In this paper we consider one-dimensional two-phase Stefan problems for a class of parabolic equations with nonlinear heat source terms and with nonlinear flux conditions on the fixed boundary. Here, both timedependent and time-independent source terms and boundary conditions are treated. We investigate the large time behavior of solutions to our problems by using the theory for dynamical syste...

We prove the existence and uniqueness, local in time, of a solution for a one-phase Stefan problem of a non-classical heat equation for a semiinfinite material with temperature boundary condition at the fixed face. We use the Friedman-Rubinstein integral representation method and the Banach contraction theorem in order to solve an equivalent system of two Volterra integral equations.

We study a two-phase modified Stefan problem modeling solid combustion and nonequilibrium phase transition. The problem is known to exhibit a variety of non-trivial dynamical scenarios. We develop a priori estimates and establish well-posedness of the problem in weighted spaces of continuous functions. The estimates secure sufficient decay of solutions that allows for an analysis in Hilbert spa...

The study of the basic model for incompressible two-phase flows with phase transitions in the case of equal densities, initiated in the paper Prüss, Shibata, Shimizu, and Simonett [16], is continued here with a stability analysis of equilibria and results on asymptotic behaviour of global solutions. The results parallel those for the thermodynamically consistent Stefan problem with surface tens...

We consider a stochastic perturbation of the Stefan problem. The noise is Brownian in time and smoothly correlated in space. We prove existence and uniqueness and characterize the domain of existence.