A non linear oblique derivative boundary
value problem for the heat equation
Part1 : Basic existence results for the evolution
        problem and steady solutions


         F.Mehats
         J.M.Roquejoffre



We study the heat equation $B_t -\Delta B =0$ in the half-plane
with the nonlinear oblique derivative condition $B_X = K B B_Z$ on the
boundary, where $(B_X,B_Z)$
are respectively the normal and the tangential derivatives of $B$.

This work is divided into two parts. In this first part,
we introduce self-similar solutions
which verify an elliptic 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 smooth. About the evolution problem
itself, we prove here the global existence of a weak solution for the
Cauchy problem associated to it.



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