A nonlinear oblique derivative boundary value problem for the heat equation Part 2: Singular self-similar solutions

This paper continues the study started in 12]. In the upper half-plane, consider the elliptic equation

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, submitted to the nonlinear oblique derivative boundary condition U_x = UU_z on the axis x = 0. The solution of this problem appears to be the self-similar solution of the heat equation with the same boundary condition. As goes to 0, the function U converges to the non trivial solution U of the corresponding degenerate problem. Moreover there exists z₀ > 0 such that U vanishes for z ≥ z₀, is C^∞ on ]0, z₀× ₊, is continuous on the boundary x = 0 and is discontinuous on the half-axis {z = 0, x> 0}.