-
Previous Article
Infinitely many solutions for nonlinear Schrödinger system with non-symmetric potentials
- CPAA Home
- This Issue
-
Next Article
Oscillatory integrals related to Carleson's theorem: fractional monomials
Layer solutions for an Allen-Cahn type system driven by the fractional Laplacian
1. | College of Mathematics and Econometrics, Hunan University, Changsha 410082, China |
References:
[1] |
S. Alama, L. Bronsard and C. Gui, Stationary layered solutions in $\R^{2}$ 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] |
L. Bronsard, C. Gui and M. Schatzman, A three layered minimizer in $\R^{2}$ 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>3.3.CO;2-6. |
[3] |
L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Communications in Partial Differential Equations, 32 (2007), 1245-1260.
doi: 10.1080/03605300600987306. |
[4] |
X. Cabré and J. Solá-Morales, Layer solutions in a half-space for boundary reactions, Communications on Pure and Applied Mathematics, 58 (2005), 1678-1732.
doi: 10.1002/cpa.20093. |
[5] |
X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians, I: Regularity, maximum principles, and Hamiltonian estimates, Annales de l'Institut Henri Poincare (C) Non Linear Analysis, 31 (2014), 23-53.
doi: 10.1016/j.anihpc.2013.02.001. |
[6] |
X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians II: existence, uniqueness, and qualitative properties of solutions, Transactions of the American Mathematical Society, 367 (2015), 911-941.
doi: 10.1090/S0002-9947-2014-05906-0. |
[7] |
X. Cabré and E. Cinti, Sharp energy estimates for nonlinear fractional diffusion equations, Calculus of Variations and Partial Differential Equations, 49 (2014), 233-269.
doi: 10.1007/s00526-012-0580-6. |
[8] |
X. Cabré and E. Cinti, Energy estimates and 1-D symmetry for nonlinear equations involving the half-Laplacian, Discrete and Continuous Dynamical System, 28 (2010), 1179-1206.
doi: 10.3934/dcds.2010.28.1179. |
[9] |
E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bulletin des Sciences Mathématiques, 136 (2012), 521-573.
doi: 10.1016/j.bulsci.2011.12.004. |
[10] |
E. B. Fabes, C. E. Kenig and R. P. Serapioni, The local regularity of solutions of degenerate elliptic equations, Communications in Partial Differential Equations, 7 (1982), 77-116.
doi: 10.1080/03605308208820218. |
[11] |
G. Palatucci, O. Savin and E. Valdinoci, Local and global minimizers for a variational energy involving a fractional norm, Annali di Matematica Pura ed Applicata, 192 (2013), 673-718.
doi: 10.1007/s10231-011-0243-9. |
[12] |
L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Communications on Pure and Applied Mathematics, 60 (2007), 67-112.
doi: 10.1002/cpa.20153. |
[13] |
Y. Sire and E. Valdinoci, Fractional Laplacian phase transitions and boundary reactions: a geometric inequality and a symmetry result, Journal of Functional Analysis, 256 (2009), 1842-1864.
doi: 10.1016/j.jfa.2009.01.020. |
show all references
References:
[1] |
S. Alama, L. Bronsard and C. Gui, Stationary layered solutions in $\R^{2}$ 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] |
L. Bronsard, C. Gui and M. Schatzman, A three layered minimizer in $\R^{2}$ 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>3.3.CO;2-6. |
[3] |
L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Communications in Partial Differential Equations, 32 (2007), 1245-1260.
doi: 10.1080/03605300600987306. |
[4] |
X. Cabré and J. Solá-Morales, Layer solutions in a half-space for boundary reactions, Communications on Pure and Applied Mathematics, 58 (2005), 1678-1732.
doi: 10.1002/cpa.20093. |
[5] |
X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians, I: Regularity, maximum principles, and Hamiltonian estimates, Annales de l'Institut Henri Poincare (C) Non Linear Analysis, 31 (2014), 23-53.
doi: 10.1016/j.anihpc.2013.02.001. |
[6] |
X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians II: existence, uniqueness, and qualitative properties of solutions, Transactions of the American Mathematical Society, 367 (2015), 911-941.
doi: 10.1090/S0002-9947-2014-05906-0. |
[7] |
X. Cabré and E. Cinti, Sharp energy estimates for nonlinear fractional diffusion equations, Calculus of Variations and Partial Differential Equations, 49 (2014), 233-269.
doi: 10.1007/s00526-012-0580-6. |
[8] |
X. Cabré and E. Cinti, Energy estimates and 1-D symmetry for nonlinear equations involving the half-Laplacian, Discrete and Continuous Dynamical System, 28 (2010), 1179-1206.
doi: 10.3934/dcds.2010.28.1179. |
[9] |
E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bulletin des Sciences Mathématiques, 136 (2012), 521-573.
doi: 10.1016/j.bulsci.2011.12.004. |
[10] |
E. B. Fabes, C. E. Kenig and R. P. Serapioni, The local regularity of solutions of degenerate elliptic equations, Communications in Partial Differential Equations, 7 (1982), 77-116.
doi: 10.1080/03605308208820218. |
[11] |
G. Palatucci, O. Savin and E. Valdinoci, Local and global minimizers for a variational energy involving a fractional norm, Annali di Matematica Pura ed Applicata, 192 (2013), 673-718.
doi: 10.1007/s10231-011-0243-9. |
[12] |
L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Communications on Pure and Applied Mathematics, 60 (2007), 67-112.
doi: 10.1002/cpa.20153. |
[13] |
Y. Sire and E. Valdinoci, Fractional Laplacian phase transitions and boundary reactions: a geometric inequality and a symmetry result, Journal of Functional Analysis, 256 (2009), 1842-1864.
