Asymptotic stability for odd perturbations of the stationary kink in the variable-speed \phi^4 model

We consider the \phi ^4 model in one space dimension with propagation speeds that are small deviations from a constant function. In the constant-speed case, a stationary solution called the kink is known explicitly, and the recent work of Kowalczyk, Martel, and Muñoz established the asymptotic stability of the kink with respect to odd perturbations in the natural energy space. We show that a stationary kink solution exists also for our class of nonconstant propagation speeds, and extend the asymptotic stability result by taking a perturbative approach to the method of Kowalczyk, Martel, and Muñoz. This requires an understanding of the spectrum of the linearization around the variable-speed kink.