Large time behavior of solutions for a PDE model for compressible elastic curve

Chiharu Kosugi; Toyohiko Aiki

doi:10.3934/dcdss.2023161

Article Contents

2023, Volume 16, Issue 12: 3733-3745. Doi: 10.3934/dcdss.2023161

This issue Previous Article A nonlocal Gray-Scott model: Well-posedness and diffusive limit Next Article Kobayashi-Warren-Carter system of singular type under dynamic boundary condition

Large time behavior of solutions for a PDE model for compressible elastic curve

Chiharu Kosugi^1, , and
Toyohiko Aiki^2,

1.
Graduate School of Sciences and Technology for Innovation, Yamaguchi University, Japan
2.
Department of Mathematical and Physical Sciences, Japan Women's University, Japan

^*Corresponding author: Chiharu Kosugi
^*Corresponding author: Chiharu Kosugi

Dedicated to Professor Pierluigi Colli for his 65th birthday

Received: May 2023

Revised: August 2023

Early access: August 28, 2023

Published online: August 28, 2023

This work was supported by JSPS KAKENHI Grant Number JP22K03377.

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Abstract

In this paper, we consider the initial and boundary value problem for the beam equation with the viscosity term and the compressible stress function, which was proposed to deal with the nonlinear strain. This problem is regarded as a mathematical model to represent dynamics of the closed curve created with elastic materials. We have already proved the existence and uniqueness of the weak solution and strong solution to the problem. Our aim of the present paper is to establish theorems concerned with large time behavior of weak solutions and strong solutions, respectively. The key of the proof is the uniform estimate for the position of the gravity center of the closed curve.

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Mathematics Subject Classification: Primary: 35G31, 35Q74; Secondary: 74B20.

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Figure 1.1. Compressible stress function $ f $

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Figure 1.2. Unknown function $ u $

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References

[1]	T. Aiki and C. Kosugi, Numerical scheme for ordinary differential equations describing shrinking and stretching motion of elastic materials, Advances in Mathematical Sciences and Applications, 29 (2020), 459-494.
[2]	T. Aiki and C. Kosugi, Existence and uniqueness of weak solutions for the model representing motions of curves made of elastic materials, Bulletin of Irkutsk State University. Series Mathematics, 36 (2021), 44-46. doi: 10.26516/1997-7670.2021.36.44.
[3]	T. Aiki, N. H. Kröger and A. Muntean, A macro-micro elasticity-diffusion system modeling absorption-induced swelling in rubber foams: Proof of the strong solvability, Quarterly of Applied Mathematics, 79 (2021), 545-579. doi: 10.1090/qam/1592.
[4]	D. Furihata and T. Matsuo, Discrete Variational Derivative Method: A Structure Preserving Numerical Method for Partial Differential Equations, Chapman & Hall CRC, 2011. doi: 10.1201/b10387.
[5]	G. A. Holzapfel, Nonliner Solid Mechanics: A Continuum Approach for Engineering, John Wiley & Sons Publ., 2000.
[6]	C. Kosugi, Solvability of a PDE model with nonlinear stress function having singularity for compressible elastic curve, to appear, Advances in Mathematical Sciences and Applications.
[7]	C. Kosugi, T. Aiki, M. Anthonissen and M. Okumura, Numerical results for ordinary and partial differential equations describing motions of elastic materials, Advances in Mathematical Sciences and Applications, 30 (2021), 387-414.
[8]	R. W. Ogden, Large deformation isotropic elasticity: On the correlation of theory and experiment for compressible rubber like solids, Proc. R. Soc. Lond. Ser. A, Math. Phys. Sci., 328 (1972), 567-583. doi: 10.1098/rspa.1972.0096.
[9]	S. Okabe, The motion of elastic planar closed curves under the area-preserving condition, Indiana Univ. Math. J., 56 (2007), 1871-1912. doi: 10.1512/iumj.2007.56.3015.
[10]	R. Racke and C. Shang, Global attractors for nonlinear beam equation, Proc. Roy. Soc. Edinburgh Sect. A, 142 (2012), 1087-1107. doi: 10.1017/S030821051000168X.
[11]	J. C. Simo and C. Miehe, Associative coupled thermoplasticity at finite strains: Formulation, numerical analysis and implementation, Comput. Methods in App. Mech. Engi., 98 (1992), 41-104. doi: 10.1016/0045-7825(92)90170-O.
[12]	H. Takeda and S. Yoshikawa, On the initial value problem of the semilinear beam equations with weak damping Ⅰ: Smoothing effect, J. Math. Anal. Appl., 401 (2013), 244-258. doi: 10.1016/j.jmaa.2012.12.015.
[13]	H. Takeda and S. Yoshikawa, On the initial value problem of the semilinear beam equations with weak damping Ⅱ: Aymptotic profiles, J. Differential Equations, 253 (2012), 3061-3080. doi: 10.1016/j.jde.2012.07.014.