The model discussed in this paper describes the evolution of the size-distribution of a population of cells in time. It is assumed that there is a degree of stochasticity in the growth process of each individual cell in the population. This manifests itself as a dispersion term in the differential equation for the evolution of the size-distribution of the overall population. We study the stability of the Steady Size-Distributions (SSDs) of the model (the spatial components of separable solutions) and show that given a set of parameters where an SSD exists, it is unique and globally asymptotically stable.

Using variational methods we establish existence of nontrivial solutions for the following class of anisotropic critical problem $$ {(P_{\lambda})} \qquad \left \{ \begin{array}{l} - \displaystyle \sum_{i=1}^{N} \frac{\partial}{\partial x_{i}} \Big ( \Big | \frac{\partial u}{\partial x_{i}} \Big |^{p_{i}-2}\frac{\partial u}{\partial x_{i}} \Big )= \lambda f(u) + g(u) , \quad \mbox{in} \quad \Omega \\ u \geq 0, \quad \mbox{in} \quad \Omega \\ u=0, \quad \mbox{on} \quad \partial \Omega , \end{array} \right. $$ where $\Omega$ is a bounded domain with smooth boundary in $\mathbb{R}^N$, $\lambda$ is a positive parameter, $g(u)$ behaves like $|u|^{p^*-2}u$, $p^{*}$ is the critical exponent for this class of problem and $f$ is a continuous function verifying some adequate assumptions.

An unconditional existence result of a solution for a steady fluid-structure problem is stated. More precisely, we consider an incompressible fluid in a thin film, ruled by the Reynolds equation coupled with a surface deformation modelled by a nonlinear non local Hertz law. The viscosity is supposed to depend nonlinearly on the fluid pressure. Due to the apparition of a mushy region, the two-phase flow satisfies a free boundary problem defined by a pressure-saturation model. Such a problem has been studied with simpler free boundaries models (variational inequality), or with boundary conditions imposing small data assumptions. We show that up to a realistic hypothesis on the asymptotic pressure-viscosity behaviour it is possible to obtain an unconditional solution of the problem.

We study inertial manifolds for a semilinear evolution equation $du/dt+Au=F(t,u)$ in a complex Banach space. It is known that various conditions ensure existence of inertial manifolds for the equation, however, Miklavčič gave a sharp but simple condition so as to show the existence of inertial manifolds. In this paper, we show smoothness of inertial manifolds using the sharp condition with additional assumptions on $F$, and also apply to a scalar reaction diffusion equation $u_t-u_{xx}=f(t,x,u,u_x)$ with the Dirichlet boundary conditions.

This paper is devoted to a study of the unique continuation property for stochastic parabolic equations. Due to the adapted nature of solutions in the stochastic situation, classical approaches to treat the unique continuation problem for deterministic equations do not work. Our method is based on a suitable partial Holmgren coordinate transform and a stochastic version of Carleman estimate.

We consider a S-N-S Josephson junction modeled through the Ginzburg-Landau theory. When the normal material is sufficiently thin and the applied magnetic field is below the critical field of vortex nucleation, we prove to leading order that jump boundary conditions of the type predicted by de Gennes are satisfied across the junction.

We prove a priori bounds for derivatives of solutions $w$ of a class of quasilinear equations of the form \begin{equation*} \mathrm {div} \mathcal{A} ( x,w ) \nabla w+\vec{\gamma} ( x,w ) \cdot \nabla w+f ( x,w ) =0, \end{equation*} where $x \! = \! ( x_{1},\dots ,x_{n} ) $, and where $f$, $\vec{\gamma} = ( \gamma^{i} ) _{1\leq i\leq n}$ and $\mathcal{A}= ( a_{ij} ) _{1\leq i,j\leq n}$ are $\mathcal{C}^{\infty }$. The rank of the square symmetric matrix $\mathcal{A}$ is allowed to degenerate, as all but one eigenvalue of $\mathcal{A}$ are permitted to vanish to infinite order. We estimate derivatives of $w$ of arbitrarily high order in terms of just $w$ and its first derivatives. These estimates will be applied in a subsequent work to establish existence, uniqueness and regularity of weak solutions of the Dirchlet problem.