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The classical $C^0$ linearization theorem for the non-autonomous differentialequations states the existence of a $C^0$ topological conjugacy between thenonlinear system and its linear part. That is, there exists a homeomorphism(equivalent function) $H$ sending the solutions of the nonlinear system ontothose of its linear part. It is proved in the previous literature that theequivalent function $H$ and its inverse $G=H^{-1}$ are both H\"{o}ldercontinuous if the nonlinear perturbation is Lipschitzian. Questions: is itpossible to improve the regularity? Is the regularity sharp? To answer thisquestion, we construct a counterexample to show that the equivalent function$H$ is exactly Lipschitzian, but the inverse $G=H^{-1}$ is merely H\"{o}ldercontinuous. Furthermore, we propose a conjecture that such regularity of thehomeomorphisms is sharp (it could not be improved anymore). We prove that theconjecture is true for the systems with linear contraction. Furthermore, wepresent the special cases of linear perturbation, which are closely related tothe spectrum.

Sharpness of $C^0$ conjugacy for the non-autonomous differential equations with Lipschitzian perturbation