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!
Best Answer
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.