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### Open Access Journals

NHM

We analyze the evolution of multi-dimensional normal graphs over
the unit sphere under volume preserving mean curvature flow and
derive a non-linear partial differential equation in polar
coordinates. Furthermore, we construct finite difference numerical
schemes and present numerical results for the evolution of
non-convex closed plane curves under this flow, to observe that
they become convex very fast.

DCDS-S

In this manuscript, we consider a Cahn-Hilliard/Allen-Cahn equation is introduced in [17].
We give an existence of the solution, slightly improved from [18]. We also present a stochastic version of this equation in [3].

DCDS-B

We consider a generalized Stochastic Cahn-Hilliard equation with
multiplicative white noise posed on bounded convex domains in
$R^d$, $d=1,2,3$, with piece-wise smooth boundary, and
introduce an additive time dependent white noise term in the
chemical potential. Since the Green's function of the problem is
induced by a convolution semigroup, we present the equation in a
weak stochastic integral formulation and prove existence of
solution when $d\leq 2$ for general domains, and for $d=3$ for
domains with minimum eigenfunction growth, without making use of
any explicit expression of the spectrum and the eigenfunctions.
The analysis is based on stochastic integral calculus, Galerkin
approximations and the asymptotic spectral properties of the
Neumann Laplacian operator. Existence is also derived for some
non-convex cases when the boundary is smooth.

DCDS

Motivated by the physical theory of Critical Dynamics the
Cahn-Hilliard equation on a bounded space domain is considered and
forcing terms of general type are introduced. For such a rescaled
equation the limiting inter-face problem is studied and the
following are derived: (i) asymptotic results indicating that the
forcing terms may slow down the equilibrium locally or globally,
(ii) the sharp interface limit problem in the multidimensional
case demonstrating a local influence in phase transitions of
terms that stem from the chemical potential, while free energy
independent terms act on the rest of the domain, (iii) a limiting
non-homogeneous linear diffusion equation for the one-dimensional
problem in the case of deterministic forcing term that follows
the white noise scaling.

DCDS

Gierer-Meinhardt system is a mathematical model to describe biological pattern formation due to activator and inhibitor. Turing pattern is expected in the presence of local self-enhancement and long-range inhibition. The long-time behavior of the solution, however, has not yet been clarified mathematically. In this paper, we study the case when its ODE part takes periodic-in-time solutions, that is, $\tau=\frac{s+1}{p-1}$. Under some additional assumptions on parameters, we show that the solution exists global-in-time and absorbed into one of these ODE orbits. Thus spatial patterns eventually disappear if those parameters are in a region without local self-enhancement or long-range inhibition.

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