# American Institute of Mathematical Sciences

October  2021, 41(10): 4737-4765. doi: 10.3934/dcds.2021055

## A variational approach to three-phase traveling waves for a gradient system

 Department of Mathematics, National Taiwan University, Taipei, 10617, Taiwan

* Corresponding author: Hung-Yu Chien, andy1010@gmail.com

Received  February 2020 Revised  January 2021 Published  October 2021 Early access  March 2021

Fund Project: C.-C. Chen and H.-Y. Chien are supported by the grants 108-2115-M-002-011-MY3 of MOST, Taiwan. C.-C. Huang is supported by the grants 109-2811-M-002-623 and 108-2115-M-002-012-MY2, MOST

In this paper, we use a variational approach to study traveling wave solutions of a gradient system in an infinite strip. As the even-symmetric potential of the system has three local minima, we prove the existence of a traveling wave that propagates from one phase to the other two phases, where these phases corresponds to the three local minima of the potential. To control the asymptotic behavior of the wave at minus infinity, we successfully find a certain convexity condition on the potential, which guarantees the convergence of the wave to a constant state but not to a one-dimensional homoclinic solution or other equilibria. In addition, a non-trivial steady state in $\mathbb R^2$ is established by taking a limit of the traveling wave solutions in the strip as the width of the strip tends to infinity.

