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 and Continuous Dynamical Systems, 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. DangP. 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. FuscoF. 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. DangP. 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. FuscoF. 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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