\left\{\begin{array}{rll} {\partial}_t U &= {\partial}^0_{xx}U + g(U) & \quad t>0 \,,\, 0<x<0 \\ U(t,0)& =0 & \quad t>0 \\ U(t,0) & = 0 & \quad t>0 \\ U(0,x) &=u(x) & \quad 0<x<0 \end{array}\right.$$ 