Citation: Chiun-Chuan Chen, Hung-Yu Chien, Chih-Chiang Huang. A variational approach to three-phase traveling waves for a gradient system. Discrete and Continuous Dynamical Systems, 2021, 41 (10) : 4737-4765. doi: 10.3934/dcds.2021055
##### References:
 [1] S. Alama, L. Bronsard and C. Gui, Stationary layered solutions in $\mathbb R^2$ for an Allen-Cahn system with multiple well potential, Calculus of Variations and Partial Differential Equations, 5 (1997), 359-390.  doi: 10.1007/s005260050071. [2] F. Alessio, A. Calamai and P. Montecchiari, Saddle-type solutions for a class of semilinear elliptic equations, Adv. Differential Equations, 12 (2007), 361–380, URL http://projecteuclid.org/euclid.ade/1355867455. [3] N. D. Alikakos, On the structure of phase transition maps for three or more coexisting phases, in Geometric Partial Differential Equations Proceedings, CRM Series, 15, Ed. Norm., Pisa, 2013, 1–31. doi: 10.1007/978-88-7642-473-1_1. [4] N. D. Alikakos and G. Fusco, Entire solutions to equivariant elliptic systems with variational structure, Archive for Rational Mechanics and Analysis, 202 (2011), 567-597.  doi: 10.1007/s00205-011-0441-z. [5] N. D. Alikakos and N. I. Katzourakis, Heteroclinic travelling waves of gradient diffusion systems, Transactions of the American Mathematical Society, 363 (2011), 1365-1365.  doi: 10.1090/S0002-9947-2010-04987-6. [6] 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.  doi: 10.1016/0001-6160(79)90196-2. [7] S. Baldo, Minimal interface criterion for phase transitions in mixtures of Cahn-Hilliard fluids, Annales Inst. H. Poincaré Anal. Non Linéaire, 7 (1990), 67–90, URL http://eudml.org/doc/78216. doi: 10.1016/S0294-1449(16)30304-3. [8] M. Bertsch, C. B. Muratov and I. Primi, Traveling wave solutions of harmonic heat flow, Calculus of Variations and Partial Differential Equations, 26 (2006), 489-509.  doi: 10.1007/s00526-006-0016-2. [9] F. Béthuel, H. Brezis and F. Hélein, Ginzburg-Landau Vortices, Birkhäuser Boston, 1994. doi: 10.1007/978-1-4612-0287-5. [10] F. Béthuel and G. Orlandi, Ginzburg-Landau functionals, phase transitions and vorticity, in Noncompact Problems at the Intersection of Geometry, Analysis, and Topology, Contemp. Math., 350, American Mathematical Society, Providence, RI, 2004, 35–47. doi: 10.1090/conm/350/06336. [11] L. Bronsard, C. Gui and M. Schatzman, A three-layered minimizer in $\mathbb R^2$ for a variational problem with a symmetric three-well potential, Communications on Pure and Applied Mathematics, 49 (1996), 677-715.  doi: 10.1002/(SICI)1097-0312(199607)49:7<677::AID-CPA2>3.0.CO;2-6. [12] L. Bronsard and F. Reitich, On three-phase boundary motion and the singular limit of a vector-valued Ginzburg-Landau equation, Archive for Rational Mechanics and Analysis, 124 (1993), 355-379.  doi: 10.1007/BF00375607. [13] J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system. I. Interfacial free energy, The Journal of Chemical Physics, 28. doi: 10.1002/9781118788295.ch4. [14] C.-N. Chen, C.-C. Chen and C.-C. Huang, Traveling waves for the FitzHugh-Nagumo system on an infinite channel, Journal of Differential Equations, 261 (2016), 3010-3041.  doi: 10.1016/j.jde.2016.05.014. [15] C.-N. Chen and Y. S. Choi, Traveling pulse solutions to FitzHugh-Nagumo equations, Calculus of Variations and Partial Differential Equations, 54 (2015), 1-45.  doi: 10.1007/s00526-014-0776-z. [16] X. Chen, Generation and propagation of interfaces for reaction-diffusion equations, Journal of Differential Equations, 96 (1992), 116-141.  doi: 10.1016/0022-0396(92)90146-E. [17] X. Chen, J.-S. Guo, F. Hamel, H. Ninomiya and J.-M. Roquejoffre, Traveling waves with paraboloid like interfaces for balanced bistable dynamics, Annales de l'Institut Henri Poincaré. Annales: Analyse Non Linéaire Analysis, 24 (2007), 369-393.  doi: 10.1016/j.anihpc.2006.03.012. [18] G. Dal Maso, An Introduction to Gamma-Convergence, Progress in Nonlinear Differential Equations and their Applications, 8. Birkhäuser Boston, Inc., Boston, MA, 1993. doi: 10.1007/978-1-4612-0327-8. [19] H. Dang, P. C. Fife and L. A. Peletier, Saddle solutions of the bistable diffusion equation, Zeitschrift für Angewandte Mathematik und Physik ZAMP, 43 (1992), 984–998. doi: 10.1007/BF00916424. [20] P. de Mottoni and M. Schatzman, Geometrical evolution of developed interfaces, Transactions of the American Mathematical Society, 347 (1995), 1533–1589, URL http://www.jstor.org/stable/2154960. doi: 10.1090/S0002-9947-1995-1672406-7. [21] P. C. Fife and J. B. McLeod, The approach of solutions of nonlinear diffusion equations to travelling front solutions, Archive for Rational Mechanics and Analysis, 65 (1975), 335-361.  