This might be a way to go to start going about it:
Conway's Game of Life is Turing complete: it is possible to simulate a universal turing machine within the Game of Life.
Deciding whether a Turing machine will halt or continue infinitely for an input is the "Halting Problem". It is not possible to have a general algorithm that decides the Halting Problem for all possible inputs to a Turing machine simulated on the Game of Life.
Thus the Halting Problem is also undecidable for arbitrary inputs on particular subsets of initial patterns on the Game of Life: specifically those which implement a Turing machine simulation.