# American Institute of Mathematical Sciences

September 2019, 18(5): 2457-2472. doi: 10.3934/cpaa.2019111

## Global asymptotic stability of traveling waves to the Allen-Cahn equation with a fractional Laplacian

 School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, China

* Corresponding author

Received  May 2018 Revised  November 2018 Published  April 2019

In this paper, we study the asymptotic stability of traveling wave fronts to the Allen-Cahn equation with a fractional Laplacian. The main tools that we used are super- and subsolutions and squeezing methods.

Citation: Luyi Ma, Hong-Tao Niu, Zhi-Cheng Wang. Global asymptotic stability of traveling waves to the Allen-Cahn equation with a fractional Laplacian. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2457-2472. doi: 10.3934/cpaa.2019111
##### References:
 [1] X. Cabré, N. Cónsul and J. Mandé, Traveling wave solutions in a half-space for boundary reactions, Anal. PDE, 8 (2015), 333-364. doi: 10.2140/apde.2015.8.333. [2] X. Cabré and J-M. Roquejoffre, The influence of fractional diffusion in Fisher-KPP equations, Comm. Math. Phys., 320 (2013), 679-722. doi: 10.1007/s00220-013-1682-5. [3] X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians, I: Regularity, maximum principles, and Hamiltonian estimates, Ann. Inst. H. Poincaré, 31 (2014), 23-53. doi: 10.1016/j.anihpc.2013.02.001. [4] X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians II: Existence, uniqueness, and qualitative properties of solutions, Trans. Amer. Math. Soc., 367 (2015), 911-941. doi: 10.1090/S0002-9947-2014-05906-0. [5] L. Caffarelli, A. Mellet and Y. Sire, Traveling waves for a boundary reaction-diffusion equation, Adv. Math., 230 (2012), 433-457. doi: 10.1016/j.aim.2012.01.020. [6] L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260. doi: 10.1080/03605300600987306. [7] H. Chan and J. Wei, Traveling wave solutions for bistable fractional Allen-Cahn equations with a pyramidal front, J. Differential Equations, 262 (2017), 4567-4609. doi: 10.1016/j.jde.2016.12.010. [8] X. Chen, Existence, uniqueness, and asymptotic stability of traveling waves in nonlocal evolution equations, Adv. Differential Equations, 2 (1997), 125-160. [9] P. C. Fife and J. B. Mcleod, The approach of solutions of nonlinear diffusion equations to travelling front solutions, Arch. Ration. Mech. Anal., 65 (1977), 335-361. doi: 10.1007/BF00250432. [10] C. Gui and T. Huan, Traveling wave solutions to some reaction diffusion equations with fractional Laplacians, Calc. Var. Partial Differential Equations, 54 (2015), 251-273. doi: 10.1007/s00526-014-0785-y. [11] C. Gui and M. Zhao, Traveling wave solutions of Allen-Cahn equation with a fractional Laplacian, Ann. Inst. H. Poincaré, 32 (2015), 785-812. doi: 10.1016/j.anihpc.2014.03.005. [12] N. S. Landkof, Foundations of Modern Potential Theory, Die Grundlehren der mathematischen Wissenschaften, Springer-Verlag, New York, 1972. doi: 978-3-642-65185-4. [13] R. H. Martin and H. L. Smith, Abstract functional-differential equations and reaction-diffusion systems, Trans. Amer. Math. Soc., 321 (1990), 1-44. doi: 10.1090/S0002-9947-1990-0967316-X. [14] H. Matano, M. Nara and M. Taniguchi, Stability of planar waves in the Allen-Cahn equation, Comm. Partial Differential Equations, 34 (2009), 976-1002. doi: 10.1080/03605300902963500. [15] A. Mellet, J-M. Roquejoffre and Y. Sire, Existence and asymptotics of fronts in non local combustion models, Commun. Math. Sci., 12 (2014), 1-11. doi: 10.4310/CMS.2014.v12.n1.a1. [16] C. B. Muratov, F. Posta and S. Y. Shvartsman, Autocrine signal transmission with extracellular ligand degradation, Phys. Biol., 6 (2009). doi: 10.1088/1478-3975/6/1/016006. [17] Y. Necb, A. A. Nepomnyashchy and V. A. Volpert, Exact solutions in front propagation problems with superdiffusion, Physica D, 239 (2010), 134-144. doi: 10.1016/j.physd.2009.10.011. [18] H. Ninomiya and M. Taniguchi, Existence and global stability of traveling curved fronts in the Allen-Cahn equations, J. Differential Equations, 213 (2005), 204-233. doi: 10.1016/j.jde.2004.06.011. [19] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44. New York: Springer-Verlag, 1983. doi: 10.1007/978-1-4612-5561-1. [20] H. L. Smith and X. Zhao, Global asymptotic stability of traveling waves in delayed reaction-diffusion equations, SIAM J. Math. Anal., 31 (2000), 514-534. doi: 10.1137/S0036141098346785. [21] M. Taniguchi, Traveling fronts of pyramidal shapes in the Allen-Cahn equations, SIAM J. Math. Anal., 39 (2007), 319-344. doi: 10.1137/060661788. [22] M. Taniguchi, The uniqueness and asymptotic stability of pyramidal traveling fronts in the Allen-Cahn equations, J. Differential Equations, 246 (2009), 2103-2130. doi: 10.1016/j.jde.2008.06.037. [23] Z.-C. Wang, W.-T. Li and S. Ruan, Existence and stability of traveling wave fronts in reaction advection diffusion equations with nonlocal delay, J. Differential Equations, 238 (2007), 153-200. doi: 10.1016/j.jde.2007.03.025. [24] Q. Ye, Z. Li, M. Wang and Y. Wu, Introduction of Reaction-Diffusion Equations, Science Press, Beijing, 2011.