doi: 10.1007/BF00250432. [22] D. Gilbarg and N. S. Trudinger, Ellitic Partial Differential Equations of Second Order, 1998 edition, Springer, 2001. [23] C. Gui and M. Schatzman, Symmetric quadruple phase transitions, Indiana University Mathematics Journal, 57 (2008), 781-836.  doi: 10.1512/iumj.2008.57.3089. [24] M. E. Gurtin, On phase transitions with bulk, interfacial, and boundary Energy, Archive for Rational Mechanics and Analysis, 96 (1986), 243-264.  doi: 10.1007/BF00251908. [25] M. E. Gurtin and H. Matano, On the structure of equilibrium phase transitions within the gradient theory of fluids, Quarterly of Applied Mathematics, 46 (1988), 301-317.  doi: 10.1090/qam/950604. [26] F. Hamel, Bistable transition fronts in $\mathbb R^N$, Advances in Mathematics, 289 (2016), 279-344.  doi: 10.1016/j.aim.2015.11.033. [27] F. Hamel, R. Monneau and J.-M. Roquejoffre, Existence and qualitative properties of multidimensional conical bistable fronts, Discrete and Continuous Dynamical Systems, 13 (2005), 1069-1096.  doi: 10.3934/dcds.2005.13.1069. [28] S. Heinze, A Variational Approach to Traveling Waves, Technical Report 85, Max Planck Institute for Mathematical Sciences. [29] C. D. Levermore and J. X. Xin, Multidimensional stability of traveling waves in a bistable reaction-diffusion equation, II, Communications in Partial Differential Equations, 17 (1992), 1901-1924.  doi: 10.1080/03605309208820908. [30] M. Lucia, C. B. Muratov and M. Novaga, Existence of traveling waves of invasion for Ginzburg-Landau-type problems in infinite cylinders, Archive for Rational Mechanics and Analysis, 188 (2008), 475-508.  doi: 10.1007/s00205-007-0097-x. [31] H. Matano, M. Nara and M. Taniguchi, Stability of planar waves in the Allen-Cahn equation, Communications in Partial Differential Equations, 34 (2009), 976-1002.  doi: 10.1080/03605300902963500. [32] L. Modica, The gradient theory of phase transitions and the minimal interface criterion., Arch. Ration. Mech. Anal., 98 (1987), 123-142.  doi: 10.1007/BF00251230. [33] Y. Morita and H. Ninomiya, Monostable-type traveling waves of bistable reaction-diffusion equations in the multi-dimensional space, Bulletin of the Institute of Mathematics Academia Sinica (New Series), 3 (2008), 567-584. [34] Y. Morita and H. Ninomiya, Traveling wave solutions and entire solutions of reaction-diffusion equations, Sugaku Expositions, 23 (2010), 213-233. [35] C. B. Muratov and M. Novaga, Front propagation in infinite cylinders. I. A variational approach, Communications in Mathematical Sciences, 6 (2008), 799-826.  doi: 10.4310/CMS.2008.v6.n4.a1. [36] C. B. Muratov, A global variational structure and propagation of disturbances in reaction-diffusion systems of gradient type, Discrete and Continuous Dynamical Systems - Series B, 4 (2004), 867-892.  doi: 10.3934/dcdsb.2004.4.867. [37] H. Ninomiya and M. Taniguchi, Existence and global stability of traveling curved fronts in the Allen-Cahn equations, Journal of Differential Equations, 213 (2005), 204-233.  doi: 10.1016/j.jde.2004.06.011. [38] R. S. Palais, The principle of symmetric criticality, Communications in Mathematical Physics, 69 (1979), 19-30.  doi: 10.1007/BF01941322. [39] P. Sternberg, Vector-Valued local minimizers of nonconvex variational problems, Rocky Mountain Journal of Mathematics, 21 (1991), 799-807.  doi: 10.1216/rmjm/1181072968. [40] P. Sternberg and W. P. Zeimer, Local minimisers of a three-phase partition problem with triple junctions, Proc. Roy. Soc. Edinburgh Sect. A, 124 (1994), 1059-1073.  doi: 10.1017/S0308210500030110. [41] M. Taniguchi, Traveling fronts of pyramidal shapes in the Allen-Cahn equations, SIAM Journal on Mathematical Analysis, 39 (2007), 319-344.  doi: 10.1137/060661788. [42] M. Taniguchi, The uniqueness and asymptotic stability of pyramidal traveling fronts in the Allen-Cahn equations, Journal of Differential Equations, 246 (2009), 2103-2130.  doi: 10.1016/j.jde.2008.06.037. [43] M. Taniguchi, An $(N-1)$-dimensional convex compact set gives an $N$-dimensional traveling front in the Allen-Cahn equation, SIAM Journal on Mathematical Analysis, 47 (2015), 455-476.  doi: 10.1137/130945041. [44] M. Taniguchi, Convex compact sets in $R^{N-1}$ give traveling fronts of cooperation-diffusion systems in $R^N$, Journal of Differential Equations, 260 (2016), 4301-4338.  doi: 10.1016/j.jde.2015.11.010.

