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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:
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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.
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| [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. |
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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. |
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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. |
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D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag, Berlin, 1981.
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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. |
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