Related papers: Solutions with single radial interface of the gene…
By using a suitable transform related to Sobolev inequality, we investigate the sharp constants and optimizers in radial space for the following weighted Caffarelli-Kohn-Nirenberg-type inequalities: \begin{equation*}…
We consider a class of particular solutions to the (2+1)-dimensional nonlinear partial differential equation (PDE) $u_t +\partial_{x_2}^n u_{x_1} - u_{x_1} u =0$ (here $n$ is any integer) reducing it to the ordinary differential equation…
We analyze a diffuse interface model that couples a viscous Cahn-Hilliard equation for the phase variable with a diffusion-reaction equation for the nutrient concentration. The system under consideration also takes into account some…
Fourth-order semilinear parabolic equations of the Cahn--Hilliard-type (01) u_t + \D^2 u = \g u \pm \D (|u|^{p-1}u) in \Omega \times \re_+, are considered in a smooth bounded domain $\O \subset \ren$ with Navier-type boundary conditions on…
We study the Emden-Fowler equation $-\Delta u=|u|^{p-1}u$ on the hyperbolic space ${\mathbb H}^n$. We are interested in radial solutions, namely solutions depending only on the geodesic distance from a given point. The critical exponent for…
The motion of two contiguous incompressible and viscous fluids is described within the diffuse interface theory by the so-called Model H. The system consists of the Navier-Stokes equations, which are coupled with the Cahn-Hilliard equation…
A deep relationship [arXiv:2503.17816v1] between real linear second order ordinary differential equations $u''\left(x\right)+h\left(x\right)u\left(x\right)=0$, with differentiable $h(x)$, and two dimensional hyperbolic geometry is…
Topology changes in multi-phase fluid flows are difficult to model within a traditional sharp interface theory. Diffuse interface models turn out to be an attractive alternative to model two-phase flows. Based on a…
We propose a new weak solution concept for (two-phase) mean curvature flow which enjoys both (unconditional) existence and (weak-strong) uniqueness properties. These solutions are evolving varifolds, just as in Brakke's formulation, but are…
We consider a two-dimensional, two-layer, incompressible, steady flow, with vorticity which is constant in each layer, in an infinite channel with rigid walls. The velocity is continuous across the interface, there is no surface tension or…
Motivated by its relation to models of flame propagation, we study globally Lipschitz solutions of $\Delta u=f(u)$ in $\mathbb{R}^n$, where $f$ is smooth, non-negative, with support in the interval $[0,1]$. In such setting, any "blow-down"…
The Cahn-Hilliard equation is related with a number of interesting physical phenomena like the spinodal decomposition, phase separation and phase ordering dynamics. On the other hand this equation is very stiff an the difficulty to solve it…
We consider a singular parabolic equation of form \[ u_t = u_{xx} + \frac{\alpha}{2}(\mathrm{sgn}\,u_x)_x \] with periodic boundary conditions. Solutions to this kind of equations exhibit competition between smoothing due to one-dimensional…
In this work, we study the co-dimensional one interface limit and geometric motions of parabolic Ginzburg--Landau systems with potentials of high-dimensional wells. The main result generalizes the one by Lin et al. (Comm. Pure Appl. Math.,…
This paper is concerned with radially symmetric solutions of the parabolic-elliptic version of the Keller-Segel system with flux limitation, as given by \begin{equation} \left\{ \begin{array}{l} \displaystyle u_t=\nabla \cdot…
We prove that any limit-interface corresponding to a locally uniformly bounded, locally energy-bounded sequence of stable critical points of the van der Waals--Cahn--Hilliard energy functionals with perturbation parameter tending to 0 is…
In this paper we study the linear Weingarten equation defined by the fully non-linear PDE $$a\, \mbox{div}\frac{Du}{\sqrt{1+|Du|^2}}+b\, \frac{\mbox{det}D^2u}{(1+|Du|^2)^2}=\phi\left(\frac{1}{\sqrt{1+|Du|^2}}\right)$$ in a domain…
The degenerate Cahn-Hilliard equation is a standard model to describe living tissues. It takes into account cell populations undergoing short-range attraction and long-range repulsion effects. In this framework, we consider the usual…
Global self-similar solutions to the parabolic Hardy-H\'enon equation $$ u_t=\Delta u^m+|x|^{\sigma}u^p, \quad (x,t)\in\mathbb{R}^N\times(0,\infty), $$ are classified in the range of exponents $m\geq1$, $p>m$ and $\sigma>\max\{-2,-N\}$. The…
This paper is concerned with radially symmetric solutions of systems of the form \[ u_t = -\nabla V(u) + \Delta_x u \] where space variable $x$ and and state-parameter $u$ are multidimensional, and the potential $V$ is coercive at infinity.…