A Free Boundary-Value Problem Related to the Combustion of a Solid

We demonstrate the existence, uniqueness and continuous dependence upon the data for the solution $(u,v,s)$ of the free boundary value problem: \[\begin{gathered} u_t = \alpha u_{xx} ,\quad v _t = \beta v _{xx} ,\quad - \infty < x < s\left( t \right),\quad 0 < t\leqq T, \\ u\left( {x,0} \right) = \varphi \left( x \right),\quad v \left( {x,0} \right) = \psi \left( x \right),\quad - \infty < x\leqq s\left( 0 \right), \\ \alpha u_x \left( {s\left( t \right),t} \right) = - \left( {\gamma + u\left( {s\left( t \right),t} \right)} \right)\dot s\left( t \right),\quad \beta v _x \left( {s\left( t \right),t} \right) = \left( {\mu - v \left( {s\left( t \right),t} \right)} \right)\dot s\left( t \right),\quad 0 < t\leqq T, \\ \dot s\left( t \right) = v\,\exp \,\left\{ { - \delta/v \left( {s\left( t \right),t} \right)} \right\}f\left( {u\left( {s\left( t \right),t} \right)} \right),\quad 0 < t\leqq T, \\ \end{gathered} \]$s\left( 0 \right) = 0$; where $\alpha ,\beta ,\delta ,\mu $, and v are positive constants related ...