All global bifurcation diagrams of stationary solutions to a phase field model

Tatsuki Mori; Sohei Tasaki; Tohru Tsujikawa; Shoji Yotsutani

doi:10.3934/dcdsb.2022183

Article Contents

2023, Volume 28, Issue 12: 6306-6344. Doi: 10.3934/dcdsb.2022183 open access

This issue Previous Article Modeling bacterial traveling wave patterns with exact cross-diffusion and population growth Next Article Front propagation and blocking for the competition-diffusion system in a domain of half-lines with a junction

All global bifurcation diagrams of stationary solutions to a phase field model

1.
Department of Mathematical Engineering, Faculty of Engineering, Musashino University, Japan
2.
Department of Mathematics, Faculty of Science, Hokkaido University, Japan
3.
University of Miyazaki, Japan
4.
Ryukoku University, Japan

^*Corresponding author: Tohru Tsujikawa
^*Corresponding author: Tohru Tsujikawa

Dedicated to the memory of Professor Masayasu Mimura

Received: April 2022

Revised: August 2022

Early access: September 16, 2022

Published online: September 16, 2022

T. Mori was supported by Grant-in-Aid. for Grant-in-Aid for Young Scientists (B) 22K13962. S. Tasaki was supported by Grant-in-Aid. for Scientific Research (C) 19K03645 and Grant-in-Aid for Scientific Research on Innovative Areas 17H06327. T. Tsujikawa was supported by Grant-in-Aid. for Scientific Research (C) 17K05334 and 22K03378. S. Yotsutani was supported by Grant-in-Aid. for Scientific Research (C) 19K03593. This work was supported by Joint Research Center for Science and Technology of Ryukoku University in 2021.

Abstract / Introduction Full Text(HTML) Figure(22) Related Papers Cited by

Abstract

We are concerned with bifurcation diagrams of stationary solutions to a phase field model proposed by Fix and followed by Caginalp. We show all the global bifurcation diagrams of stationary solutions to the model in the 1-dimension case. We see that bifurcation diagrams are surprisingly rich in variety depending on the latent heat and the initial total enthalpy. For instance, bifurcation diagrams include the secondary bifurcation point where symmetric breaking occurs, and curves which connect a limit of boundary layer solutions to the other limit of internal layer solutions.

Keywords:

Mathematics Subject Classification: Primary: 34C23, 34B10; Secondary: 37G40.

Citation:

FullText(HTML)

Figure 1.1. Shape of solutions $ \varphi(x; \overline{u}, d) $ for several $ (\overline{u}, d) \in \mathcal{G} $.

Download: Full-size image PowerPoint slide

Figure 2.1. Curves of $ \mathcal{B}^{\, \ell, 0} $.

Download: Full-size image PowerPoint slide

Figure 2.2. Curves of $ \mathcal{B}^{\, \ell, m} $ for $ \ell \geq 1 $.

Download: Full-size image PowerPoint slide

Figure 2.3. Curves of $ \mathcal{B}^{\, \ell, m} $ for $ 3/4 < \ell < 1 $.

Download: Full-size image PowerPoint slide

Figure 2.4. Curves of $ \mathcal{B}^{\, \ell, m} $ for $ \ell = 3/4 $.

Download: Full-size image PowerPoint slide

Figure 2.5. Curves of $ \mathcal{B}^{\, \ell, m} $ for $ 2/3 < \ell < 3/4 $.

Download: Full-size image PowerPoint slide

Figure 2.6. Curves of $ \mathcal{B}^{\, \ell, m} $ for $ \ell = 2/3 $.

Download: Full-size image PowerPoint slide

Figure 2.7. Curves of $ \mathcal{B}^{\, \ell, m} $ for $ 1/4 < \ell < 2/3 $.

Download: Full-size image PowerPoint slide

Figure 2.8. Curves of $ \mathcal{B}^{\, \ell, m} $ for $ \ell = 1/4 $.

Download: Full-size image PowerPoint slide

Figure 2.9. Curves of $ \mathcal{B}^{\, \ell, m} $ for $ (\sqrt{3}-1)/3 < \ell < 1/4 $.

Download: Full-size image PowerPoint slide

Figure 2.10. Curves of $ \mathcal{B}^{\, \ell, m} $ for $ \ell = (\sqrt{3}-1)/3 $.

