On a Backward-Forward Parabolic Equation and Its Regularization

The steady-state diffusion of heat in a fluid which flows rapidly but not unidirectionally in a semi-infinite channel leads to a singular perturbation of a backwards-forwards heat equation with Neumann conditions on the channel walls. The temperature is prescribed on the whole channel entrance for the perturbed problems, but only at the points where the fluid flow is inward for the limiting problem. Conditions are obtained for uniqueness, and the additional arbitrary constants needed to obtain a well-posed problem when uniqueness does not hold are characterized. Conditions for the convergence of the solution of the perturbed problem to that of the limiting problem are also found.