Layered solutions for a fractional inhomogeneous Allen–Cahn equation

We consider the problem

$$\varepsilon^{2s} (-\partial_{xx})^s \tilde{u}(\tilde{x}) -V(\tilde{x})\tilde{u}(\tilde{x})(1-\tilde{u}^2(\tilde{x}))=0 \quad{\rm in} \mathbb{R},$$

where \({(-\partial_{xx})^s}\) denotes the usual fractional Laplace operator, \({\varepsilon > 0}\) is a small parameter and the smooth bounded function V satisfies \({{\rm inf}_{\tilde{x} \in \mathbb{R}}V(\tilde{x}) > 0}\). For \({s\in(\frac{1}{2},1)}\), we prove the existence of separate multi-layered solutions for any small \({\varepsilon}\), where the layers are located near any non-degenerate local maximal points and non-degenerate local minimal points of function V. We also prove the existence of clustering-layered solutions, and these clustering layers appear within a very small neighborhood of a local maximum point of V.

     

The Allen–Cahn equation on closed manifolds

We study global variational properties of the space of solutions to \(-\varepsilon ^2\Delta u + W'(u)=0\) on any closed Riemannian manifold M. Our techniques are inspired by recent advances in the variational theory of minimal hypersurfaces and extend a well-known analogy with the theory of phase transitions. First, we show that solutions at the lowest positive energy level are either stable or obtained by min–max and have index 1. We show that if \(\varepsilon \) is not small enough, in terms of the Cheeger constant of M, then there are no interesting solutions. However, we show that the number of min–max solutions to the equation above goes to infinity as \(\varepsilon \rightarrow 0\) and their energies have sublinear growth. This result is sharp in the sense that for generic metrics the number of solutions is finite, for fixed \(\varepsilon \), as shown recently by G. Smith. We also show that the energy of the min–max solutions accumulate, as \(\varepsilon \rightarrow 0\), around limit-interfaces which are smooth embedded minimal hypersurfaces whose area with multiplicity grows sublinearly. For generic metrics with \(\mathrm{Ric}_M>0\), the limit-interface of the solutions at the lowest positive energy level is an embedded minimal hypersurface of least area in the sense of Mazet–Rosenberg. Finally, we prove that the min–max energy values are bounded from below by the widths of the area functional as defined by Marques–Neves.     

Boundary interface for the Allen–Cahn equation

We consider the Allen–Cahn equation

Large deviation principle for a stochastic Allen–Cahn equation

The Allen–Cahn equation is a prototype model for phase separation processes, a fundamental example of a nonlinear spatial dynamic and an important approximation of a geometric evolution equation by a reaction–diffusion equation. Stochastic perturbations, especially in the case of additive noise, to the Allen–Cahn equation have attracted considerable attention. We consider here an alternative random perturbation determined by a Brownian flow of spatial diffeomorphism that was introduced by Röger and Weber (Stoch Partial Differ Equ Anal Comput 1(1):175–203, 2013). We first provide a large deviation principle for stochastic flows in spaces of functions that are Hölder continuous in time, which extends results by Budhiraja et al. (Ann Probab 36(4):1390–1420, 2008). From this result and a continuity argument we deduce a large deviation principle for the Allen–Cahn equation perturbed by a Brownian flow in the limit of small noise. Finally, we present two asymptotic reductions of the large deviation functional.     

Global solution branches for a nonlocal Allen–Cahn equation

We consider the Neumann problem of a 1D stationary Allen–Cahn equation with nonlocal term. Our previous paper [4] obtained a local branch of asymmetric solutions which bifurcates from a point on the branch of odd-symmetric solutions. This paper derives the global behavior of the branch of asymmetric solutions, and moreover, determines the set of all solutions to the nonlocal Allen–Cahn equation. Our proof is based on a level set analysis for an integral map associated with the nonlocal term.     

A Harnack inequality for the parabolic Allen–Cahn equation

We prove a differential Harnack inequality for the solution of the parabolic Allen–Cahn equation \( \frac{\partial f}{\partial t}=\triangle f-(f^3-f)\) on a closed n-dimensional manifold. As a corollary, we find a classical Harnack inequality. We also formally compare the standing wave solution to a gradient estimate of Modica from the 1980s for the elliptic equation.     

An efficient reproducing kernel method for solving the Allen–Cahn equation

In this paper, an efficient reproducing kernel method combined with the finite difference method and the Quasi-Newton method is proposed to solve the Allen–Cahn equation. Based on the Legendre polynomials, we construct a new reproducing kernel function with polynomial form. We prove that the semi-scheme can preserve the energy dissipation property unconditionally. Numerical experiments are given to show the efficiency and validity of the proposed scheme.     

