Electronic Journal of Differential Equations,
Vol. 2002(2002), No. 100, pp. 1-30.
Title: On a parabolic-hyperbolic Penrose-Fife phase-field system
Authors: Pierluigi Colli (Univ. degli Studi di Pavia, Italy)
Maurizio Grasselli (Politecnico di Milano, Italy)
Akio Ito (Kinki Univ. ,Takayaumenobe, Japan)
Abstract:
The initial and boundary value problem is studied for a non-conserved
phase-field system derived from the Penrose-Fife model for the
kinetics of phase transitions. Here the evolution of the order
parameter is governed by a nonlinear hyperbolic equation which
is characterized by the presence of an inertial term
with small positive coefficient. This feature is a consequence
of the assumption that the response of the phase variable
to the generalized force which drives the system toward equilibrium
states is not instantaneous but delayed. The resulting model consists
of a nonlinear parabolic equation for the absolute temperature
coupled with the hyperbolic equation for the phase. Existence of
a weak solution is obtained as well as the convergence of any family
of weak solutions of the parabolic-hyperbolic model to the weak
solution of the standard Penrose-Fife phase-field model as the inertial
coefficient goes to zero. In addition, continuous dependence estimates
are proved for the parabolic-hyperbolic system as well as for the
standard model.
Submitted October 11, 2002. Published November 26, 2002.
Math Subject Classifications: 35G25, 35Q99, 80A22.
Key Words: Phase-field; Penrose-Fife model; existence of solutions;
nonlinear partial differential equations;
continuous dependence on the data.