Existence and uniqueness theory for the coupled vertex problem

Establish a full existence-and-uniqueness theory for the coupled time-fractional heat equation on a three-bond metric star graph with Kirchhoff vertex conditions and nontrivial initial data on the incoming bond.

Background

The paper establishes at-most-one-solution uniqueness for the coupled star-graph problem under the standing regularity hypothesis. It does not provide the corresponding existence argument, despite identifying an explicit Green-function construction as the expected route. The unresolved task is to develop a theory analogous to existing well-posedness results for single intervals but adapted to fractional diffusion with graph vertex conditions.