On a nonlinear heat equation with a time delay

A nonlinear forward-backward heat equation with a regularization term was proposed by Barenblatt et al. [1, 2] to model the heat and mass exchange in stably stratified turbulent shear flow. It was proven to be well-posed in the case of given initial and Neumann boundary conditions. However, the solution was found to have an unphysical discontinuity with certain smooth initial functions. In this paper, a nonlinear heat equation with a time delay originally used by Barenblatt et al. [1, 2] to derive their model is investigated. The same type of initial-boundary value problem is shown to have a unique smooth global solution when the initial function is reasonably smooth. Numerical examples are used to demonstrate that its solution forms step-like profiles in finite times. A semi-discretization of the initial-boundary value problem is proved to have a unique asymptotically and globally stable equilibrium.