Download: Full-size image PowerPoint slide

Figure 2.11. Curves of $ \mathcal{B}^{\, \ell, m} $ for $ 0 < \ell < (\sqrt{3}-1)/3 $.

Download: Full-size image PowerPoint slide

Figure 3.1. Curves of $ \, \mathcal{B}^{\, \ell, m}_{+}\, $ and $ \, \mathcal{B}^{\, \ell, m}_{-} (m \!\geq\! 0)\, $ for $ \ell \geq 1 $.

Download: Full-size image PowerPoint slide

Figure 3.2. Curves of $ \, \mathcal{B}^{\, \ell, m}_{+}\, $ and $ \, \mathcal{B}^{\, \ell, m}_{-} (m \!\geq\! 0)\, $ for $ 2/3 < \ell < 1 $.

Download: Full-size image PowerPoint slide

Figure 3.3. Curves of $ \, \mathcal{B}^{\, \ell, m}_{+}\, $ and $ \, \mathcal{B}^{\, \ell, m}_{-} (m \!\geq\! 0)\, $ for $ \ell = 2/3 $.

Download: Full-size image PowerPoint slide

Figure 3.4. Curves of $ \, \mathcal{B}^{\, \ell, m}_{+}\, $ and $ \, \mathcal{B}^{\, \ell, m}_{-} (m \!\geq\! 0)\, $ for $ 0 < \ell < 2/3 $.

Download: Full-size image PowerPoint slide

Figure 3.5. Graphs of $ \ell/2 $, $ \sqrt{3} |3\ell - 2|/18 $ and $ \sqrt{3} (1-\ell)^{3/2}/9 $.

Download: Full-size image PowerPoint slide

Figure 6.1. $ (z^3 - (1-\ell)z)/2 $, $ \Phi(\overline{u}, \hat{d}(\overline{u}); \ell) $, $ \Phi(\overline{u}, 0; \ell) $ for $ \ell \geq 1 $.

Download: Full-size image PowerPoint slide

Figure 6.2. $ (z^3 \!-\! (1 \!-\! \ell)z)/2 $, $ \Phi(\overline{u}, \hat{d}(\overline{u}); \ell) $, $ \Phi(\overline{u}, 0; \ell) $ for $ 2/3 < \ell < 1 $.

Download: Full-size image PowerPoint slide

Figure 6.3. $ (z^3 - (1-\ell)z)/2 $, $ \Phi(\overline{u}, \hat{d}(\overline{u}); \ell) $, $ \Phi(\overline{u}, 0; \ell) $ for $ \ell = 2/3 $.

Download: Full-size image PowerPoint slide

Figure 6.4. $ (z^3 - (1-\ell)z)/2 $, $ \Phi(\overline{u}, \hat{d}(\overline{u}); \ell) $, $ \Phi(\overline{u}, 0; \ell) $ for $ 0 < \ell < 2/3 $.

Download: Full-size image PowerPoint slide

Figure A.1. $ \alpha(\overline{u}) $, $ \beta(\overline{u}) $, $ \gamma(\overline{u}) $ and $ (z^3 - z)/2 $.

