In this paper we deal with well-posedness and long-time dynamics of solutions of the following infinite-dimensional version of an energy-damped Krasovskii system (N. N. Krasovskii, Stability of Motion: Applications of Lyapunov’s Second Method to Differential Systems and Equations with Delay, Stanford University Press, 1963) \(\begin{aligned} u_{tt} +A u+\varphi (\Vert u_t(t)\Vert ^{2}+\Vert A^{1/2} u(t)\Vert ^{2}) u_t+f(u)=0,\quad (*) \end{aligned}\) where A is a positive selfadjoint operator densely defined on a Hilbert space H, f is a nonlinear operator, \(\varphi \in C^1({\mathbb {R}}^+)\) is a nonnegative real function, and \(\Vert \cdot \Vert \) represents the norm in H. The existence and uniqueness of mild and regular global solutions to (*) is established in its natural weak phase space \({\mathcal {H}}=D(A^{1/2})\times H\) . Considering that \(\varphi (s)\) is strictly increasing with \(\varphi (0)\ge 0\) , we prove that the gradient dynamic system \(({\mathcal {H}},S_t)\) associated to (*) is asymptotically smooth, and consequently, the existence of a compact global attractor \({\mathfrak {A}}\) and an estimate for its Kolmogorov’s \(\varepsilon \) -entropy are established. In addition, assuming the strong condition that \(\varphi (0)>0\) , we prove that the dynamic system \(({\mathcal {H}},S_t)\) is quasi-stable, which implies that the global attractor \({\mathfrak {A}}\) has properties such as finite dimension and smoothness. Furthermore, the existence of an exponential attractor \({\mathfrak {A}}_{\exp }\) is also established. The results used from the abstract attractor theory are from the books by Chueshov and Lasiecka (Long-Time Behavior of Second Order Evolution Equations with Nonlinear Damping, Mem. Amer. Math. Soc. 195, no. 912, Providence, 2008) and (Von Karman Evolution Equations: Well-Posedness and Long-Time Dynamics, Springer Monographs in Mathematics, Springer, New York, 2010).
