Exponential decay for semilinear wave equations with viscoelastic damping and delay feedback

Exponential decay for nonlinear abstract evolution equations with a countably infinite number of time‐dependent time delays

“…We present a semilinear wave equation with infinite memory damping and delay feedback. The effect of viscoelastic damping in wave equations with an extra delayed damping term has been analyzed in Alabau-Boussouira et al 5 for the linear case and in Paolucci and Pignotti 11 for the nonlinear one. We refer to Alabau-Boussouira et al 21 for the undelayed system.…”

“…Stability of delayed linear and nonlinear wave equation have been studied in many papers (see, for instance, previous studies 5,7,16,17 ). We also mention previous studies 11,13,[18][19][20] for the study of plate equations, also named either Euler-Bernoulli or Timoshenko model, and previous studies 4,5,11,12,21 for viscoelastic wave-type equation. In particular, we refer to Alabau-Boussouira 22 for a Timoshenko model for beams in one dimension.…”

“…Time delay effects are very common in physical and biological models, and it is well-known that the introduction of a delay term in the model can induce instability (see, e.g., Datko et al and Nicaise and Pignotti [1][2][3] ). From a mathematical point of view, it is very useful studying the stability behavior of solution to such systems, which is the main topic of a large number of papers (see, for instance, previous studies [3][4][5][6][7][8][9][10][11][12][13][14] ). In this paper, we consider H an Hilbert space endowed with inner product ⟨• , •⟩ H and norm || • || H and let U : [0, + ∞) → H satisfy the following abstract equation:…”

Stability of Multi-dimensional Nonlinear Piezoelectric Beam with Viscoelastic Infinite Memory

“…freitas [7] studied the long time dynamics of a kind of nonlinear piezoelectric beams with fractional damping and thermal effects, and fre [8] considered a nonlinear piezoelectric system with delay effect, in which the global attractor and exponential attractors are studied. For other kinds of PDEs on such issue, we refer to pao [19] for a semilinear wave equations with viscoelastic damping and delay feedback and ch2 [4] for two nonlinear systems including Korteweg-de Vries-Burgers and Kuramoto-Sivashinsky equations with memory, and the references therein.…”