Abstract
We consider a class of problems in which traces of the first spacial derivative of the solution appear in the coefficients of a parabolic partial differential equation. Such problems arise in free boundary problems and the determination of unknown coefficients in various parabolic differential equations. Existence, uniqueness and continuous dependence on the data are demonstrated. The analysis involves a priori estimates and the Schauder fixed point theorem. Overall, the article generalizes some of the results of Cannon and Yin (1989, Journal of Differential Equations, 79, 266–288).