Refuses to break up

What do you do when your significant other doesn't want to be with you but also doesn't want to break up? He shows no signs of even liking me let alone loving me, caught him messaging other women and told him if he wants to break up then let's do that. He refused, said he wanted to stay together. I feel like he's just staying as we have children and it would be awkward for him if we broke up as he'd actually have to look after them. He constantly hides his phone too. He says that messaging other women didn't matter as it was nothing physical but it matters to me...