Asymptotic stability of kink with internal modes under odd perturbation

We give a sufficient condition, in the spirit of Kowalczyk–Martel–Munoz–Van Den Bosch (Ann PDE 7(1):Paper No. 10, 98, 2021), for the local asymptotic stability of kinks under odd perturbations. In particular, we allow the existence of quite general configuration of internal modes. The extension of our result to moving kinks remains an open problem.