Lax-Milgram and the existence of solution to parabolic equation

Question

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. May I get some help?

Answer 1

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.