Related papers: Long-time behavior of solutions to the generalized…
We study the long-time behavior of the solution of a damped BBM equation $u_t + u_x - u_{xxt} + uu_x + \mathscr{L}_{\gamma}(u) = 0$. The proposed dampings $\mathscr{L}_{\gamma}$ generalize standards ones, as parabolic…
We prove comparison, uniqueness and existence results for viscosity solutions to a wide class of fully nonlinear second order partial differential equations $F(x, u, du, d^{2}u)=0$ defined on a finite-dimensional Riemannian manifold $M$.…
We study the gradient flow of the Allen-Cahn equation with fixed boundary contact angle in Euclidean domains for initial data with bounded energy. Under general assumptions, we establish both interior and boundary convergence properties for…
We study the asymptotics of Allen-Cahn-type bistable reaction-diffusion equations which are additively perturbed by a stochastic forcing (time white noise). The conclusion is that the long time, large space behavior of the solutions is…
We study a diffuse interface model describing the motion of two viscous fluids driven by the surface tension in a Hele-Shaw cell. The full system consists of the Cahn-Hilliard equation coupled with the Darcy's law. We address the physically…
We study irreversible A-B reaction kinetics at a fixed interface separating two immiscible bulk phases, A and B. We consider general dynamical exponent $z$, where $x_t\sim t^{1/z}$ is the rms diffusion distance after time $t$. At short…
Reaction-diffusion equations (RDEs) model the spatiotemporal evolution of a density field $u(\vec{x},t)$ according to diffusion and net local changes. Usually, the diffusivity is positive for all values of $u,$ which causes the density to…
In this paper, we show that bounded weak solutions of the Cauchy problem for general degenerate parabolic equations of the form \begin{equation} \notag u_t \,+\; \mbox{div}\,f(x,t,u) \;=\; \mbox{div}\,(\;\!|\,u\,|^{\alpha} \, \nabla u…
We study the global behavior of finite energy solutions to the $d$-dimensional focusing nonlinear Schr\"odinger equation (NLS), $i \partial_t u+\Delta u+ |u|^{p-1}u=0, $ with initial data $u_0\in H^1,\; x \in R^n$. The nonlinearity power…
This paper introduces a generalized matrix-valued Allen--Cahn model, where the unknown matrix-valued field belongs to $\mathbb{R}^{m_1\times m_2}$ with dimension $m_1\geq m_2$. By taking different values of $m_1$ and $m_2$, this model…
This paper presents a full classification of the short-time behavior of the interfaces in the Cauchy problem for the nonlinear second order degenerate parabolic PDE \[ u_t-\Delta u^m +b u^\beta=0, \ x\in \mathbb{R}^N, 0<t<T \] with…
We extend the doubly degenerate Cahn-Hilliard (DDCH) models for isotropic surface diffusion, which yield more accurate approximations than classical degenerate Cahn-Hilliard (DCH) models, to the anisotropic case. We consider both weak and…
The Functionalized Cahn-Hilliard equation has been proposed as a model for the interfacial energy of phase-separated mixtures of amphiphilic molecules. We study the existence of a nonnegative weak solutions of a gradient flow of the…
We analyze a diffuse interface model for multi-phase flows of $N$ incompressible, viscous Newtonian fluids with different densities. In the case of a bounded and sufficiently smooth domain existence of weak solutions in two and three space…
For \alpha \in (1,2) we prove that the initial-value problem \partial_t u+D^\alpha\partial_x u+\partial_x(u^2/2)=0 on \mathbb{R}_x\times\mathbb{R}_t; u(0)=\phi, is globally well-posed in the space of real-valued L^2-functions. We use a…
We study in this work a continuum model derived from 1D attachment-detachment-limited (ADL) type step flow on vicinal surface, $ u_t=-u^2(u^3)_{hhhh}$, where $u$, considered as a function of step height $h$, is the step slope of the…
We study the long-time behavior of the unique viscosity solution $u$ of the viscous Hamilton-Jacobi Equation $u_t-\Delta u + |Du|^m = f\hbox{in }\Omega\times (0,+\infty)$ with inhomogeneous Dirichlet boundary conditions, where $\Omega$ is a…
New results are obtained for global regularity and long-time behavior of the solutions to the 2D Boussinesq equations for the flow of an incompressible fluid with positive viscosity and zero diffusivity in a smooth bounded domain. Our first…
We consider the varifold associated to the Allen--Cahn phase transition problem in $\mathbb R^{n+1}$(or $n+1$-dimensional Riemannian manifolds with bounded curvature) with integral $L^{q_0}$ bounds on the Allen--Cahn mean curvature (first…
We consider an Allen-Cahn equation with nonlinear diffusion, motivated by the study of the scaling limit of certain interacting particle systems. We investigate its singular limit and show the generation and propagation of an interface in…