doi: 10.1016/j.jfa.2009.01.020. |
[1] |
Ahmad Makki, Alain Miranville. Existence of solutions for anisotropic Cahn-Hilliard and Allen-Cahn systems in higher space dimensions. Discrete and Continuous Dynamical Systems - S, 2016, 9 (3) : 759-775. doi: 10.3934/dcdss.2016027 |
[2] |
Eleonora Cinti. Saddle-shaped solutions for the fractional Allen-Cahn equation. Discrete and Continuous Dynamical Systems - S, 2018, 11 (3) : 441-463. doi: 10.3934/dcdss.2018024 |
[3] |
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 and Applied Analysis, 2019, 18 (5) : 2457-2472. doi: 10.3934/cpaa.2019111 |
[4] |
Georgia Karali, Yuko Nagase. On the existence of solution for a Cahn-Hilliard/Allen-Cahn equation. Discrete and Continuous Dynamical Systems - S, 2014, 7 (1) : 127-137. doi: 10.3934/dcdss.2014.7.127 |
[5] |
Xinlong Feng, Huailing Song, Tao Tang, Jiang Yang. Nonlinear stability of the implicit-explicit methods for the Allen-Cahn equation. Inverse Problems and Imaging, 2013, 7 (3) : 679-695. doi: 10.3934/ipi.2013.7.679 |
[6] |
Tatsuki Mori, Kousuke Kuto, Tohru Tsujikawa, Shoji Yotsutani. Representation formulas of solutions and bifurcation sheets to a nonlocal Allen-Cahn equation. Discrete and Continuous Dynamical Systems, 2020, 40 (8) : 4907-4925. doi: 10.3934/dcds.2020205 |
[7] |
Christos Sourdis. On the growth of the energy of entire solutions to the vector Allen-Cahn equation. Communications on Pure and Applied Analysis, 2015, 14 (2) : 577-584. doi: 10.3934/cpaa.2015.14.577 |
[8] |
Paul H. Rabinowitz, Ed Stredulinsky. On a class of infinite transition solutions for an Allen-Cahn model equation. Discrete and Continuous Dynamical Systems, 2008, 21 (1) : 319-332. doi: 10.3934/dcds.2008.21.319 |
[9] |
Maicon Sônego, Arnaldo Simal do Nascimento. Stable transition layer induced by degeneracy of the spatial inhomogeneities in the Allen-Cahn problem. Discrete and Continuous Dynamical Systems - B, 2022, 27 (6) : 3297-3311. doi: 10.3934/dcdsb.2021185 |
[10] |
Juan Wen, Yaling He, Yinnian He, Kun Wang. Stabilized finite element methods based on multiscale enrichment for Allen-Cahn and Cahn-Hilliard equations. Communications on Pure and Applied Analysis, 2022, 21 (6) : 1873-1894. doi: 10.3934/cpaa.2021074 |
[11] |
Changchun Liu, Hui Tang. Existence of periodic solution for a Cahn-Hilliard/Allen-Cahn equation in two space dimensions. Evolution Equations and Control Theory, 2017, 6 (2) : 219-237. doi: 10.3934/eect.2017012 |
[12] |
Gianni Gilardi. On an Allen-Cahn type integrodifferential equation. Discrete and Continuous Dynamical Systems - S, 2013, 6 (3) : 703-709. doi: 10.3934/dcdss.2013.6.703 |
[13] |
Giorgio Fusco. Layered solutions to the vector Allen-Cahn equation in $\mathbb{R}^2$. Minimizers and heteroclinic connections. Communications on Pure and Applied Analysis, 2017, 16 (5) : 1807-1841. doi: 10.3934/cpaa.2017088 |
[14] |
Giorgio Fusco, Francesco Leonetti, Cristina Pignotti. On the asymptotic behavior of symmetric solutions of the Allen-Cahn equation in unbounded domains in $\mathbb{R}^2$. Discrete and Continuous Dynamical Systems, 2017, 37 (2) : 725-742. doi: 10.3934/dcds.2017030 |
[15] |
Michał Kowalczyk, Yong Liu, Frank Pacard. Towards classification of multiple-end solutions to the Allen-Cahn equation in $\mathbb{R}^2$. Networks and Heterogeneous Media, 2012, 7 (4) : 837-855. doi: 10.3934/nhm.2012.7.837 |
[16] |
Lishan Lin. A priori bounds and existence result of positive solutions for fractional Laplacian systems. Discrete and Continuous Dynamical Systems, 2019, 39 (3) : 1517-1531. doi: 10.3934/dcds.2019065 |
[17] |
Quan Wang, Dongming Yan. On the stability and transition of the Cahn-Hilliard/Allen-Cahn system. Discrete and Continuous Dynamical Systems - B, 2020, 25 (7) : 2607-2620. doi: 10.3934/dcdsb.2020024 |
[18] |
Christopher P. Grant. Grain sizes in the discrete Allen-Cahn and Cahn-Hilliard equations. Discrete and Continuous Dynamical Systems, 2001, 7 (1) : 127-146. doi: 10.3934/dcds.2001.7.127 |
[19] |
Alain Miranville, Ramon Quintanilla, Wafa Saoud. Asymptotic behavior of a Cahn-Hilliard/Allen-Cahn system with temperature. Communications on Pure and Applied Analysis, 2020, 19 (4) : 2257-2288. doi: 10.3934/cpaa.2020099 |
[20] |
Shixing Li, Dongming Yan. On the steady state bifurcation of the Cahn-Hilliard/Allen-Cahn system. Discrete and Continuous Dynamical Systems - B, 2019, 24 (7) : 3077-3088. doi: 10.3934/dcdsb.2018301 |
2021 Impact Factor: 1.273
Tools
Metrics
Other articles
by authors
[Back to Top]