In this paper, we establish some new results on the well-posedness and longtime dynamics for the second-order evolution equation with degenerate energy damping: \(u_{tt}+Au+\varphi (\Vert A^{\frac{1}{2}}u\Vert ^2+\Vert u_t\Vert ^2)u_t+f(u)=0\) , which is a infinite dimensional version of the Krasovskii model, where A is a positive self-adjoint operator defined densely in a Hilbert space H, \(\varphi \) is a nonnegative real function, and \(\Vert \cdot \Vert \) represents the norm of H. The main contributions are that: (i) We prove the existence and the energy decay of the global solutions of the model as well as the dissipativity of the associated dynamical system under the relaxed condition: \(\varphi \in C (\mathbb {R}^+)\cap C^1(0,+\infty )\) , with \(\varphi (0)=0, \varphi (s)>0, \forall s>0\) , which removes the longstanding restrictions on this issue in literature: \(\varphi (s)\) is either uniformly Lipschitz continuous or \(\varphi (s)\sim s^q\) with \(q\ge 1/2\) . (ii) Especially when \(\varphi (s)=\gamma s^q\) , with \(\gamma >0, q\in (0,2]\) , we establish the existence of the compact global attractor of the associated dynamical system, which extends the range of the exponent q from either \(q\in [1,1+\delta ), 0<\delta \ll 1\) (for related beam model) or \(q\in [1/2,1]\) (for related wave model) in literature before to \(q\in (0,2]\) . The methods proposed here allow our overcoming the degeneration of the nonlocal energy damping, extending the range of the exponent q and improving greatly the results in literature before.
