Best book for Linear stability analysis

Wow! A serious control systems post by someone new! And I can't answer it! Dangit!

I assume you mean partial differential equations. All of the solutions that I have seen to PDEs have been in some engineering class or another, in the context of that specific class.

Further, I have a sneaking suspicion that while some partial differential equations involving time can be solved for stability, many can't. It probably boils down to whether you can separate the variables and somehow treat the time-based part separately.

I'm sorry I can't be more helpful -- please post your answer if you find it.