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We consider the second order initial value problem in a Hilbert space, which is a singular perturbation of a first order initial value problem. The difference of the solution and its singular limit is estimated in terms of the small parameter $\varepsilon .$ The coefficients are commuting self-adjoint operators and the estimates hold also for the semilinear problem.
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[2] Fattorini, H.O.: Second Order Linear Differential Equations in Banach Spaces. North Holland, 1985. MR 0797071 | Zbl 0564.34063
[3] Najman, B.: Time singular limit of semilinear wave equations with damping. J. Math. Anal. Appl. 174 (1991), 95–117. DOI 10.1006/jmaa.1993.1104 | MR 1212920
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