  We consider an infinite system of quasilinear first-order partial
differential equations, generalized to contain spacial integration, which
describes an incompressible fluid mixture of infinite components in a line
segment whose motion is driven by unbounded and space-time dependent
evaporation rates. We prove unique existence of the solution to the
initial-boundary value problem, with conservation-of-fluid condition at the
boundary. The proof uses a map on the space of collection of characteristics,
and a representation based on a non-Markovian point process with
last-arrival-time dependent intensity.