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##### References:
 [1] S. Alama, L. Bronsard and C. Gui, Stationary layered solutions in $\mathbb R^2$ for an Allen-Cahn system with multiple well potential, Calculus of Variations and Partial Differential Equations, 5 (1997), 359-390.  doi: 10.1007/s005260050071. [2] F. Alessio, A. Calamai and P. Montecchiari, Saddle-type solutions for a class of semilinear elliptic equations, Adv. Differential Equations, 12 (2007), 361–380, URL http://projecteuclid.org/euclid.ade/1355867455. [3] N. D. Alikakos, On the structure of phase transition maps for three or more coexisting phases, in Geometric Partial Differential Equations Proceedings, CRM Series, 15, Ed. Norm., Pisa, 2013, 1–31. doi: 10.1007/978-88-7642-473-1_1. [4] N. D. Alikakos and G. Fusco, Entire solutions to equivariant elliptic systems with variational structure, Archive for Rational Mechanics and Analysis, 202 (2011), 567-597.  doi: 10.1007/s00205-011-0441-z. [5] N. D. Alikakos and N. I. Katzourakis, Heteroclinic travelling waves of gradient diffusion systems, Transactions of the American Mathematical Society, 363 (2011), 1365-1365.  doi: 10.1090/S0002-9947-2010-04987-6. [6] 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.  doi: 10.1016/0001-6160(79)90196-2. [7] S. Baldo, Minimal interface criterion for phase transitions in mixtures of Cahn-Hilliard fluids, Annales Inst. H. Poincaré Anal. Non Linéaire, 7 (1990), 67–90, URL http://eudml.org/doc/78216. doi: 10.1016/S0294-1449(16)30304-3. [8] M. Bertsch, C. B. Muratov and I. Primi, Traveling wave solutions of harmonic heat flow, Calculus of Variations and Partial Differential Equations, 26 (2006), 489-509.  doi: 10.1007/s00526-006-0016-2. [9] F. Béthuel, H. Brezis and F. Hélein, Ginzburg-Landau Vortices, Birkhäuser Boston, 1994. doi: 10.1007/978-1-4612-0287-5. [10] F. Béthuel and G. Orlandi, Ginzburg-Landau functionals, phase transitions and vorticity, in Noncompact Problems at the Intersection of Geometry, Analysis, and Topology, Contemp. Math., 350, American Mathematical Society, Providence, RI, 2004, 35–47. doi: 10.1090/conm/350/06336. [11] L. Bronsard, C. Gui and M. Schatzman, A three-layered minimizer in $\mathbb R^2$ for a variational problem with a symmetric three-well potential, Communications on Pure and Applied Mathematics, 49 (1996), 677-715.  doi: 10.1002/(SICI)1097-0312(199607)49:7<677::AID-CPA2>3.0.CO;2-6. [12] L. Bronsard and F. Reitich, On three-phase boundary motion and the singular limit of a vector-valued Ginzburg-Landau equation, Archive for Rational Mechanics and Analysis, 124 (1993), 355-379.  doi: 10.1007/BF00375607. [13] J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system. I. Interfacial free energy, The Journal of Chemical Physics, 28. doi: 10.1002/9781118788295.ch4. [14] C.-N. Chen, C.-C. Chen and C.-C. Huang, Traveling waves for the FitzHugh-Nagumo system on an infinite channel, Journal of Differential Equations, 261 (2016), 3010-3041.  doi: 10.1016/j.jde.2016.05.014. [15] C.-N. Chen and Y. S. Choi, Traveling pulse solutions to FitzHugh-Nagumo equations, Calculus of Variations and Partial Differential Equations, 54 (2015), 1-45.  doi: 10.1007/s00526-014-0776-z. [16] X. Chen, Generation and propagation of interfaces for reaction-diffusion equations, Journal of Differential Equations, 96 (1992), 116-141.  doi: 10.1016/0022-0396(92)90146-E. [17] X. Chen, J.-S. Guo, F. Hamel, H. Ninomiya and J.-M. Roquejoffre, Traveling waves with paraboloid like interfaces for balanced bistable dynamics, Annales de l'Institut Henri Poincaré. Annales: Analyse Non Linéaire Analysis, 24 (2007), 369-393.  doi: 10.1016/j.anihpc.2006.03.012. [18] G. Dal Maso, An Introduction to Gamma-Convergence, Progress in Nonlinear Differential Equations and their Applications, 8. Birkhäuser Boston, Inc., Boston, MA, 1993. doi: 10.1007/978-1-4612-0327-8. [19] H. Dang, P. C. Fife and L. A. Peletier, Saddle solutions of the bistable diffusion equation, Zeitschrift für Angewandte Mathematik und Physik ZAMP, 43 (1992), 984–998. doi: 10.1007/BF00916424. [20] P. de Mottoni and M. Schatzman, Geometrical evolution of developed interfaces, Transactions of the American Mathematical Society, 347 (1995), 1533–1589, URL http://www.jstor.org/stable/2154960. doi: 10.1090/S0002-9947-1995-1672406-7. [21] P. C. Fife and J. B. McLeod, The approach of solutions of nonlinear diffusion equations to travelling front solutions, Archive for Rational Mechanics and Analysis, 65 (1975), 335-361.  