"a dying Earth" or "the dying Earth"
Solution 1:
By writing a dying Earth, the author is ensuring that the context of this work is understood as though there are many other possible kinds of Earth contexts that the reader may live in. Maybe it is less clear because there's only one Earth, so consider the following example:
Scooby Doo and his friends have arrived in a London under siege by aliens.
This isn't inviting the reader to consider other cities and towns named London around the world, but instead to imagine a context in which London, UK is under siege by aliens.