  Hein and Pr\"{u}ss [J. Differential Equations, 261(2016)4709-4727] presented
a version of Hartman-Grobman type $C^{0}$ linearization result for semilinear
hyperbolic evolution equations. They showed that the linearising map
(homomorphism) and its inverse are H\"{o}lder continuous. An important
question: is it possible to improve the regularity of the homomorphisms? In the
present paper, we prove that if the mild solutions of semilinear system are
bounded, then the regularity of the homomorphisms is Lipchitzian, but the
inverse is merely H\"{o}lder continuous. We also give a generalized local
linearization result in this paper. Finally, some applications end the paper.
As pointed out by Backes [J. Differential Equations, 297 (2021) 536-574], even
if the diffeomorphism $F$ is $C^{\infty}$, the homomorphism can fail to be
locally Lipschitz. The homomorphisms are in general only locally H\"older
continuous. However, by establishing two effective dichotomy integral
inequalities, we prove that the conjugacy is Lipchitzian, but the inverse is
H\"{o}lder continuous. Our result is the first one to observe the higher
regularity of homomorphisms in the Hartman-Grobman theorem.
