In this paper, we investigate the well-posedness, complete regularity and longtime dynamics for the Kirchhoff-type wave equations with degenerate energy fractional damping on a bounded domain \(\Omega \subset {\mathbb {R}}^N: u_{tt}-\phi (\Vert \nabla u\Vert ^2)\Delta u+[E_U(t)]^q(-\Delta )^\theta u_t+f(u)=0\) , together with the Dirichlet boundary condition, where \(\theta \in (0,1)\) is a dissipative index determining the strength of the dissipation, \(q>0\) and \(E_U(t)\) is the energy of the system which degenerates at the point \(U=(u,u_t)=(0,0)\) . This type of dissipations is connected with energy damping models proposed by Balakrishnan et al. (Stabilization of flexible structures, 1988; Proceedings Damping 89, Flight Dynamics Lab and Air Force Wright Aeronautical Labs, WPAFB, 1989). The main results are as follows: (i) the existence and the uniqueness of the global weak solution in natural energy space \({\mathcal {H}}\) ; (ii) the complete regularity of the weak solutions as \(t>0\) ; (iii) the existence of the global attractor of the related solution semigroup \(S^\theta (t)\) for each \(\theta \) . The most distinct and challenging aspect of this paper is the “parabolic like" nature of the dynamics under the degenerate energy level nonlinear damping. The method used here allows overcoming the difficulties arising from the degenerated energy damping and improving greatly the results on this issue for this type of models in literature before.
