# Lax-Milgram and the existence of solution to parabolic equation

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!

• You might wish to have a look at the Hille-Yosida theorem and the Lumer-Phillps theorem. Aug 9, 2021 at 15:24
• 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. Aug 9, 2021 at 15:31
• Thanks! I will read it! Aug 9, 2021 at 15:34