Global attractors for a system of elasticity with small delays

Abstract

This paper is concerned with dynamics of elasticity systems featuring a nonlinear foundation of critical growth and delay effects. The existence of global attractors for such systems has been studied recently. Our main contribution establishes the upper-semicontinuity of attractors with respect to a small parameter multiplying the delay term. Our results are new even for the analogous scalar wave equation.

A non‐homogeneous weakly damped Lamé system with time‐dependent delay

“…Then, under assumption (4. 19), E 𝜀 (t) ≤ c 3 (E 𝜀 (0) + 1)e −𝛾t , t ≥ 0. Analogously, under assumption (4.21), E 𝜀 (t) ≤ c 5 (E 𝜀 (0) + 1)(t + 1) −p+1 , t ≥ 0.…”

“…Our main results, Theorems 3.1 and 4.1, establish the well-posedness and energy stability to system (1.5), respectively. We refer the reader to previous works 5, [19][20][21][22] for other studies on the stability of Lamé system with or without delay.…”

Dynamics of a nonlinear Lamé lattice system with nonlinear damping

“…The dynamics of systems of the form (1) have been studied by many authors, see for instance [5,11,29,36] and the references therein. Recently, the existence of attractors for Lamé systems were investigated in [3,6,7,17,19,28,40].…”

Dinâmica assintótica do sistema de Lamé fracamente amortecido com retardo dependente do tempo

“…É importante notar que o sistema de Lamé desempenha um papel relevante na modelagem para ondas sísmicas. Ver, por exemplo, (ACHENBACH, 1973;BOCANEGRA-RODRÍGUEZ et al, 2021;CERVENY, 2014;MöLLHOFF;BEAN;MEREDITH, ;PUJOL, 2003).…”

“…Os nossos principais resultados são os teoremas 12 e 13, que estabelecem a boa colocação e a estabilidade da energia do sistema (1.7), respectivamente. Indicamos ao leitor (ARAÚJO et al, 2021;BELISHEV;LASIECKA, 2002;BOCANEGRA-RODRÍGUEZ et al, 2021;MESQUITA;YAMAMOTO, 1989) para outros estudos sobre o sistema de Lamé com ou sem retardo.…”