A free boundary value problem related to the combustion of a solid: flux boundary conditions

We demonstrate the existence, uniqueness, and continuous dependence upon the data for the solution $\left ( u, v, s \right )$ of the free boundary value problem: \[ {u_t} = \alpha {u_{xx}}, \qquad {v_t} = \beta {v_{xx,}} \qquad 0 < x < s\left ( t \right ), \qquad 0 < t \le T\], \[ u\left ( x, 0 \right ) = \phi \left ( x \right ), \qquad v\left ( x, 0 \right ) = \psi \left ( x \right ), \qquad 0 \le x \le s\left ( 0 \right ) = b\]. \[ - \alpha {u_x}\left ( 0, t \right ) = f\left ( t \right ), \qquad - \beta {v_x}\left ( 0, t \right ) = g\left ( t \right ), \qquad 0 < t \le T\], \[ \alpha {u_x}\left ( s\left ( t \right ), t \right ) = - \left ( \gamma + u\left ( s\left ( t \right ), t \right ) \right )\dot s\left ( t \right ), \qquad \beta {v_x}\left ( s\left ( t \right ), t \right ) = \left ( \mu - v\left ( s\left ( t \right ), t \right ) \right )\dot s\left ( t \right ), \qquad 0 < t \le T\], \[ \dot s\left ( t \right ) = \nu \left ( v\left ( s\left ( t \right ), t \right ) \right ) exp\left ( - \delta /v\left ( s\left ( t \right ), t \right ) \right )F\left ( u\left ( s\left ( t \right ), t \right ) \right ), \qquad 0 < t \le T\], where $\alpha , \beta , \gamma , \delta$, and $\mu$ are positive constants related to the physical constants.