## Estimation of the conductivity in the one-phase Stefan problem: Basic results.(English)Zbl 0848.35140

This paper is concerned with establishing an existence and uniqueness theory and a priori estimates for the problem of finding $$u$$ in the Stefan problem; $u_t= a(t) u_{xx},\quad a(t) u_x(0, t)= g(t),\quad 0< x< s(t),\quad 0< t\leq T,$
$u(s(t), t)= f(t),\quad 0\leq t\leq T,\quad u(x, 0)= \vartheta(x),\quad 0\leq x\leq b,\;b> 0$ for a given $$a(t)$$. For a given boundary function $$s$$ the problem is shown to reduce to an equivalent system of integral equations. Based on those integral equations a fixed point argument is used to prove the existence of a unique solution.

### MSC:

 35R30 Inverse problems for PDEs 35R35 Free boundary problems for PDEs 80A22 Stefan problems, phase changes, etc.