The solution of Cahn-Allen equation based on Bernoulli sub-equation method

A Generalization of the Allen-cahn Equation

Our aim in this paper is to study generalizations of the Allen-Cahn equation based on a modification of the Ginzburg-Landau free energy proposed in [25]. In particular, the free energy contains an additional term called Willmore regularization. We prove the existence, uniqueness and regularity of solutions, as well as the existence of the global attractor. Furthermore, we study the convergence ...

Boundary Interface for the Allen-cahn Equation

We consider the Allen-Cahn equation ε∆u + u− u = 0 in Ω, ∂u ∂ν = 0 on ∂Ω, where Ω is a smooth and bounded domain in R such that the mean curvature is positive at each boundary point. We show that there exists a sequence εj → 0 such that the Allen-Cahn equation has a solution uεj with an interface which approaches the boundary as j → +∞.

Haar Wavelet Method for Solving Cahn-Allen Equation

Higher Dimensional Catenoid, Liouville Equation and Allen-cahn Equation

We build a family of entire solutions to the Allen-Cahn equation in RN+1 for N ≥ 3, whose level set approaches the higher dimensional catenoid in a compact region and has two logarithmic ends governed by the solutions to the Liouville equation.

Tightness for a Stochastic Allen–cahn Equation

We study an Allen–Cahn equation perturbed by a multiplicative stochastic noise that is white in time and correlated in space. Formally this equation approximates a stochastically forced mean curvature flow. We derive a uniform bound for the diffuse surface area, prove the tightness of solutions in the sharp interface limit, and show the convergence to phase-indicator functions.