-
Previous Article
Transition fronts and stretching phenomena for a general class of reaction-dispersion equations
- DCDS Home
- This Issue
-
Next Article
A non-local bistable reaction-diffusion equation with a gap
On the asymptotic behavior of symmetric solutions of the Allen-Cahn equation in unbounded domains in $\mathbb{R}^2$
University of L'Aquila, DISIM, via Vetoio, Coppito, 67010 L'Aquila, Italy |
We consider a Dirichlet problem for the Allen-Cahn equation in a smooth, bounded or unbounded, domain $Ω\subset\mathbb{R}^n.$ Under suitable assumptions, we prove an existence result and a uniform exponential estimate for symmetric solutions. In dimension $n=2$ an additional asymptotic result is obtained. These results are based on a pointwise estimate obtained for local minimizers of the Allen-Cahn energy.
References:
| [1] |
N. D. Alikakos and G. Fusco,
Asymptotic and rigidity results for symmetric solutions of the elliptic system $Δ u=W_u(u)$, Ann Sc. Norm. Sup. Pisa, 15 (2016), 809-836.
|
| [2] |
H. Dang, P. C. Fife and L. A. Peletier,
Saddle solutions of bistable diffusion equation, Z. Angew. Math. Phys., 43 (1992), 984-998.
doi: 10.1007/BF00916424. |
| [3] |
M. Efendiev and F. Hamel,
Asymptotic behavior of solutions of semilinear elliptic equations in unbounded domains: Two approaches, Adv. Math., 228 (2011), 1237-1261.
doi: 10.1016/j.aim.2011.06.013. |
| [4] |
G. Fusco,
Equivariant entire solutions to the elliptic system $Δ u=W_u(u)$ for general $G-$invariant potentials, Calc. Var. Part. Diff. Eqs., 49 (2014), 963-985.
doi: 10.1007/s00526-013-0607-7. |
| [5] |
G. Fusco,
On some elementary properties of vector minimizers of the Allen-Cahn energy, Comm. Pure Appl. Anal., 13 (2014), 1045-1060.
doi: 10.3934/cpaa.2014.13.1045. |
| [6] |
G. Fusco, F. Leonetti and C. Pignotti,
A uniform estimate for positive solutions of semilinear elliptic equations, Transactions AMS, 363 (2011), 4285-4307.
doi: 10.1090/S0002-9947-2011-05356-0. |
| [7] |
D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag, Berlin, 1981.
|
| [8] |
P. Smyrnelis,
personal comunication. |
show all references
References:
| [1] |
N. D. Alikakos and G. Fusco,
Asymptotic and rigidity results for symmetric solutions of the elliptic system $Δ u=W_u(u)$, Ann Sc. Norm. Sup. Pisa, 15 (2016), 809-836.
|
| [2] |
H. Dang, P. C. Fife and L. A. Peletier,
Saddle solutions of bistable diffusion equation, Z. Angew. Math. Phys., 43 (1992), 984-998.
doi: 10.1007/BF00916424. |
| [3] |
M. Efendiev and F. Hamel,
Asymptotic behavior of solutions of semilinear elliptic equations in unbounded domains: Two approaches, Adv. Math., 228 (2011), 1237-1261.
doi: 10.1016/j.aim.2011.06.013. |
| [4] |
G. Fusco,
Equivariant entire solutions to the elliptic system $Δ u=W_u(u)$ for general $G-$invariant potentials, Calc. Var. Part. Diff. Eqs., 49 (2014), 963-985.
doi: 10.1007/s00526-013-0607-7. |
| [5] |
G. Fusco,
On some elementary properties of vector minimizers of the Allen-Cahn energy, Comm. Pure Appl. Anal., 13 (2014), 1045-1060.
doi: 10.3934/cpaa.2014.13.1045. |
| [6] |
G. Fusco, F. Leonetti and C. Pignotti,
A uniform estimate for positive solutions of semilinear elliptic equations, Transactions AMS, 363 (2011), 4285-4307.
doi: 10.1090/S0002-9947-2011-05356-0. |
| [7] |
D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag, Berlin, 1981.
|
| [8] |
P. Smyrnelis,
personal comunication. |
| [1] |
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 |
| [2] |
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 |
| [3] |
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 |
| [4] |
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 |
| [5] |
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 |
| [6] |
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 |
| [7] |
Haydi Israel. Well-posedness and long time behavior of an Allen-Cahn type equation. Communications on Pure and Applied Analysis, 2013, 12 (6) : 2811-2827. doi: 10.3934/cpaa.2013.12.2811 |
| [8] |
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 |
| [9] |
Matthieu Alfaro, Hiroshi Matano. On the validity of formal asymptotic expansions in Allen-Cahn equation and FitzHugh-Nagumo system with generic initial data. Discrete and Continuous Dynamical Systems - B, 2012, 17 (6) : 1639-1649. doi: 10.3934/dcdsb.2012.17.1639 |
| [10] |
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 |
| [11] |
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 |
| [12] |
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 |
| [13] |
Mitsunori Nara. Large time behavior of the solutions with spreading fronts in the Allen-Cahn equations on $ \mathbb R^n $. Communications on Pure and Applied Analysis, , () : -. doi: 10.3934/cpaa.2022116 |
| [14] |
Hongmei Cheng, Rong Yuan. Multidimensional stability of disturbed pyramidal traveling fronts in the Allen-Cahn equation. Discrete and Continuous Dynamical Systems - B, 2015, 20 (4) : 1015-1029. doi: 10.3934/dcdsb.2015.20.1015 |
| [15] |
Murat Uzunca, Ayşe Sarıaydın-Filibelioǧlu. Adaptive discontinuous galerkin finite elements for advective Allen-Cahn equation. Numerical Algebra, Control and Optimization, 2021, 11 (2) : 269-281. doi: 10.3934/naco.2020025 |
| [16] |
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 |
| [17] |
Ciprian G. Gal, Maurizio Grasselli. The non-isothermal Allen-Cahn equation with dynamic boundary conditions. Discrete and Continuous Dynamical Systems, 2008, 22 (4) : 1009-1040. doi: 10.3934/dcds.2008.22.1009 |
| [18] |
Zhuoran Du, Baishun Lai. Transition layers for an inhomogeneous Allen-Cahn equation in Riemannian manifolds. Discrete and Continuous Dynamical Systems, 2013, 33 (4) : 1407-1429. doi: 10.3934/dcds.2013.33.1407 |
| [19] |
Charles-Edouard Bréhier, Ludovic Goudenège. Analysis of some splitting schemes for the stochastic Allen-Cahn equation. Discrete and Continuous Dynamical Systems - B, 2019, 24 (8) : 4169-4190. doi: 10.3934/dcdsb.2019077 |
| [20] |
Yan Hu. Layer solutions for an Allen-Cahn type system driven by the fractional Laplacian. Communications on Pure and Applied Analysis, 2016, 15 (3) : 947-964. doi: 10.3934/cpaa.2016.15.947 |
2021 Impact Factor: 1.588
Tools
Metrics
Other articles
by authors
[Back to Top]