Spoilers for The Borrower Arrietty, with facts from both the book and the movie versions of Howl's Moving Castle. At the end of the movie, the ending is ambiguous (Japanese) or he lives, (American). I posit that the answer is both, that is, he died in Japan, but he lived on in another dimension, the dimension from HMC. However, because he didn't know if actual magic existed in his world (oh, yes, there were the Borrowers, but did that mean magic?), so he "died" in Japan, and decided to live the rest of his life in the dimension where he could learn magic, and where his heart was "safe".