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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.

This paper is concerned with well-posedness and long-time dynamics for a class extensible beams with nonlocal Balakrishnan–Taylor and frictional damping. The related model describes vibrations in nonlinear extensible beams arising in connection with models of oscillation in pipes and supersonic panel flutter. Our main results feature the study of the nonlinear dynamical system generated by the problem. The main novelty is to explore the global $$L^q$$ -regularity ( $$q\ge 2$$ ) in time of the nonlocal Balakrishnan–Taylor term and show how it generates a dissipative term that plays an important role in the asymptotic behavior of solutions, mainly in what concerns to achieve the useful property of quasi-stability in the theory of infinite-dimensional dynamical systems.

Synadenium grantii is a Euphorbiaceae plant commonly found in Brazil, known as Janaúba or Leitosinha, whose latex is traditionally used for several purposes. However, it is not known whether the nematicidal action of this plant latex occurs due to the action of proteases. The present work aims to evaluate the nematicidal activity of proteases from Synadenium grantii latex on Meloidogyne incognita and Panagrellus redivivus. S. grantii latex used in the present study was collected from specimens found in Universidade Federal de Viçosa, Viçosa, Minas Gerais, Brazil. The drained latex was collected in Eppendorf microtubes and immediately stored on ice at 4 °C. After this extraction, the latex was frozen (-20 °C) during 2 hours, thawed at room temperature (25 °C) and centrifuged at 10,000 g at 4 °C for 30 minutes to remove larger particles and concentrate the proteases. After the centrifugation, assays of enzymatic activity were performed in order to know in which of the phases the enzymes were found. S. grantii latex presented protease, but no chitinase activity. The results show that there was a significant difference (p

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 [Formula: see text] satisfies the condition [Formula: see text] 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 [Formula: see text] is positive, convex, increasing, and satisfies [Formula: see text]. 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].