# American Institute of Mathematical Sciences

February  2017, 37(2): 725-742. doi: 10.3934/dcds.2017030

## 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

* Corresponding author: Giorgio Fusco

Received  June 2015 Revised  November 2015 Published  November 2016

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.

Citation: 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 & Continuous Dynamical Systems - A, 2017, 37 (2) : 725-742. doi: 10.3934/dcds.2017030
##### 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.

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##### 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.
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