## Traveling wave dynamics for Allen-Cahn equations with strong irreversibility

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- by Goro Akagi, Christian Kuehn and Ken-Ichi Nakamura PDF
- Trans. Amer. Math. Soc.
**375**(2022), 3173-3238 Request permission

## Abstract:

Constrained gradient flows are studied in fracture mechanics to describe*strongly irreversible*(or

*unidirectional*) evolution of cracks. The present paper is devoted to a study on the long-time behavior of non-compact orbits of such constrained gradient flows. More precisely, traveling wave dynamics for a one-dimensional fully nonlinear Allen-Cahn type equation involving the positive-part function is considered. Main results of the paper consist of a construction of a one-parameter family of

*degenerate*traveling wave solutions (even identified when coinciding up to translation) and exponential stability of such traveling wave solutions with some basin of attraction, although they are unstable in a usual sense.

## References

- Franz Achleitner and Christian Kuehn,
*Traveling waves for a bistable equation with nonlocal diffusion*, Adv. Differential Equations**20**(2015), no. 9-10, 887–936. MR**3360395** - Goro Akagi and Messoud Efendiev,
*Allen-Cahn equation with strong irreversibility*, European J. Appl. Math.**30**(2019), no. 4, 707–755. MR**3977346**, DOI 10.1017/s0956792518000384 - G. Akagi and M. Efendiev,
*Lyapunov stability of non-isolated equilibria for strongly irreversible Allen-Cahn equations*, In preparation. - Goro Akagi and Masato Kimura,
*Unidirectional evolution equations of diffusion type*, J. Differential Equations**266**(2019), no. 1, 1–43. MR**3870555**, DOI 10.1016/j.jde.2018.05.022 - S. M. Allen and J. W. Cahn,
*A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening*, Acta Metallurgica**27**(1979), 1085–1095. - Luigi Ambrosio and Vincenzo Maria Tortorelli,
*Approximation of functionals depending on jumps by elliptic functionals via $\Gamma$-convergence*, Comm. Pure Appl. Math.**43**(1990), no. 8, 999–1036. MR**1075076**, DOI 10.1002/cpa.3160430805 - Luigi Ambrosio and V. M. Tortorelli,
*On the approximation of free discontinuity problems*, Boll. Un. Mat. Ital. B (7)**6**(1992), no. 1, 105–123 (English, with Italian summary). MR**1164940** - Tsutomu Arai,
*On the existence of the solution for $\partial \varphi (u^{\prime } (t))+\partial \psi (u(t))\ni f(t)$*, J. Fac. Sci. Univ. Tokyo Sect. IA Math.**26**(1979), no. 1, 75–96. MR**539774** - D. G. Aronson and H. F. Weinberger,
*Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation*, Partial differential equations and related topics (Program, Tulane Univ., New Orleans, La., 1974) Lecture Notes in Math., Vol. 446, Springer, Berlin, 1975, pp. 5–49. MR**0427837** - Viorel Barbu,
*Nonlinear semigroups and differential equations in Banach spaces*, Editura Academiei Republicii Socialiste România, Bucharest; Noordhoff International Publishing, Leiden, 1976. Translated from the Romanian. MR**0390843**, DOI 10.1007/978-94-010-1537-0 - Viorel Barbu,
*Existence theorems for a class of two point boundary problems*, J. Differential Equations**17**(1975), 236–257. MR**380532**, DOI 10.1016/0022-0396(75)90043-1 - H. Brézis,
*Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert*, North-Holland Mathematics Studies, No. 5, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1973 (French). MR**0348562** - Haïm Brézis,
*Monotonicity methods in Hilbert spaces and some applications to nonlinear partial differential equations*, Contributions to nonlinear functional analysis (Proc. Sympos., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1971) Academic Press, New York, 1971, pp. 101–156. MR**0394323** - Luis Caffarelli and Alessio Figalli,
*Regularity of solutions to the parabolic fractional obstacle problem*, J. Reine Angew. Math.**680**(2013), 191–233. MR**3100955**, DOI 10.1515/crelle.2012.036 - Xinfu Chen,
*Existence, uniqueness, and asymptotic stability of traveling waves in nonlocal evolution equations*, Adv. Differential Equations**2**(1997), no. 1, 125–160. MR**1424765** - Adam Chmaj,
*Existence of traveling waves in the fractional bistable equation*, Arch. Math. (Basel)**100**(2013), no. 5, 473–480. MR**3057133**, DOI 10.1007/s00013-013-0511-6 - Yihong Du and Zongming Guo,
*Spreading-vanishing dichotomy in a diffusive logistic model with a free boundary, II*, J. Differential Equations**250**(2011), no. 12, 4336–4366. MR**2793257**, DOI 10.1016/j.jde.2011.02.011 - Yihong Du and Zongming Guo,
*The Stefan problem for the Fisher-KPP equation*, J. Differential Equations**253**(2012), no. 3, 996–1035. MR**2922661**, DOI 10.1016/j.jde.2012.04.014 - Yihong Du and Zhigui Lin,
*Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary*, SIAM J. Math. Anal.**42**(2010), no. 1, 377–405. MR**2607347**, DOI 10.1137/090771089 - Yihong Du, Bendong Lou, and Maolin Zhou,
*Nonlinear diffusion problems with free boundaries: convergence, transition speed, and zero number arguments*, SIAM J. Math. Anal.**47**(2015), no. 5, 3555–3584. MR**3400577**, DOI 10.1137/140994848 - Yihong Du, Hiroshi Matsuzawa, and Maolin Zhou,
*Spreading speed and profile for nonlinear Stefan problems in high space dimensions*, J. Math. Pures Appl. (9)**103**(2015), no. 3, 741–787. MR**3310273**, DOI 10.1016/j.matpur.2014.07.008 - Lawrence C. Evans,
*Partial differential equations*, Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 1998. MR**1625845** - Paul C. Fife and J. B. McLeod,
*The approach of solutions of nonlinear diffusion equations to travelling front solutions*, Arch. Rational Mech. Anal.**65**(1977), no. 4, 335–361. MR**442480**, DOI 10.1007/BF00250432 - R. A. Fisher,
*The wave of advance of advantageous genes*, Ann. Eugenics**7**(1937), 353–369. - G. A. Francfort,
*Quasistatic brittle fracture seen as an energy minimizing movement*, GAMM-Mitt.**29**(2006), no. 2, 172–191. MR**2268765**, DOI 10.1002/gamm.201490029 - G. A. Francfort and J.-J. Marigo,
*Revisiting brittle fracture as an energy minimization problem*, J. Mech. Phys. Solids**46**(1998), no. 8, 1319–1342. MR**1633984**, DOI 10.1016/S0022-5096(98)00034-9 - Alessandro Giacomini,
*Ambrosio-Tortorelli approximation of quasi-static evolution of brittle fractures*, Calc. Var. Partial Differential Equations**22**(2005), no. 2, 129–172. MR**2106765**, DOI 10.1007/s00526-004-0269-6 - C. H. S. Hamster and H. J. Hupkes,
*Stability of traveling waves for reaction-diffusion equations with multiplicative noise*, SIAM J. Appl. Dyn. Syst.**18**(2019), no. 1, 205–278. MR**3907917**, DOI 10.1137/17M1159518 - Yuki Kaneko and Hiroshi Matsuzawa,
*Spreading and vanishing in a free boundary problem for nonlinear diffusion equations with a given forced moving boundary*, J. Differential Equations**265**(2018), no. 3, 1000–1043. MR**3788634**, DOI 10.1016/j.jde.2018.03.026 - Yuki Kaneko and Hiroshi Matsuzawa,
*Spreading speed and sharp asymptotic profiles of solutions in free boundary problems for nonlinear advection-diffusion equations*, J. Math. Anal. Appl.**428**(2015), no. 1, 43–76. MR**3326976**, DOI 10.1016/j.jmaa.2015.02.051 - Yuki Kaneko and Yoshio Yamada,
*Spreading speed and profiles of solutions to a free boundary problem with Dirichlet boundary conditions*, J. Math. Anal. Appl.**465**(2018), no. 2, 1159–1175. MR**3809350**, DOI 10.1016/j.jmaa.2018.05.056 - Yuki Kaneko and Yoshio Yamada,
*A free boundary problem for a reaction-diffusion equation appearing in ecology*, Adv. Math. Sci. Appl.**21**(2011), no. 2, 467–492. MR**2953128** - Ja. I. Kanel′,
*Stabilization of solutions of the Cauchy problem for equations encountered in combustion theory*, Mat. Sb. (N.S.)**59 (101)**(1962), no. suppl., 245–288 (Russian). MR**0157130** - A. Kolmogorov, I. Petrovskii, and N. Piskunov,
*Study of a diffusion equation that is related to the growth of a quality of matter, and its application to a biological problem*, Byul. Mosk. Gos. Univ. Ser. A Mat. Mekh.**1**(1937), 1–26. - Christian Kuehn,
*PDE dynamics*, Mathematical Modeling and Computation, vol. 23, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2019. An introduction. MR**3938717** - Christian Kuehn,
*Travelling waves in monostable and bistable stochastic partial differential equations*, Jahresber. Dtsch. Math.-Ver.**122**(2020), no. 2, 73–107. MR**4107157**, DOI 10.1365/s13291-019-00206-9 - Peter Laurence and Sandro Salsa,
*Regularity of the free boundary of an American option on several assets*, Comm. Pure Appl. Math.**62**(2009), no. 7, 969–994. MR**2527810**, DOI 10.1002/cpa.20268 - J. Nagumo, S. Arimoto, and S. Yoshizawa,
*An active pulse transmission line simulating nerve axon*, Proc. IRE**50**(1962), 2061–2070. - Björn Sandstede,
*Stability of travelling waves*, Handbook of dynamical systems, Vol. 2, North-Holland, Amsterdam, 2002, pp. 983–1055. MR**1901069**, DOI 10.1016/S1874-575X(02)80039-X - Jacques Simon,
*Compact sets in the space $L^p(0,T;B)$*, Ann. Mat. Pura Appl. (4)**146**(1987), 65–96. MR**916688**, DOI 10.1007/BF01762360 - Aizik I. Volpert, Vitaly A. Volpert, and Vladimir A. Volpert,
*Traveling wave solutions of parabolic systems*, Translations of Mathematical Monographs, vol. 140, American Mathematical Society, Providence, RI, 1994. Translated from the Russian manuscript by James F. Heyda. MR**1297766**, DOI 10.1090/mmono/140

