If you got him cornered and he got away from you, note what he did to get out and take advantage of it the next time you get him cornered. If he jumped over you, then be less aggressive next time and be in the right position to SRK him out of the air.
Also, while I haven't seen you play, it's always important to note that you don't go for the cross-up when you have people cornered. Lots of people have it in their muscle memory to cross people up because it works, but you're giving up ALL your positioning for it and if he blocks it right, well now you're in the corner.
Also, if someone is always chucking fireballs, don't always jump forward over it. Neutral jumping fireballs is important, because he can't immediately throw a fireball afterwards, he has to wait until the first fireball leaves the screen so you can move forward a little bit. Throw fireballs back, block a few, make it so that he can't rely on the same timing for all his fireballs so he has to think a little bit.