An optimal estimate for the difference of solutions of two abstract evolution equations

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Local and global smoothing effects for some linear hyperbolic equations with a strong dissipation (2014, methodology): “In order to treat the most general possible situation, we follow a multiplier technique similar to that used in [9] to study a quite different question in which the continuous spectrum of the generator has a major importance.”
Decay properties of solutions to the Cauchy problem for the damped wave equation with absorption (2006, background): “See also Marcati and Nishihara [23] in 1-dimensional space, Hosono and Ogawa [15] in 2-dimensional space and Narazaki [25] in general space dimension and Ikehata [16], Ikehata and Nishihara [19], Chill and Haraux [3] in the abstract setting.”
Energy decay estimates for wave equations with a fractional damping (2012, methodology): “The advantage of this approach is that we do not assume a special shape of initial velocity like Ponce or Xu– Wang when one derives energy decay estimates, and the results can be soon generalized to the abstract theory by combining the Chill–Haraux [3] abstract theory based on the spectral analysis ”
Diffusion phenomenon in Hilbert spaces and applications (2011, methodology): “In [28] Zuazua and his coauthors consider the above system (4.1) in the setting of periodic coefficients b bounded in L∞ .”