A nonlinear oblique derivative boundary value problem for the heat equation: Analogy with the porous medium equation

Abstract

The problem under investigation is the heat equation in the upper half-plane, to which the diffusion in the longitudinal direction has been suppressed, and augmented with a nonlinear oblique derivative condition. This paper proves global existence and qualitative properties to the Cauchy problem for this model, furthering the study [18] of the self-similar solutions. The qualitative behaviour of the solutions exhibits a strong analogy with the porous medium equation: propagation with compact support and finite speed, free boundary relation and time-asymptotic convergence to self-similar solutions.