I'm interested in proving the existence of parabolic equations using Lax-Milgram theory, and I saw this question: Lax-Milgram and the existence of solution to parabolic equation. I have read the book it mentions, where the author uses a "modified Lax-Milgram theorem" which I did't see in any other contexts. I wonder whether there are any other references in this subject.
Also, in the quoted MSE question, a wiki page (https://en.wikipedia.org/wiki/Lions%E2%80%93Lax%E2%80%93Milgram_theorem) is mentioned in the comments. In the wiki page, it is said we can use classical Lax-Milgram theory to solve heat equation. However, no other details are mentioned there. I believe the "classical Lax-Milgram theory" here is different from the "modified Lax-Milgram theorem" in the aforementioned book.
I really wish to see a more detailed description on how to solve heat equation using classical Lax-Milgram theory. I think it can give us more insights on parabolic pdes. Thank you if you would like to help.