The vanishing viscosity method and a two-phase Stefan problem with nonlinear flux condition of Signorini type (Q1113024)

This paper is concerned with the enthalpy formulation of the two-phase Stefan problem with nonlinear flux condition which is described by the system \[ u_ t-\Delta \beta (u)=f\quad in\quad Q=(0,T)\times \Omega, \] \[ -\partial \beta (u)/\partial n\in \gamma (\beta (u)-g)\quad in\quad \Sigma =(0,T)\times \partial \Omega, \] \[ u(0,x)=u_ 0(x)\quad in\quad \Omega. \] Here \(0<T<\infty\); \(\Omega\) is a bounded domain in \(R^ N\) (N\(\geq 1)\) with smooth boundary \(\partial \Omega\); \(\beta\) : \(R\to R\) is a nondecreasing Lipschitz continuous function such that \(\beta =0\) on [0,1] and \(\beta\) is bi-Lipschitz continuous on (-\(\infty,0]\) and on [0,\(\infty)\); \(\gamma\) is a maximal monotone graph in \(R\times R\) of the form \(\gamma (r)=0\) for \(r>0\), \(=(-\infty,0]\) for \(r=0\) and \(=\Phi\) for \(r<0\); f, g, \(u_ 0\) are given data and they are functions defined on Q, \(\Sigma\), \(\Omega\), respectively. In this case the flux boundary condition is the so-called Signorini type. Under a condition on the data which ensure that the free boundary does not touch the fixed boundary, it as proved that a weak solution can be constructed by the standard vanishing viscosity method and such a solution is unique.