In contract to OU student I hated my adidas shoes with a passion. Worst £85 I ever spent! Quite liked Asics but adore Brookes and am on my 7th pair of their Adrenaline shoes.
If you're going to train properly for a half marathon you need to get yourself to a running store and get fitted for some proper shoes, try on lots of pairs and see what feels best. Try and get a pair on offer. Often last season's shoes have big reductions but there's nothing wrong with them, I always run in shoes a couple of years behind the current model and get them for around half price on Amazon.