Related papers: Multiple-interface solutions of one dimensional ge…

We consider the generalized parabolic Cahn-Hilliard equation $$ u_t=-\Delta\left[\Delta u -W'(u)\right]+W''(u)\left[\Delta u -W'(u)\right] \qquad \forall\, (t, x)\in \widetilde{{\mathbb R}}\times{\mathbb R}^n, $$ where $n=2$ or $n\geq 4$,…

We consider the parabolic one-dimensional Allen-Cahn equation $$u_t= u_{xx}+ u(1-u^2)\quad (x,t)\in \mathbb{R}\times (-\infty, 0].$$ The steady state $w(x) =\tanh (x/\sqrt{2})$, connects, as a "transition layer" the stable phases $-1$ and…

We consider the parabolic Allen-Cahn equation in $\mathbb{R}^n$, $n\ge 2$, $$u_t= \Delta u + (1-u^2)u \quad \hbox{ in } \mathbb{R}^n \times (-\infty, 0].$$ We construct an ancient radially symmetric solution $u(x,t)$ with any given number…

In this paper we construct entire solutions to the Cahn-Hilliard equation $-\Delta(-\Delta u+W^{'}(u))+W^{"}(u)(-\Delta u+W^{'}(u))=0$ in the Euclidean plane, where $W(u)$ is the standard double-well potential $\frac{1}{4} (1-u^2)^2$. Such…

We consider the wave equation $\varepsilon^2(-\partial_t^2 + \Delta)u + f(u) = 0$ for $0<\varepsilon\ll 1$, where $f$ is the derivative of a balanced, double-well potential, the model case being $f(u) = u-u^3$. For equations of this form,…

We obtain new integral representations, expressed as contour integrals in the complex Fourier plane, for the solution of fully nonhomogeneous interface problems for the linearized Cahn-Hilliard equation with arbitrary initial data on the…

Consider the Allen-Cahn equation $u_t=\varepsilon^2\Delta u-F'(u)$, where $F$ is a double well potential with wells of equal depth, located at $\pm1$. There are a lot of papers devoted to the study of the limiting behavior of the solutions…

We consider a diffuse interface model describing a ternary system constituted by a conductive diblock copolymer and a homopolymer acting as solvent. The resulting dynamics is modeled by two Cahn--Hilliard--Oono equations for the copolymer…

We consider a class of semilinear elliptic system of the form $-\Delta u(x,y)+\nabla W(u(x,y))=0,\quad (x,y)\in\R^{2}$ where $W:\R^{2}\to\R$ is a double well non negative symmetric potential. We show, via variational methods, that if the…

The generalized Allen-Cahn equation, \[ u_t=\varepsilon^2(D(u)u_x)_x-\frac{\varepsilon^2}2D'(u)u_x^2-F'(u), \] with nonlinear diffusion, $D = D(u)$, and potential, $F = F(u)$, of the form \[ D(u) = |1-u^2|^{m}, \quad \text{or} \quad D(u) =…

We investigate the Allen-Cahn system \begin{equation*} \Delta u-W_u(u)=0,\quad u:\mathbb{R}^2\rightarrow\mathbb{R}^2, \end{equation*} where $W\in C^2(\mathbb{R}^2,[0,+\infty))$ is a potential with three global minima. We establish the…

For the two dimensional Allen-Cahn system with a triple-well potential, previous results established the existence of a minimizing solution $u:\mathbb{R}^2\rightarrow\mathbb{R}^2$ with a triple junction structure at infinity. We show that…

We study the existence of solutions $u:\R^{3}\to\R^{2}$ for the semilinear elliptic systems \begin{equation}\label{eq:abs} -\Delta u(x,y,z)+\nabla W(u(x,y,z))=0, \end{equation} where $W:\R^{2}\to\R$ is a double well symmetric potential. We…

This paper considers a one-dimensional generalized Allen-Cahn equation of the form \[ u_t = \varepsilon^2 (D(u)u_x)_x - f(u), \] where $\varepsilon>0$ is constant, $D=D(u)$ is a positive, uniformly bounded below diffusivity coefficient that…

A general class of probability density functions \[u(x,t)=Ct^{-\alpha d}\left (1-\left (\frac{\|x\|}{ct^{\alpha}}\right )^{\beta}\right )_+^{\gamma},\quad x\in \mathbb{R}^d,t>0,\] is considered, containing as particular case the Barenblatt…

A method for finding the general solution to the partial differential equations: \ $F(u_x,u_y)=0$; \ $F(f(x)\:u_x,u_y)=0$ \ (or \ $F(u_x,h(y)\:u_y)=0$) \ is presented, founded on a Legendre like transformation and a theorem for Pfaffian…

This paper studies minimizing solutions to a two dimensional Allen-Cahn system on the upper half plane, subject to Dirichlet boundary conditions, \begin{equation*} \Delta u-\nabla_u W(u)=0, \quad u: \mathbb{R}_+^2\to \mathbb{R}^2,\ u=u_0…

We present previous results on the general solution of the multidimensional Hamilton-Jacobi equation $\frac{\partial u}{\partial t} - \frac{\partial u}{\partial x_a} \frac{\partial u}{\partial x_a}= 0$ and methods that were used to find…

We consider Cahn-Hilliard equations with external forcing terms. Energy decreasing and mass conservation might not hold. We show that level surfaces of the solutions of such generalized Cahn-Hilliard equations tend to the solutions of a…

We present a systematic study of entire symmetric solutions $u:R^n\rightarrow R^m$ of the vector Allen-Cahn equation $\Delta u-W_u(u)=0, x \in R^n$, where $W:R^m\rightarrow R$ is smooth, symmetric, nonnegative with a finite number of zeros…