Can anybody please let me know an idea about solving the following PDE for a given initial condition $T_0=0$ and boundary conditions $T(0,w)=u_0$?
\begin{equation*} \nabla_w T_t(t,w) + T(t,w) . \nabla_w L_w + L_w . \nabla_w T(t,w) = 0 \end{equation*} where $L_w = \sum_{i,j} w_{i,j} x_{i,j}$ and $x \in \{0,1\}$ and $0 \leq w\leq 1$.
The above formula is gradient of heat equation on a edge-weighted graph. $x_{ij}$ are present if the edge is in graph. $w$ is the edge-weight of graph. and finally, $L_w$ is the weighted Laplacian of the graph