## Additional Information

**Goro Akagi**- Affiliation: Mathematical Institute and Graduate School of Science, Tohoku University, 6-3 Aoba, Aramaki, Aoba-ku, Sendai 980-8578, Japan
- MR Author ID: 725232
- ORCID: 0000-0001-7686-652X
- Email: goro.akagi@tohoku.ac.jp
**Christian Kuehn**- Affiliation: Department of Mathematics, Technical University of Munich, Boltzmannstraße 3, D-85748 Garching bei München, Germany
- MR Author ID: 875693
- ORCID: 0000-0002-7063-6173
- Email: ckuehn@ma.tum.de
**Ken-Ichi Nakamura**- Affiliation: Institute of Science and Engineering, Kanazawa University, Kakuma-machi, Kanazawa-shi, Ishikawa 920-1192, Japan
- MR Author ID: 613791
- ORCID: 0000-0001-9930-4001
- Email: k-nakamura@se.kanazawa-u.ac.jp
- Received by editor(s): July 30, 2020
- Received by editor(s) in revised form: September 24, 2021
- Published electronically: February 4, 2022
- Additional Notes: The first author was supported by the Alexander von Humboldt Foundation and by the Carl Friedrich von Siemens Foundation and by JSPS KAKENHI Grant Number JP21KK0044, JP21K18581, JP20H01812, JP18K18715, JP16H03946 and JP20H00117, JP17H01095. The second author acknowledges support via a Lichtenberg Professorship of the VolkswagenStiftung
- © Copyright 2022 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**375**(2022), 3173-3238 - MSC (2020): Primary 35C07; Secondary 35R35, 47J35
- DOI: https://doi.org/10.1090/tran/8583
- MathSciNet review: 4402659