An implicit two-phase free boundary problem for the heat equation (Q1762798)

The authors prove existence and uniqueness of a classical solution to the following free boundary problem \[ u_{xx}-u_t=0, \quad 0<x<s(t), \] \[ \gamma v_{xx}-v_t=0, \quad s(t)<x<1, \] \[ s(0)=b, \quad 0<b<1, \] \[ u(x,0)=\phi(x), \quad 0<x<b, \] \[ v(x,0)=\psi(x), \quad b<x<1, \] \[ u_x(0,t)=f(t), \quad v(1,t)=F(t), \] \[ u(s(t),t)=v(s(t),t)=0, \] \[ u_x(s(t),t)-v_x(s(t),t)=\lambda s(t). \]