Passing from bulk to bulk-surface evolution in the Allen–Cahn equation

In this paper we formulate a boundary layer approximation for an Allen–Cahn-type equation involving a small parameter ${\varepsilon}$ . Here, ${\varepsilon}$ is related to the thickness of the boundary layer and we are interested in the limit ${\varepsilon \to 0}$ in order to derive nontrivial boundary conditions. The evolution of the system is written as an energy balance formulation of the L²-gradient flow with the corresponding Allen–Cahn energy functional. By transforming the boundary layer to a fixed domain we show the convergence of the solutions to a solution of a limit system. This is done by using concepts related to Γ- and Mosco convergence. By considering different scalings in the boundary layer we obtain different boundary conditions.     

Global attractor for the Cahn–Hilliard/Allen–Cahn system

We study the coupled Cahn–Hilliard/Allen–Cahn problem with constraints, which describes the isothermal diffusion-driven phase transition phenomena in binary systems. Our aim is to show the existence–uniqueness result and to construct the global attractor for the related dynamical system.     

Tropical discretization: ultradiscrete Fisher–KPP equation and ultradiscrete Allen–Cahn equation

A systematic approach to the construction of ultradiscrete analogues for differential systems is presented. This method is tailored to first-order differential equations and reaction–diffusion systems. The discretizing method is applied to Fisher–KPP equation and Allen–Cahn equation. Stationary solutions, travelling wave solutions and entire solutions of the resulting ultradiscrete systems are constructed.     

Minimisers of the Allen–Cahn equation on hyperbolic graphs

We investigate minimal solutions of the Allen–Cahn equation on a Gromov-hyperbolic graph. Under some natural conditions on the graph, we show the existence of non-constant uniformly-bounded minimal solutions with prescribed asymptotic behaviours. For a phase field model on a hyperbolic graph, such solutions describe energy-minimising steady-state phase transitions that converge towards prescribed phases given by the asymptotic directions on the graph.     

Perelmanʼs W-entropy for the Fokker–Planck equation over complete Riemannian manifolds

We establish a monotonicity theorem and a rigidity theorem for the Perelman W-entropy of the Fokker–Planck equation on complete Riemannian manifolds with non-negative m-dimensional Bakry–Emery Ricci curvature. Moreover, we give a probabilistic and kinetic interpretation of the W-entropy for the Fokker–Planck equation on complete Riemannian manifolds.     

An efficient explicit full-discrete scheme for strong approximation of stochastic Allen–Cahn equation

In Becker and Jentzen (2019) and Becker et al. (2017), an explicit temporal semi-discretization scheme and a space–time full-discretization scheme were, respectively, introduced and analyzed for the additive noise-driven stochastic Allen–Cahn type equations, with strong convergence rates recovered. The present work aims to propose a different explicit full-discrete scheme to numerically solve the stochastic Allen–Cahn equation with cubic nonlinearity, perturbed by additive space–time white noise. The approximation is easily implementable, performing the spatial discretization by a spectral Galerkin method and the temporal discretization by a kind of nonlinearity-tamed accelerated exponential integrator scheme. Error bounds in a strong sense are analyzed for both the spatial semi-discretization and the spatio-temporal full discretization, with convergence rates in both space and time explicitly identified. It turns out that the obtained convergence rate of the new scheme is, in the temporal direction, twice as high as existing ones in the literature. Numerical results are finally reported to confirm the previous theoretical findings.     

Arbitrarily high-order accurate and energy-stable schemes for solving the conservative Allen–Cahn equation

In this paper, three high-order accurate and unconditionally energy-stable methods are proposed for solving the conservative Allen–Cahn equation with a space–time dependent Lagrange multiplier. One is developed based on an energy linearization Runge–Kutta (EL–RK) method which combines an energy linearization technique with a specific class of RK schemes, the other two are based on the Hamiltonian boundary value method (HBVM) including a Gauss collocation method, which is the particular instance of HBVM, and a general class of cases. The system is first discretized in time by these methods in which the property of unconditional energy stability is proved. Then the Fourier pseudo-spectral method is employed in space along with the proofs of mass conservation. To show the stability and validity of the obtained schemes, a number of 2D and 3D numerical simulations are presented for accurately calculating geometric features of the system. In addition, our numerical results are compared with other known structure-preserving methods in terms of numerical accuracy and conservation properties.     

Symmetry of some entire solutions to the Allen–Cahn equation in two dimensions

In this paper, we prove even symmetry and monotonicity of certain solutions of Allen–Cahn equation in a half plane. We also show that entire solutions with finite Morse index and four ends must be evenly symmetric with respect to two orthogonal axes. A classification scheme of general entire solutions with finite Morse index is also presented using energy quantization.     

Discrete maximum-norm stability of a linearized second-order finite difference scheme for Allen–Cahn equation

In this paper, we use finite difference methods for solving the Allen–Cahn equation that contains small perturbation parameters and strong nonlinearity. We consider a linearized second-order three-level scheme in time and a second-order finite difference approach in space, and establish discrete boundedness stability in maximum norm: if the initial data are bounded by 1, then the numerical solutions in later times can also be bounded uniformly by 1. It is shown that the main result can be obtained under certain restrictions on the time step.