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In this paper we study linear parabolic equations on a finite orientedstarshaped network; the equations are coupled by transmission conditions set atthe inner node, which do not impose continuity on the unknown. We consider thisproblem as a parabolic approximation of a set of first order linear transportequations on the network and we prove that, when the diffusion coefficientvanishes, the family of solutions converges to the unique solution to the firstorder equations and satisfies suitable transmission conditions at the innernode, which are determined by the parameters appearing in the parabolictransmission conditions.

Vanishing viscosity approximation for linear transport equations on finite starshaped networks