A nonlinear oblique derivative boundary value problem for the heat equation - Part 1: Basic results

We study the heat equation B-t - Delta B = 0 in the half-plane with the non linear oblique derivative condition B-x = KBBz on the boundary, where (B-x, B-z) are respectively the normal and the tangential derivatives of B. The ultimate goal is to let K --> +infinity in the equations. In this first part, we introduce self-similar solutions which verify an ell iptic equation with the same nonlinear boundary condition. The main part of this first paper concerns this self-similar problem. It is well-posed and its solution is shown to be smooth, by means of boundary integral estimates . The originality of the approach is the robustness of the estimates with r espect to K. The evolution problem itself admits global classical solutions which converge, as times tends to +infinity, to the self-similar solution. (C) Elsevier, Paris.