Large-time solutions of the two-phase Stefan problem with delay (Q1206926)

The paper is devoted to the study of the global in time unique solvability of two-phase, one-dimensional Stefan problems where the source terms of the governing equations are given by delay functionals \[ g_ i(t,x)=\int^ 0_{-r}f_ i(s,x,u(t,x),u(t+s,x))ds, \] here \(i=1,2\) correspond to the phases, \(r\) is prescribed and \(u\) is the solution. After a short discussion of this problem within the framework of classical Stefan problems and the physical background the main results are stated. Beside the established unique existence of solution which is global in time regularity results are formulated in dependence of the provided regularity of the time-dependent Dirichlet data. Furthermore, an asymptotic steady-state counterpart of the problem is constructed. In proving the global in time existence for the problems with delay the authors use the analysis of large-time solutions to standard Stefan problems (without delay).