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Allen-Cahn Equation

The Allen-Cahn equation is a reaction-diffusion equation associated with a double-well potential. For W(u)=(1-u^2)^2/4, its gradient-flow normalization is

partial_tu=Deltau-W^'(u)=Deltau+u-u^3.

Stationary solutions satisfy Deltau=u^3-u. The corresponding energy is int(||del u||^2/2+W(u))dx. The one-dimensional stationary transition profile is u(x)=tanh(x/sqrt(2)). The de Giorgi conjecture asks when bounded entire stationary solutions with a strictly positive directional derivative in one fixed direction depend on only one linear coordinate.

See also

de Giorgi Conjecture, Partial Differential Equation

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References

del Pino, M.; Kowalczyk, M.; and Wei, J. "On De Giorgi's Conjecture and Beyond." Proc. Nat. Acad. Sci. USA 109, 6845-6850, 2012. https://doi.org/10.1073/pnas.1202687109.

Cite this as:

Weisstein, Eric W. "Allen-Cahn Equation." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Allen-CahnEquation.html

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