Decay rates of solutions to a von Kármán system for viscoelastic plates with memory

Abstract

Abstract.We consider the dynamical von Karman equations for viscoelastic plates under the presence of a long-range memory. We find uniform rates of decay (in time) of the energy, provided that suitable assumptions on the relaxation functions are given. Namely, if the relaxation decays exponentially, then the first-order energy also decays exponentially. When the relaxation g satisfies -cig1+p(t) < g\t) <-c0g(t)1+r, 0 < g"(t) < c2g1+i (t), and g,g1+p g i'(K) with p> 2, then the energy decays as n~+t)F • ^ new Liapunov functional is built for this problem.

Existence and uniform decay for a non‐linear viscoelastic equation with strong damping

“…It is important to mention that if = = 0 in ( * ), then, we presume that following similar arguments to the ones introduced by Munoz Rivera et al in Reference [5], we can obtain the exponential decay, at least for regular solutions, assuming that the kernel of the memory decays exponentially. However, when = 0 and = 0 in ( * ) and, furthermore, keeping in mind that we are dealing with weak solutions, the arguments presented in References [5] and [11] are not applicable in the present case in view of the non-linearity of the considered equation.…”

“…However, when = 0 and = 0 in ( * ) and, furthermore, keeping in mind that we are dealing with weak solutions, the arguments presented in References [5] and [11] are not applicable in the present case in view of the non-linearity of the considered equation. This forced us to introduce the dissipative term assuming that ¿0 in ( * ).…”

“…In this direction, we can cite the work of Lagnese [4], who studied the viscoelastic plate equation and showed that the energy decays to zero as time goes to inÿnity by introducing a dissipative mechanism on the boundary of the system and the work of Munoz Rivera et al [5], who proved that the ÿrst-and second-order energy, associated with the solutions of the viscoelastic plate equation, decay exponentially provided the kernel of the memory also decays exponentially, that is, when the unique dissipation mechanism is given by the relaxation function.…”

Decay properties for the solutions of a partial differential equation with memory

“…In particular, our results are natural and more precise than those available in the literature, cf. [6,7,2,3,1], in fact, they are optimal.…”

Energy decay of dissipative plate equations with memory-type boundary conditions

“…In the presence of the time dependent coefficient θ(t), Mustafa [35] and Mustafa and Messaoudi [36] established for the wave equation a general energy decay result depending on both h and θ . On the other hand, when the unique damping mechanism is given by memory conditions, we refer to Lagnese [20] and Rivera et al [32] who considered internal viscoelastic damping and proved that the energy decays exponentially if the relaxation function g decays exponentially and polynomially if g decays polynomially. The same results were obtained by Alabau-Boussouira et al [3] for a more general abstract equation.…”