Contents 
We consider a starshaped network $\mathcal{R}$ of $n+1$ edges $e_j$, of length $l_j>0$, $j\in\{0,..,n\}$, connected at one vertex that we assume to be the origin $0$ of all the edges.
We consider on this plane 1D network a heat equation with a different diffusion coefficient on each string, given by the following system
$$\left\{
\begin{array}{lll}
u_{j,t}(x,t)(c_j(x)u_{j,x}(x,t))_x=g_j(x,t),&\quad\forall j\in\left\{ 0,\cdots,n \right\}, (x,t)\in (0,l_j)\times (0,T),\\
u_j(l_j,t)=h_j(t),&\quad \,\forall j\in\left\{ 0,\cdots,n \right\}, t\in(0,T),\\
u(x,0) = u^0(x), &\quad x\in\mathcal{R},
\end{array}\right.\qquad (1)
$$
under the assumptions of continuity and of Kirschoff law at the vertex $0$, given by
$$u_j(0,t)=u_i(0,t)=:u(0,t),\quad\forall i,j\in\left\{0,...,n\right\},\,0 
