We examine a model for the propagation of heat through a one-dimensional object with an interior ''flaw". The flaw is modeled as a nonlinear relationship between the flux and temperature jump at an interior point of the object. Under realistic hypotheses, the resulting nonlinear initial boundary value problem is shown to have a unique and suitably smooth solution.
Bryan, Kurt M. and Caudill, Lester, "Solvability of a Parabolic Boundary Value Problem with Internal Jump Condition" (2000). Mathematical Sciences Technical Reports (MSTR). 99.