I'm reading Partial Differential Equations by Evans (available in pdf https://math24.files.wordpress.com/2013/02/partial-differential-equations-by-evans.pdf, which is one of several copies that come up when I google "PDE Evans"), and a lot of the arguments appear to go like this:
- We are searching for a function $u$ that satisfies a particular PDE
- We define some other function, say $w$, in terms of $u$
- We show that if a specific $u$ solves the first PDE, a specific $w$ will satisfy the second
- We derive a $w$ that satisfies the second
- We conclude the corresponding $u$ satisfies the first.
An example of this occurs on pg 194 (pg 202 of the linked pdf) of the first edition, on the "Cole-Hopf transformation", but arguments of this shape (modulo my likely confusion) appear throughout the book.
My confusion is, this appears to be a logical fallacy of the form "If A then B, therefore B then A". That is, it feels like we're taking "if u satisfies then w will" and assuming that this means "if w satisfies then u will".
One idea I had is that, maybe we aren't saying this itself the proof, but rather this is helping us find a good candidate for u, then, if it just so happens it satisfies the original PDE then we've solved it, and the author isn't showing us the tedious checking portion?
(Meta point: my apologies for not reproducing an example of such an actual argument here, if there are suggestions for handling such external material I'm happy to look into them.)
