Kobayashi-Warren-Carter system of singular type under dynamic boundary condition

Ryota Nakayashiki; Ken Shirakawa

doi:10.3934/dcdss.2023162

Article Contents

2023, Volume 16, Issue 12: 3746-3783. Doi: 10.3934/dcdss.2023162

This issue Previous Article Large time behavior of solutions for a PDE model for compressible elastic curve Next Article Second-order sufficient conditions for sparse optimal control of singular Allen–Cahn systems with dynamic boundary conditions

Kobayashi-Warren-Carter system of singular type under dynamic boundary condition

Ryota Nakayashiki^1, and
Ken Shirakawa^2, ,

1.
Department of General Education, Salesian Polytechnic, Japan
2.
Department of Mathematics, Faculty of Education, Chiba University, Japan

^*Corresponding author: Ken Shirakawa
^*Corresponding author: Ken Shirakawa

Dedicated to Professor Pierluigi Colli on the occasion of his 65th birthday

Received: May 2023

Revised: May 2023

Early access: September 2, 2023

Published online: September 2, 2023

This work is supported by Grant-in-Aid for Scientific Research (C) No. 20K03672, JSPS.

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Abstract

In this paper, we consider a coupled system, known as Kobayashi–Warren–Carter system, abbreviated as the KWC system. KWC system consists of an Allen–Cahn type equation and a singular diffusion equation, and it was proposed by [Kobayashi et al, Phys. D, 140,141–150 (2000)] as a possible mathematical model of grain boundary motion. The focus of this work is on the dynamic boundary condition imposed in our KWC system, and the mathematical interest is in a conflicting situation between: the continuity of the transmission condition included in the dynamic boundary condition; and the discontinuity encouraged by the singular diffusion equation. On this basis, we will prove the Main Theorem concerned with the existence of solution to our KWC system with energy-dissipation. Additionally, as a sub-result, we will prove a key-lemma that is to give a certain mathematical interpretation for the conflicting situation.

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Mathematics Subject Classification: Primary: 35K51, 35K55, 35K61, 35K67; Secondary: 82C26.

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