Let $\Omega \subset \mathbb{R}^n$ and $T>0$. Consider the following system of two equations where the unknowns are $u$ and $v$ \begin{equation} a(x,t) \partial_t u + \Delta (v \Delta u) - \nabla \cdot(v \nabla u) = f(x,t), \ \ \ \ \ \text{ in } \Omega \end{equation} \begin{equation} \partial_t v - \nabla \cdot(D(x) \nabla v) + b(x,t) v = g(x,t) , \ \ \ \ \ \text{ in } \Omega \end{equation} \begin{equation} u = 0 , \ \ \ \Delta u =0, \ \ \ (D \nabla v) \cdot n = 0, \ \ \ \ \text{ on } \partial \Omega. \end{equation} \begin{equation} u(x,0) = u_0(x) , \ \ \ c(x,0) = c_0(x), \ \ \ \ \text{ in } \partial \Omega. \end{equation} The functions $a,f, D,b, c_0, u_0$ are all given functions with the following assumptions: $a,b \in L^\infty(0, T, L^\infty(\Omega))$, $D \in L^\infty(\Omega)$, $c_0, u_0, f \in L^2(0,T, L^2(\Omega))$ and $g \in L^2(0,T, (H^1(\Omega))')$. Where $(H^1(\Omega))'$ is the dual of $H^1(\Omega)$ I want to prove the existence and uniqueness of the solution $(u,v)$.
I derived the weak formulations of both equations, and since the second equation is independent of $u$ I treated it separately. Using Faedo-Galerkin approach, I managed to prove the existence and uniqueness of $v \in H^1(0,T, H^1(\Omega), (H^1(\Omega))')$ which means $v \in L^2(0,T, H^1(\Omega))$ and $\partial_t v \in L^2(0,T, (H^1(\Omega))')$.
Now my problem is with the first equation. I will treat $v$ as a parameter in the first equation since we already proved it exists. However, what regularity should it be? Usually parameters are in $L^\infty(0,T, L^\infty(\Omega))$ but $v$ is not.
Another thing I want to do is split the first equation into two equations by introducing $w = v \Delta u$. Does this make sense?
Any kind of help would be appreciated!