We prove existence and uniqueness of a solution for the stochastic Allen-Cahn equation with logarithmic potential multiplicative Wiener noise, under homogeneous Neumann boundary condition. The is obtained in variational sense by means an approximated involving Yosida regularization nonlinearity. noise assumed to vanish at extremal points physical relevant domain satisfy suitable Lipschitz-conti...

In this paper, a numerical solution of nonlinear partial differential equation, Benjamin-Bona-Mahony (BBM) and Cahn-Hilliard equation is presented by using Adomain Decomposition Method (ADM) and Variational Iteration Method (VIM). The results reveal that the two methods are very effective, simple and very close to the exact solution.

New types of stationary solutions of a one-dimensional driven sixthorder Cahn-Hilliard type equation that arises as a model for epitaxially growing nano-structures such as quantum dots, are derived by an extension of the method of matched asymptotic expansions that retains exponentially small terms. This method yields analytical expressions for far-field behavior as well as the widths of the hu...

This paper studies finite element approximations of the stochastic Allen–Cahn equation with gradient-type multiplicative noise that is white in time and correlated in space. The sharp interface limit—as the diffuse interface thickness vanishes—of the stochastic Allen–Cahn equation is formally a stochastic mean curvature flow which is described by a stochastically perturbed geometric law of the ...

An error bound is proved for a fully practical piecewise linear nite element approximation, using a backward Euler time discretization, of the Cahn-Hilliard equation with a logarithmic free energy.

For all n ≥ 1, we are interested in bounded solutions of the AllenCahn equation ∆u + u − u3 = 0 which are defined in all Rn+1 and whose zero set is asymptotic to a given minimal cone. In particular, in dimension n+1 ≥ 8, we prove the existence of stable solutions of the Allen-Cahn equation whose zero sets are not hyperplanes.

Understanding the entire solutions of nonlinear elliptic equations in RN such as (0.1) Δu+ f(u) = 0 in R , is a basic problem in PDE research. This is the context of various classical results in literature like the Gidas-Ni-Nirenberg theorems on radial symmetry, Liouville type theorems, or the achievements around De Giorgi’s conjecture. In those results, the geometry of level sets of the soluti...

In this paper, we establish a Stroock-Varadhan support theorem for the global mild solution to a d (d ≤ 3)-dimensional stochastic Cahn-Hilliard partial differential equation driven by a space-time white noise.