I'm trying to solve the evolution problem below using the standard Galerkin method:
$$\begin{cases}
\dot y(x,t)=\Delta y(x,t) +b(t) \nabla y(x,t), \ (x,t)\in \Omega\times (0,T) \\
y=0, \ \text{on}\ \partial \Omega, \ \ \ y(0)=y_0 \in L^2(\Omega)
\end{cases}$$
In Theorem 1.2, page 102 of Optimal Control of Systems, By Lions, the author concludes that the evolution problem
$$ \langle \dot y(t), \varphi \rangle_{V^*,V} + a(t,y(t),\varphi)=0, \quad \forall \varphi\in V$$
possesses a unique solution $y$, provided that the bilinear form $a:[0,T]\times V^2 \to \mathbb R$ is as follows:
I am uncertain about the constants $c$ and $\lambda$ in equations (1.1) and (1.2). Can they depend on $t$?
In the example above, the function $b$ belongs to $L^2(\Omega)$, so I'm unable to find constants $c$ and $\lambda$ independent of $t$ unless I assume that $b\in L^{\infty}(0,T)$.
I have noticed that some authors establish the existence of a solution to PDEs similar to the above example by deriving a priori estimates and applying the standard Galerkin method, as in Breiten: Lemma 1 or Addou