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!



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


