A free boundary problem for heat equation arising in infiltration. (Q1862706)

The authors study the following free boundary problem: \[ u_{xx}-u_t=0,\quad 0<x<s(t),\,0<t<T, \] \[ u(x,0)=\phi (x),\quad 0<x<b, \quad u(0,t)=f_1(t),\quad 0<t<T \] \[ u(s(t),t)=f_2(s(t),t),\quad 0<t<T, \quad u_x(s(t),t)=g(s(t),t),\quad 0<t<T \] where the unknowns are \(s\) (required to be \(C^1[0,T]\)) and \(u\). After having studied some qualitative properties, the authors use a time discretisation method to prove existence (and uniqueness) of a solution with nondecreasing free boundary.