doi: 10.1007/BF00250432. [22] D. Gilbarg and N. S. Trudinger, Ellitic Partial Differential Equations of Second Order, 1998 edition, Springer, 2001. [23] C. Gui and M. Schatzman, Symmetric quadruple phase transitions, Indiana University Mathematics Journal, 57 (2008), 781-836.  doi: 10.1512/iumj.2008.57.3089. [24] M. E. Gurtin, On phase transitions with bulk, interfacial, and boundary Energy, Archive for Rational Mechanics and Analysis, 96 (1986), 243-264.  doi: 10.1007/BF00251908. [25] M. E. Gurtin and H. Matano, On the structure of equilibrium phase transitions within the gradient theory of fluids, Quarterly of Applied Mathematics, 46 (1988), 301-317.  doi: 10.1090/qam/950604. [26] F. Hamel, Bistable transition fronts in $\mathbb R^N$, Advances in Mathematics, 289 (2016), 279-344.  doi: 10.1016/j.aim.2015.11.033. [27] F. Hamel, R. Monneau and J.-M. Roquejoffre, Existence and qualitative properties of multidimensional conical bistable fronts, Discrete and Continuous Dynamical Systems, 13 (2005), 1069-1096.  doi: 10.3934/dcds.2005.13.1069. [28] S. Heinze, A Variational Approach to Traveling Waves, Technical Report 85, Max Planck Institute for Mathematical Sciences. [29] C. D. Levermore and J. X. Xin, Multidimensional stability of traveling waves in a bistable reaction-diffusion equation, II, Communications in Partial Differential Equations, 17 (1992), 1901-1924.  doi: 10.1080/03605309208820908. [30] M. Lucia, C. B. Muratov and M. Novaga, Existence of traveling waves of invasion for Ginzburg-Landau-type problems in infinite cylinders, Archive for Rational Mechanics and Analysis, 188 (2008), 475-508.  doi: 10.1007/s00205-007-0097-x. [31] H. Matano, M. Nara and M. Taniguchi, Stability of planar waves in the Allen-Cahn equation, Communications in Partial Differential Equations, 34 (2009), 976-1002.  doi: 10.1080/03605300902963500. [32] L. Modica, The gradient theory of phase transitions and the minimal interface criterion., Arch. Ration. Mech. Anal., 98 (1987), 123-142.  doi: 10.1007/BF00251230. [33] Y. Morita and H. Ninomiya, Monostable-type traveling waves of bistable reaction-diffusion equations in the multi-dimensional space, Bulletin of the Institute of Mathematics Academia Sinica (New Series), 3 (2008), 567-584. [34] Y. Morita and H. Ninomiya, Traveling wave solutions and entire solutions of reaction-diffusion equations, Sugaku Expositions, 23 (2010), 213-233. [35] C. B. Muratov and M. Novaga, Front propagation in infinite cylinders. I. A variational approach, Communications in Mathematical Sciences, 6 (2008), 799-826.  doi: 10.4310/CMS.2008.v6.n4.a1. [36] C. B. Muratov, A global variational structure and propagation of disturbances in reaction-diffusion systems of gradient type, Discrete and Continuous Dynamical Systems - Series B, 4 (2004), 867-892.  doi: 10.3934/dcdsb.2004.4.867. [37] H. Ninomiya and M. Taniguchi, Existence and global stability of traveling curved fronts in the Allen-Cahn equations, Journal of Differential Equations, 213 (2005), 204-233.  doi: 10.1016/j.jde.2004.06.011. [38] R. S. Palais, The principle of symmetric criticality, Communications in Mathematical Physics, 69 (1979), 19-30.  doi: 10.1007/BF01941322. [39] P. Sternberg, Vector-Valued local minimizers of nonconvex variational problems, Rocky Mountain Journal of Mathematics, 21 (1991), 799-807.  doi: 10.1216/rmjm/1181072968. [40] P. Sternberg and W. P. Zeimer, Local minimisers of a three-phase partition problem with triple junctions, Proc. Roy. Soc. Edinburgh Sect. A, 124 (1994), 1059-1073.  doi: 10.1017/S0308210500030110. [41] M. Taniguchi, Traveling fronts of pyramidal shapes in the Allen-Cahn equations, SIAM Journal on Mathematical Analysis, 39 (2007), 319-344.  doi: 10.1137/060661788. [42] M. Taniguchi, The uniqueness and asymptotic stability of pyramidal traveling fronts in the Allen-Cahn equations, Journal of Differential Equations, 246 (2009), 2103-2130.  doi: 10.1016/j.jde.2008.06.037. [43] M. Taniguchi, An $(N-1)$-dimensional convex compact set gives an $N$-dimensional traveling front in the Allen-Cahn equation, SIAM Journal on Mathematical Analysis, 47 (2015), 455-476.  doi: 10.1137/130945041. [44] M. Taniguchi, Convex compact sets in $R^{N-1}$ give traveling fronts of cooperation-diffusion systems in $R^N$, Journal of Differential Equations, 260 (2016), 4301-4338.  doi: 10.1016/j.jde.2015.11.010.
The union of gray regions is $\{W(u)\le\epsilon\}$. The green curves are geodesic balls. And the red circle is $\{ \mathbf p\ |\ | \mathbf p - \mathbf a| = \delta\}$
The portrait of $\partial_u\widetilde W = 0$, $\det( {{\rm{Hess\; }}}\widetilde W) = 0$, $\partial_{uu}W = 0$ and $| \mathbf p- \mathbf a| = 0.2$
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