A class of hyperbolic fractional Kirchhoff equations involving viscoelastic and dissipative terms

This article investigates a class of hyperbolic equations of the fractional Kirchhoff type with viscoelastic and nonlinear terms:

$ \left\{ \begin{array}{ll} u_{tt} +M([u]^{2}_{s})(-\Delta)^su-\int_{0}^{t}g(t-\tau)(-\Delta)^su(\tau)d\tau+|u_t|^{a-2}u_t+u_t+u = |u|^{b-2}u,\\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \text{in } \Omega\times (0,T), \\ u(x,t) = 0, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \text{on } \partial \Omega \times (0,T),\\ u(x,0) = u_0(x),\ \ \ u_t(x,0) = u_1(x)\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \text{in } \Omega, \end{array} \right. $

where $ [u]_{s} $ is the Gagliardo semi-norm of $ u $, $ \Omega \subset \mathbb{R}^N $ is a confined area featuring a smooth boundary, $ (-\Delta)^s $ is the fractional Laplacian with $ s\in(0, 1) $, $ 2 < a < 2\gamma < b < 2_s^* $, $ u_0 $ and $ u_1 $ are the initial function. First, we obtain the existence of global solutions by combining the potential wells with the Galerkin method. Moreover, employing the perturbed energy approach, we systematically study the asymptotic behavior of solutions.