Global attractors for two-phase stefan problems in one-dimensional space

Abstract

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 time-dependent 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 systems. First, we show the existence of a global attractor $𝒜$ of autonomous Stefan problem. The main purpose in the present paper is to prove that the set $𝒜$ attracts all solutions of non-autonomous Stefan problems as time tends to infinity under the assumption that time-dependent data converge to time-independent ones as time goes to infinity.