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I think it is standard and common to use Lax-Milgram theorem to prove the existence of solution to elliptic equation. However, can we use it to establish the existence of parabolic equation? I do not find some examples in standard PDE textbooks.

Suppose I have a parabolic equation $$ \partial_t u - \partial_{x_j}(a_{ij} \partial_{x_i} u) + b_i \partial_{x_i} u + c u =f(x,t)$$ on $\Omega \times [0,T]$. Then the weak formulation should be $$ \int_{\Omega} \partial_{t} u \varphi + a_{ij} \partial_{x_i} u \partial_{x_j} \varphi + b_i \partial_{x_i} u \varphi + c u \varphi-f\varphi=0,$$ for all $\varphi(x) \in H^1_0(\Omega)$ and a.e. $t\in[0,T]$. But I do not know how can we define the bilinear mapping in this way. Or, is it impossible to prove the existence via Lax-Milgram? May I get some help? Thanks!

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    $\begingroup$ You might wish to have a look at the Hille-Yosida theorem and the Lumer-Phillps theorem. $\endgroup$
    – Jochen Glueck
    Aug 9, 2021 at 15:24
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    $\begingroup$ More specifically, the proof of Theorem 5.7 in these lecture notes from the Internet Seminar on Evolution Equations shows how the Lax-Milgram lemma can be used to prove well-posedness of parabolic equations. The proof essentially comes down to showing well-posedness of the elliptic problem - which is is hidden in the word "m-accretive" ini the proof - and then applying a generation theorem for semigroups, as indicated in my previous comment. $\endgroup$
    – Jochen Glueck
    Aug 9, 2021 at 15:31
  • $\begingroup$ Thanks! I will read it! $\endgroup$
    – mnmn1993
    Aug 9, 2021 at 15:34

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Finally I found the exact theorem from Chapter 3.1.2 in the book "Elliptic & Parabolic Equations" written by Zhuoqun Wu, Jingxue Yin and Chunpeng Wang.

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