Download: Full-size image PowerPoint slide

Related Papers

Cited by

References

[1]	G. Caginalp, An analysis of a phase field model of a free boundary, Arch. Rational Mech. Anal., 92 (1986), 205-245. doi: 10.1007/BF00254827.
[2]	C. M. Elliott and S. M. Zheng, Global existence and stability of solutions to the phase field equations, Free Boundary Value Problems, Internat. Ser. Numer. Math., Birkhäuser, Basel, 95 (1990), 46-58. doi: 10.1007/978-3-0348-7301-7_4.
[3]	G. J. Fix, Phase field methods for free boundary problems, Free Boundary Problems: Theory and Applications Vol.2, Res. Notes in Math., Pitman, Boston, Mass.-London, 46 (1983), 580-589.
[4]	M. E. Gurtin and H. Matano, On the structure of equilibrium phase transitions within the gradient theory of fluids, Quart. Appl. Math., 46 (1988), 301-317. doi: 10.1090/qam/950604.
[5]	D. Hilhorst, M. Iida, M. Mimura and H. Ninomiya, A reaction-diffusion system approximation to the two-phase Stefan problem, Nonlinear Anal., 47 (2001), 801-812. doi: 10.1016/S0362-546X(01)00224-3.
[6]	S. Kosugi, Y. Morita and S. Yotsutani, Stationary solutions to the one-dimensional Cahn-Hilliard equation: Proof by the complete elliptic integrals, Discrete Contin. Dyn. Syst., 19 (2007), 609-629. doi: 10.3934/dcds.2007.19.609.
[7]	K. Kuto, T. Mori, T. Tsujikawa and S. Yotsutani, Secondary bifurcation for a nonlocal Allen-Cahn equation, J. Differential Equations, 263 (2017), 2687-2714. doi: 10.1016/j.jde.2017.04.010.
[8]	K. Kuto, T. Mori, T. Tsujikawa and S. Yotsutani, Global solution branches for a nonlocal Allen-Cahn equation, J. Differential Equations, 264 (2018), 5928-5949. doi: 10.1016/j.jde.2018.01.025.
[9]	K. Kuto and T. Tsujikawa, Stationary patterns for an adsorbate-induced phase transition model. Ⅰ. Existence, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 1105-1117. doi: 10.3934/dcdsb.2010.14.1105.
[10]	K. Kuto and T. Tsujikawa, Stationary patterns for an adsorbate-induced phase transition model. Ⅱ, Nonlinear Anal., TMA, 26 (2013), 1313-1343.
[11]	Y. Lou, W.-M. Ni and S. Yotsutani, On a limiting system in the Lotka-Volterra competition with cross diffusion, Discrete Contin. Dyn. Syst., 10 (2004), 435-458. doi: 10.3934/dcds.2004.10.435.
[12]	Y. Miyamoto, T. Mori, T. Tsujikawa and S. Yotsutani, Stability for stationary solutions of a nonlocal Allen-Cahn equation, J. Differential Equations, 275 (2021), 581-597. doi: 10.1016/j.jde.2020.11.024.
[13]	T. Miyasita, Global existence and exponential attractor of solutions of Fix-Caginalp equation, Sci. Math. Jpn., 77 (2015), 339-355. doi: 10.32219/isms.77.3_339.
[14]	T. Mori, K. Kuto, T. Tsujikawa and S. Yotsutani, Exact multiplicity of stationary limiting problem of a cell polarization model, Discrete Contin. Dyn. Syst., 36 (2016), 5627-5655. doi: 10.3934/dcds.2016047.
[15]	T. Mori, K. Kuto, T. Tsujikawa and S. Yotsutani, Representation formulas of solutions and bifurcation sheets to a nonlocal Allen-Cahn equation, Discrete Contin. Dyn. Syst., 40 (2020), 4907-4925. doi: 10.3934/dcds.2020205.
[16]	O. Penrose and P. C. Fife, Thermodynamically consistent models of phase-field type for the kinetic of phase transitions, Phys. D, 43 (1990), 44-62. doi: 10.1016/0167-2789(90)90015-H.
[17]	R. Schaaf, Global Solution Branches of Two-Point Boundary Value Problems, Lecture Notes in Mathematics, 1458. Springer-Verlag, Berlin, 1990. doi: 10.1007/BFb0098346.
[18]	J. Smoller, Shock Waves and Reaction-Diffusion Equations, Second edition, Grundlehren der mathematischen Wissenschaften, 258. Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-0873-0.
[19]	J. Smoller and A. Wasserman, Global bifurcation of steady-state solutions, J. Differential Equations, 39 (1981), 269-290. doi: 10.1016/0022-0396(81)90077-2.
[20]	T. Suzuki and S. Tasaki, Stationary Fix-Caginalp equation with non-local term, Nonlinear Anal., 71 (2009), 1329-1349. doi: 10.1016/j.na.2008.12.007.
[21]	T. Wakasa, Exact eigenvalues and eigenfunctions associated with linearization for Chafee-Infante problem, Funkcial Ekvac., 49 (2006), 321-336. doi: 10.1619/fesi.49.321.
[22]	T. Wakasa and S. Yotsutani, Limiting classification on linearized eigenvalue problems for 1-dimensional Allen-Cahn equation Ⅱ - Asymptotic profiles of eigenfunctions, J. Differential Equations, 261 (2016), 5465-5498. doi: 10.1016/j.jde.2016.08.016.