Abstract
This article presents a nonlocal Penrose-Fife type phase field system with inertial term. We do not know whether we can prove the existence of solutions to the problem as in Colli-Grasselli-Ito [3] or not. In this article we introduce a time discretization scheme, then pass to the limit as the time step h approaches 0, and obtain an error estimate for the difference between the continuous solution and the discrete solution.
Reference7 articles.
1. V. Barbu; Nonlinear Semigroups and Di erential Equations in Banach spaces, Noordho International Publishing, Leyden, 1976.
2. V. Barbu; Nonlinear Di erential Equations of Monotone Types in Banach Spaces, Springer,New York, 2010.
3. P. Colli, M. Grasselli, A. Ito; On a parabolic-hyperbolic Penrose-Fife phase- eld system,,Electron. J. Differential Equations 2002, No. 100, 30 pp. (Erratum: Electron. J. Differential Equations 2002, No. 100, 32 pp.).
4. P. Colli, S. Kurima; Time discretization of a nonlinear phase eld system in general domains, Comm. Pure Appl. Anal., 18 (2019), 3161-3179.
5. J. W. Jerome; Approximations of Nonlinear Evolution Systems, Mathematics in Science and Engineering, 164, Academic Press Inc., Orlando, 1983.