The parabolic free boundary problem with Puschino dynamics is given by (see in [3]) $$ (1) { upsilon_t = Dupsilon_{xx} - (c_1 + b)upsilon + c_1 H(x - s(t)) for (x,t) in Omega^- cup Omega^+, { upsilon_x(0,t) = 0 = upsilon_x(1,t) for t > 0, { upsilon(x,0) = upsilon_0(x) for 0 leq x leq 1, { aufrac{dt}{ds} = C)upsilon(s(t),t)) for t > 0, { s(0) = s_0, 0 < s_0 < 1, $$ where $upsilon(x,t)$ and $upsilon_x(x,t)$ are assumed continuous in $Omega = (0,1) imes (0, infty)$.