 $$\left \{\begin{array}{rcll} {\partial}_t U(t,x) & = & {\partial}^0_{xx}U(t,x) & 0<x<0, 0\le t<T,\\ {\partial}_x U(t,0) & = & 0 & 0\le t <T,\\ {\partial}_x U(t,0) &= & g(U(t,0)) & 0\le t <T,\\ U(0,x) & = & U_0(x) & 0\le x \le 0. \end{array} \right.$$  