A nonautonomous Cr$C^r$‐topological equivalence involving contractions and unbounded nonlinearities

Abstract We study the smoothness of the topological equivalence between a linear equation and its unbounded nonlinear perturbation. To the best of our knowledge, without using spectral bound conditions, such a study has not been previously considered in the literature. The main result of this work fills in this gap. It shows on the positive half line that this kind of topological equivalence is of class when the linear part is a uniform contraction and the nonlinearities considered are unbounded with respect to the space variable.