I am attempting to solve the following evolution problem using the standard Galerkin method
$$\begin{cases}
\dot y(x,t)=\Delta y(x,t) +b(t) \nabla y(x,t), \ (x,t)\in \Omega\times (0,T) \\
y=0, \ \text{on}\ \partial \Omega, \ \ \ y(0)=y_0 \in L^2(\Omega)
\end{cases}$$
For reference, in Theorem 1.2, page 102 of Optimal Control of Systems, By Lions, the author concludes that the evolution problem
$$ \langle \dot y(t), v\rangle_{V^*,V} + a(t,y(t),v)=0, \quad \forall v\in V$$
possesses a unique solution $y$ in an appropriate space, provided that the bilinear form $a:[0,T]\times V^2 \to \mathbb R$ satisfies the conditions:
I am confused about the constants $c$ and $\lambda$ in (1.1) and (1.2). Are they allowed to depend on $t$?
In the example provided, the function $b$ belongs to $L^2(\Omega)$, so I cannot find such constants $c$ and $\lambda$ independent of $t$ without assuming that $b\in L^{\infty}(0,T)$.
I have observed that some authors conclude the existence of a solution to PDEs somewhat similar to the above example by establishing a priori estimates and applying the standard Galerkin method, as seen in Breiten: Lemma 1 or Addou
