Existence of solutions to the problem of nonequilibrium phase change

Abstract

In questo lavoro si considera il seguente problema di "nonequilibrium phase-change": \[ \begin{cases} C\frac{\partial u}{\partial t}-k\frac{\partial^{2}u}{\partial x^{2}}=0,0<x<1,0<t<T,\\ u(0,t)=h(t),u(1,t)=h(t),\\ u(x,0)=u(x),s(0)=s_{0},\\ s(0)=s_{0},k\left[\frac{\partial u(s(t)+,t)}{\partial x}-\frac{\partial u(s(t)-,t)}{\partial x}\right]=L\dot{s(t)=} & Lg(u(s(t),t)) \end{cases} \] e si dimostra l'esistenza di sue soluzioni deboli o classiche sotto certe condizioni.

In thls paper we consider the following problem of nonequilibrlum phase-change: \[ \begin{cases} C\frac{\partial u}{\partial t}-k\frac{\partial^{2}u}{\partial x^{2}}=0,0<x<1,0<t<T,\\ u(0,t)=h(t),u(1,t)=h(t),\\ u(x,0)=u(x),s(0)=s_{0},\\ s(0)=s_{0},k\left[\frac{\partial u(s(t)+,t)}{\partial x}-\frac{\partial u(s(t)-,t)}{\partial x}\right]=L\dot{s(t)=} & Lg(u(s(t),t)) \end{cases} \] and prove the existence of its weak or classical solutions under some conditions.