Volume 54, Number 3, May-June 2020
|Page(s)||977 - 1002|
|Published online||22 April 2020|
Time discretization of an initial value problem for a simultaneous abstract evolution equation applying to parabolic-hyperbolic phase-field systems
Department of Mathematics, Tokyo University of Science, 1-3, Kagurazaka, 162-8601 Shinjuku-ku, Tokyo, Japan
* Corresponding author: firstname.lastname@example.org
Accepted: 1 December 2019
This article deals with a simultaneous abstract evolution equation. This includes a parabolic-hyperbolic phase-field system as an example which consists of a parabolic equation for the relative temperature coupled with a semilinear damped wave equation for the order parameter (see e.g., Grasselli and Pata [Adv. Math. Sci. Appl. 13 (2003) 443–459, Comm. Pure Appl. Anal. 3 (2004) 849–881], Grasselli et al. [Comm. Pure Appl. Anal. 5 (2006) 827–838], Wu et al. [Math. Models Methods Appl. Sci. 17 (2007) 125–153, J. Math. Anal. Appl. 329 (2007) 948–976]). On the other hand, a time discretization of an initial value problem for an abstract evolution equation has been studied (see e.g., Colli and Favini [Int. J. Math. Math. Sci. 19 (1996) 481–494]) and Schimperna [J. Differ. Equ. 164 (2000) 395–430] has established existence of solutions to an abstract problem applying to a nonlinear phase-field system of Caginalp type on a bounded domain by employing a time discretization scheme. In this paper we focus on a time discretization of a simultaneous abstract evolution equation applying to parabolic-hyperbolic phase-field systems. Moreover, we can establish an error estimate for the difference between continuous and discrete solutions.
Mathematics Subject Classification: 35A35 / 47N20 / 35G30 / 35L70
Key words: Simultaneous abstract evolution equations / parabolic-hyperbolic phase-field systems / existence / time discretizations / error estimates
© EDP Sciences, SMAI 2020
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