General decay for a viscoelastic problem with not necessarily decreasing kernel

General decay of nonlinear viscoelastic Kirchhoff equation with Balakrishnan‐Taylor damping and logarithmic nonlinearity

“…More precisely, the authors got exponential decay result as time goes to infinity in the case h ′ ( t )+ γh ( t ) ≥ 0 for all t ≥ 0 provided that for some α >0. Their proof is based on the use of a new “Lyapunov functional.” In this current work, we follow up the same steps of previous results in Mesloub and Boulaaras and Boumaza and Boulaaras for a new class of Kirrchoff hyperbolic equation in bounded domains with Balakrishnan‐Taylor damping, with respect to the same conditions in the previous ones in these studies …”

“…As in previous works,) we impose the following conditions on the relaxation function h . Namely, we suppose that the kernel is a function satisfying (A1) (A2) There exists a positive differentiable function such that and for α >0, and satisfies, for some positive constant L , …”

“…Viscoelasticity and related phenomena are interested in the biological materials study. A viscoelastic material will return to its original shape after any deforming force has been removed even though it will take time to do so (see previous studies). In the absence of the Balakrishnan‐Taylor damping in our problem , ie, σ =0, problem has been extensively studied, and several results concerning existence, nonexistence, and asymptotic behavior have been established (see, eg, Nakao, Ono, and Li and Tsai for the case h ( t )=0 and Cavalcanti et al for the case ξ 1 =0 ).…”

“…In addition to these assumptions, if then, in this case, an explicit rate of decay is given. In Mesloub and Boulaaras, the exponential decay for the solution of viscoelastic wave equation with not necessarily decreasing kernel is considered. More precisely, the authors got exponential decay result as time goes to infinity in the case h ′ ( t )+ γh ( t ) ≥ 0 for all t ≥ 0 provided that for some α >0.…”

“…In recent years, problems involving viscoelasticity have been studied in many papers, and several results concerning the global existence and decay of global solution have been established (see previous studies [1][2][3][4][5][6][7]10,[12][13][14][18][19][20][21][22], where the relaxation function was assumed to be either of exponential or of polynomial decay. In Mesloub and Boulaaras,7 the authors studied a viscoelastic equation for more general decaying kernels and established some general decay results, from which the usual exponential and polynomial rates are only special cases. Later, many papers have appeared using similar techniques and obtaining similar general decay results (see, among others, Zarai et al, 22 Al-Gharabli, 23 and Boulaaras et al 24 and Zarai and Tatar 11 ).…”