On phase-field equations of Penrose–Fife type with the conserved order parameter under flux boundary condition: Global-in-time solvability and uniform boundedness

Abstract

We study an initial boundary value problem for a system of conserved phase-field model proposed by Penrose and Fife ([13,14]) under flux boundary condition for the temperature in higher space dimensions. In contrast to other works related to this problem, we are concerned with the above mentioned problem under the correct form of the flux boundary condition for the temperature, which is obliged to overcome additional difficulties in mathematical treatment. We prove that the initial boundary value problem for such a system admits a unique global-in-time strong solution in Sobolev–Slobodetskiĭ spaces. Moreover, it is shown that the temperature keeps positive and both temperature and order parameter are uniformly bounded in two dimensions through the time evolution.