General decay for Kirchhoff type in viscoelasticity with not necessarily decreasing kernel

Abstract

In this paper, a problem that arises in Kirchhoff type in viscoelasticity is considered. We obtain an asymptotic stability result of global solution, for certain class of relaxation functions not necessarily exponentially or polynomially decaying to zero. KEYWORDS exponential decay, general decay, Kirchhoff type, polynomial decay, viscoelastic equation (s) = m 0 + m 1 s , where s > 0, m 0 > 0, m 1 > 0, ≥ 1.

Existence of positive solutions of (p(x),q(x))‐Laplacian parabolic systems with right hand side defined as a multiplication of two separate functions

“…In order to discuss problem , we need some theories on which we call variable exponent Sobolev space. Firstly, we state some basic properties of spaces which will be used later (for details, see Boumaaza and Boulaaras).…”

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 ).…”

“…Tatar and Zarai 11,18 investigated exponential and polynomial decay results under some classical condition. 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.…”