This is a repost of a now-deleted MSE question that did not get any comments or answers.
$\textbf{Background}$: This question is mainly about two basic questions: How do we systematically obtain the weak formulation from the naive formulation of a PDE, and how free are we to choose the inner product to dualize with. I am studying a semilinear partial differential equation of the form \begin{equation}\partial_t u = \Delta u + F(u)\end{equation} with initial condition $u(0) = u_0$, where $\Delta$ is the Laplacian and $F$ only contains derivatives up to first order. $F$ maps an as-of-yet unspecified subset of a function space to another as-of-yet unspecified space. For example, if the problem is formulated on a suitable spatial domain $\Omega$ and $F(u) = |\nabla u|$, then $$F \colon C^\infty_0(\Omega) \rightarrow H^1_0(\Omega)$$ would be a permissible choice of domain and range. However, the core questions I will formulate should in principle make sense for any nonlinear PDE of the form $$\partial_t u = A(u).$$
$\textbf{Mathematical Setting}$: The type of solutions I work with are variational/weak solutions. In practice, this means that I fix a finite time $T>0$ and a Gelfand triple $(V,H,V^\ast)$ and aim to prove existence of solutions $u \in L^1([0,T];V) \cap C([0,T];H)$ with $u(0) = u_0 \in H$ of the $V^\ast$-valued equation $$\langle u(t),- \rangle_H = \langle u(0),-\rangle_H + \int^t_0 \, _{V^\ast}\langle \Delta u(s)+ F(u(s)),-\rangle_{V} \,\mathrm d s,$$ where $_{V^\ast}\langle -,-\rangle_{V}$ is the dual pairing on $V^\ast \times V$ and the $V^\ast$-valued integral is interpreted in the Bochner sense.
$\textbf{Example case}$: Now, as an example, consider a real Banach space $V$ continuously and densely embedded into $H = H^1_0(\Omega)$ for some convex domain with smooth boundary. As an inner product on $H$, we choose $(u,v)_{H^1_0} = \langle \nabla u, \nabla v \rangle_{L^2}$. Assume that the gradient of elements of $V$ vanishes on the boundary and that the functions in $V$ are at least twice weakly differentiable. Then we can set $$_{V^\ast}\langle \Delta u + F(u),v \rangle_{V} := -\langle \Delta u, \Delta v \rangle_{L^2} + \langle \nabla F(u), \nabla v \rangle_{L^2},$$ for all $u,v \in V$. We call this the variational extension of $\Delta u + F(u)$ from now on.
$\textbf{Question 1}$: What are the conditions on $F$ (resp. $A$), its domain and its range, in relation to $V$ and $H$, so that the inner product on $H$ uniquely determines a variational extension of $\Delta u + F(u)$ (resp. $A$)?
To make it clearer what I'm asking: Must the domain of $A$ be, for example, a dense subset of $V$? Is it necessary that $A$ maps into $H$, so that $\langle A(u),v\rangle_H$ is well-defined?
$\textbf{Question 2}$: How exactly do we define uniqueness here? One possible condition that comes to my mind is demicontinuity of the extension $$_{V^\ast}\langle A(-),-\rangle_{V} \colon V \rightarrow V^\ast$$ with $$u \mapsto \, _{V^\ast}\langle A(u),-\rangle_{V} \in V^\ast$$
$\textbf{Question 3}$: Provided there are such conditions, how do weak solutions with respect to different inner products relate to each other? More specifically, if two inner products induce equivalent norms on $H$, (in which cases) can it be assured that the equation has a solution for both dualisations?
The motivation comes from considering the inner products $(-,-)_{H^1_0}$ and $\langle -,- \rangle_{H^1}$ on $H^1_0(\Omega)$, which induce equivalent norms. In the practical cases I studied, this was the case. The existence & uniqueness theorem I applied yielded solutions for the other inner product almost trivially, once it yielded solutions for one of the inner products.
$\textbf{Question 4}$: Provided unique solutions exist for both inner products, are these solutions equal as elements of $L^1([0,T];V) \cap C([0,T];H)$ for equal initial conditions?
I am thankful for answers and hints regarding any single one of these questions!