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##### References:
 [1] X. Cabré, N. Cónsul and J. Mandé, Traveling wave solutions in a half-space for boundary reactions, Anal. PDE, 8 (2015), 333-364. doi: 10.2140/apde.2015.8.333. [2] X. Cabré and J-M. Roquejoffre, The influence of fractional diffusion in Fisher-KPP equations, Comm. Math. Phys., 320 (2013), 679-722. doi: 10.1007/s00220-013-1682-5. [3] X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians, I: Regularity, maximum principles, and Hamiltonian estimates, Ann. Inst. H. Poincaré, 31 (2014), 23-53. doi: 10.1016/j.anihpc.2013.02.001. [4] X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians II: Existence, uniqueness, and qualitative properties of solutions, Trans. Amer. Math. Soc., 367 (2015), 911-941. doi: 10.1090/S0002-9947-2014-05906-0. [5] L. Caffarelli, A. Mellet and Y. Sire, Traveling waves for a boundary reaction-diffusion equation, Adv. Math., 230 (2012), 433-457. doi: 10.1016/j.aim.2012.01.020. [6] L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260. doi: 10.1080/03605300600987306. [7] H. Chan and J. Wei, Traveling wave solutions for bistable fractional Allen-Cahn equations with a pyramidal front, J. Differential Equations, 262 (2017), 4567-4609. doi: 10.1016/j.jde.2016.12.010. [8] X. Chen, Existence, uniqueness, and asymptotic stability of traveling waves in nonlocal evolution equations, Adv. Differential Equations, 2 (1997), 125-160. [9] P. C. Fife and J. B. Mcleod, The approach of solutions of nonlinear diffusion equations to travelling front solutions, Arch. Ration. Mech. Anal., 65 (1977), 335-361. doi: 10.1007/BF00250432. [10] C. Gui and T. Huan, Traveling wave solutions to some reaction diffusion equations with fractional Laplacians, Calc. Var. Partial Differential Equations, 54 (2015), 251-273. doi: 10.1007/s00526-014-0785-y. [11] C. Gui and M. Zhao, Traveling wave solutions of Allen-Cahn equation with a fractional Laplacian, Ann. Inst. H. Poincaré, 32 (2015), 785-812. doi: 10.1016/j.anihpc.2014.03.005. [12] N. S. Landkof, Foundations of Modern Potential Theory, Die Grundlehren der mathematischen Wissenschaften, Springer-Verlag, New York, 1972. doi: 978-3-642-65185-4. [13] R. H. Martin and H. L. Smith, Abstract functional-differential equations and reaction-diffusion systems, Trans. Amer. Math. Soc., 321 (1990), 1-44. doi: 10.1090/S0002-9947-1990-0967316-X. [14] H. Matano, M. Nara and M. Taniguchi, Stability of planar waves in the Allen-Cahn equation, Comm. Partial Differential Equations, 34 (2009), 976-1002. doi: 10.1080/03605300902963500. [15] A. Mellet, J-M. Roquejoffre and Y. Sire, Existence and asymptotics of fronts in non local combustion models, Commun. Math. Sci., 12 (2014), 1-11. doi: 10.4310/CMS.2014.v12.n1.a1. [16] C. B. Muratov, F. Posta and S. Y. Shvartsman, Autocrine signal transmission with extracellular ligand degradation, Phys. Biol., 6 (2009). doi: 10.1088/1478-3975/6/1/016006. [17] Y. Necb, A. A. Nepomnyashchy and V. A. Volpert, Exact solutions in front propagation problems with superdiffusion, Physica D, 239 (2010), 134-144. doi: 10.1016/j.physd.2009.10.011. [18] H. Ninomiya and M. Taniguchi, Existence and global stability of traveling curved fronts in the Allen-Cahn equations, J. Differential Equations, 213 (2005), 204-233. doi: 10.1016/j.jde.2004.06.011. [19] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44. New York: Springer-Verlag, 1983. doi: 10.1007/978-1-4612-5561-1. [20] H. L. Smith and X. Zhao, Global asymptotic stability of traveling waves in delayed reaction-diffusion equations, SIAM J. Math. Anal., 31 (2000), 514-534. doi: 10.1137/S0036141098346785. [21] M. Taniguchi, Traveling fronts of pyramidal shapes in the Allen-Cahn equations, SIAM J. Math. Anal., 39 (2007), 319-344. doi: 10.1137/060661788. [22] M. Taniguchi, The uniqueness and asymptotic stability of pyramidal traveling fronts in the Allen-Cahn equations, J. Differential Equations, 246 (2009), 2103-2130. doi: 10.1016/j.jde.2008.06.037. [23] Z.-C. Wang, W.-T. Li and S. Ruan, Existence and stability of traveling wave fronts in reaction advection diffusion equations with nonlocal delay, J. Differential Equations, 238 (2007), 153-200. doi: 10.1016/j.jde.2007.03.025. [24] Q. Ye, Z. Li, M. Wang and Y. Wu, Introduction of Reaction-Diffusion Equations, Science Press, Beijing, 2011.
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