We consider the problem of existence of a solution u to δ<sub>t</sub>u — δ<sub>xx</sub>u = 0 in (0, T) x R<sub>+</sub> subject to the boundary condition — u<sub>x</sub>(t,0) + g(u(t, 0)) = μ on (0, T) where μ is a measure on (0, T) and g a continuous nondecreasing function. When p > 1 we study the set of self-similar solutions of δ<sub>t</sub>u — δ<sub>xx</sub>u = 0 in R<sub>+</sub> — R<sub>+</sub> such that —u<sub>x</sub>(t,0)+u<sup>p</sup> = 0 on (0,∞). At end, we present various extensions to a higher dimensional framework.
