M. A. Jorge Silva

Sharp decay rates for a class of nonlinear viscoelastic plate models

This paper is concerned with uniform stability of the energy corresponding to a class of nonlinear plate equations with memory. It is assumed that the memory kernel g satisfies the condition g′≤−H(g) of Alabau-Boussouira and Cannarsa [A general method for proving sharp energy decay rates for memory-dissipative evolution equations, C. R. Acad. Sci. Paris Ser. I 347 (2009) 867–872], where H ∈ C(ℝ+) is positive, convex, increasing, and satisfies H(0) = 0. Then, we obtain sharp energy decay rate in the sense that it recovers the decay rate assumed to the memory kernel. To this end we use a recent approach proposed by Lasiecka and Wang [Intrinsic decay rate estimates for semilinear abstract second order equations with memory, in New Prospects in Direct, Inverse and Control Problems for Evolution Equations, Springer Series INDAM, Vol. 10 (Springer, 2014), pp. 271–303].

On a decay rate for nonlinear extensible viscoelastic beams with history setting

Abstract In this paper we deal with a nonlinear extensible viscoelastic beam model whose memory term is considered in a history setting. The goal is to extend an approach on stability first provided by Guesmia and Messaoudi (J Math Anal Appl. 2014;416:212–228) to a class of viscoelastic beams/plates with nonlinear extensible and source terms. Our stability result contributes in clarifying how the constants appearing in the decay rate depend upon the nonlinearities and the size of initial data. Thus, it also complements some results dealing with this methodology.