I study the following system of equations :
$$ \left\{ \begin{aligned} \partial_t u - d_u \partial^2_{\theta \theta} u &= f(u,v) \quad u \in S(0,1)\\ \partial_t v - d_v \Delta_{x,y} v &= g(u,v) \quad v \in B(0,1) \end{aligned} \right. $$
embodied with suitable initial and boundary conditions, for instance of Neumann type:
$$ \left\{ \begin{aligned} u(\theta,0) = u_0(\theta) \\ v(x,y,0) = v_0(x,y) \\ \partial_nv(\cos(\theta),\sin(\theta),t) = u(\theta,t) \end{aligned} \right. $$
Here $S(0,1)$ denotes the unit sphere in $\mathbb{R}^2$ and $B(0,1)$ the unit ball.
I want to establish its variational formulation, and I am struggling with $u$. I don't know in which functional space I should take the test function.