I know that the wave equation doesn't satisfy a maximum principle but I have also heard that hyperbolic equations do not satisfy any maximum principle. But I don't know any reference or proof regarding that. It would be really helpful if I get a help or resource regarding the same.

And a second related question is: I have also heard in one of the lectures I attended that subsolutions cannot touch even tangentially since both the strong maximum principle and the weak maximum principle says that subsolution doesn't intersect each other. However I don't get to see this geometry behind the weak solution. Any help is very much appreciated.