Allen–Cahn equation

Remarks

Example: with $n = 1$, the function $u(x) = \tanh\!\big(x/(\sqrt{2}\,\varepsilon)\big)$ is a stationary solution. Indeed, with $s = x/(\sqrt{2}\,\varepsilon)$ one finds $u'' = -\varepsilon^{-2}\operatorname{sech}^2(s)\tanh(s) = \varepsilon^{-2}(u^3 - u)$. Non-example: the Cahn–Hilliard equation $u_t = \Delta(u^3 - u - \varepsilon^2 \Delta u)$ is of fourth order, so it is not of this form. The equation is the $L^2$ gradient flow of the energy $$E(u) = \int_\Omega \Big( \frac{1}{2}\abs{\nabla u}^2 + \frac{1}{\varepsilon^2} W(u) \Big)\, dx,$$ so $E$ does not increase along smooth solutions with Neumann boundary conditions. Some authors scale the equation as $u_t = \varepsilon^2 \Delta u - W'(u)$ instead.