [0, t 0 ], we have and, hence, …”

“…() In the case of finite history, that is, u 0 ( t ) = 0 for t < 0, see Berrimi and Messaoudi, Messaoudi, Messaoudi and Al‐Khulaifi, Muñoz Rivera et al, and Mustafa. () In particular, Rivera et al considered the interpolating cases α ∈ (0,1) and a relaxation function g , which decay exponentially to 0 at infinity, that is, …”

New general decay results for a von Karman plate equation with memory-type boundary conditions

“…As in [17,21,22], we make the following assumptions on the kernels k 1 and k 2 . (A2) The functions k i (i=1,2): R + → R + are nonincreasing and twice differentiable functions satisfying for any t ≥ 0,…”