Electron. J. Diff. Eqns., Vol. 2002(2002), No. 100, pp. 1-30.

On a parabolic-hyperbolic Penrose-Fife phase-field system

Pierluigi Colli, Maurizio Grasselli, & Akio Ito

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.

An addendum was attached on March 31, 2003. Several expressions are modified. See pages 30-32 of this article.

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.

Show me the PDF file (329K), TEX file, and other files for this article.

Pierluigi Colli
Dipartimento di Matematica ``F. Casorati'',
Universita degli Studi di Pavia,
27100 Pavia, Italy
e-mail: pier@dimat.unipv.it
Maurizio Grasselli
Dipartimento di Matematica ``F. Brioschi'',
Politecnico di Milano, 20133 Milano, Italy
e-mail: maugra@mate.polimi.it
Akio Ito
Department of Architecture
School of Engineering
Kinki University 1,
Takayaumenobe, Higashi-Hiroshima, 739-2116 Japan
e-mail: aito@hiro.kindai.ac.jp

Return to the EJDE web page