Like every pet, your Petkin needs some love, too.

To take care of them:
> Go to the apartment your Petkin lives in,
> Tap your Petkin,
> Select Interact,
> Select Water, Feed, or Affection to increase the respective meter.

Or:
> Interact with your favourite Petkin on the Dressing Room.

The actions can be repeated when their timers laps.