Getting eaten by a dragon wasn't part of Ashe's plan. Not that there was much of a plan to begin with, but it had definitely involved more chasing and less running. Ashe supposes there's one good thing about all this: if he has to be trapped in a cave with a very large predator outside, at least he's with Katsu, the company medic and Ashe's current crush.
Even better, Katsu has enough knowledge of dragons and medicine that he might just be able to hatch an escape plan. Even better than that, Ashe is discovering that Katsu might not dislike him after all. It's a small step, Ashe hopes, from not disliking him to something significantly more. That is, assuming they don't get eaten first.