We consider a tumor growth model involving a nonlinear system of partial differential equations which describes the growth of two types of cell population densities with contact inhibition. In one space dimension, it is known that global solutions exist and that they satisfy the so-called segregation property: if the two populations are initially segregated - in mathematical terms this translates into disjoint supports of their densities - this property remains true at all later times. We apply recent results on transport equations and regular Lagrangian flows to obtain similar results in the case of arbitrary space dimension.

It is observed in vitro and in vivo that when two populations of different types of cells come near to each other, the rate of proliferation of most cells decreases. This phenomenon is often called contact inhibition of growth between two cells. In this paper, we consider a simplified 1-dimensional PDE-model for normal and abnormal cells, motivated by a paper of Chaplain, Graziano and Preziosi. We show that if the two populations are initially segregated, then they remain segregated due to the contact inhibition mechanism. In this case the system of PDE's can be formulated as a free boundary problem.

Barenblatt e. a. introduced a fluid model for groundwater flow in fissurised porous media. The system consists of two diffusion equations for the groundwater levels in, respectively, the porous bulk and the system of cracks. The equations are coupled by a fluid exchange term. Numerical evidence suggests that the penetration depth of the fluid increases dramatically due to the presence of cracks and that the smallness of certain parameter values play a key role in this phenomenon. In the present paper we give precise estimates for the penetration depth in terms of the smallness of some of the parameters.

Given any wave speed c (independently of its sign!), we construct a traveling wave solution of the harmonic heat flow with values in the unit sphere and defined in an infinitely long cylinder. The wave connects two locally stable and axially symmetric steady states at + and - infinity, and has a singular point on the cylinder axis. In view of the bistable character of the potential, the result is surprising, and it is intimately related to the nonuniqueness of the harmonic map flow itself. The result is counter-intuitive if c has the "wrong sign", i.e. if the steady state with higher energy invades the cylinder. We show that for only one wave speed the traveling wave behaves locally, near its singular point, as a symmetric harmonic map.

l'attività seminariale è sostenuta dai ricercatori e collaboratori dell'IAC ed è rivolta ai ricercatori dell'area romana, essendo finalizzata alla promozione degli interessi di ricerca coltivati in istituto e alla disseminazione dei risultati conseguiti.

We consider existence and qualitative properties of traveling wave solutions of a new free boundary problem which describes fluid flow in diatomite rocks and in particular the phenomenon of hydraulic fracturing. For certain parameter values discontinuities of the traveling waves may appear near their free boundaries.

We consider the heat equation for director fields, with values in the unit sphere. A variational approach is used to construct axially symmetric traveling wave solutions defined in an infinitely long cylinder. The traveling waves have a point singularity of topological degree 0 or 1. The construction of solutions with degree 0 is based on minimization of a relaxed energy.

We prove the existence of a traveling wave solution u of the harmonic heat flow in an infinitely long cylinder of radius R, which connects two locally stable and axially symmetric steady states at + and - infinity infinity. Here u is a director field, with values in the unit sphere. The traveling wave has a singular point on the cylinder axis. As R goes to infinity we obtain a traveling wave defined in all space.

Motivated by regulartiry questions for harmonic map flow, blow up phenomena for solutions of a singular parabolic pde is studied.

A method is proposed to identify solutions of the thin-film equations with non-zero contact angle

We prove existence of solutions of a new free boundary problem described by a system of degenerate parabolic equations. The problem arises in petroleum engineering and concerns fluid flows in diatomite rocks. The unknown functions represent the pressure of the fluid and a damage parameter of the porous rock. These quantities are not necessarily continuous on the free boundary, which considerably complicates the mathematical analysis.

The detailed analysis of a 1D-model for fluid flows in porous media with piecewise constant permeability clearly shows that variable permeability may lead to the phenomenon of oil trapping.

The simplest equation for the evolution of a director field is given by its corresponding heat flow. More complicated versions arise in the theories of micromagnetism and liquid crystals. In 3D there exist finite energy solutions with point singularities (also called defects in case of liquid crystals). It the paper an example of a new nonuniqueness phenomenon is discussed: having initially an equilibrium situation with one point singularity, a solution is constructed for which the singularity is moved instantaneously to another point. This suggests that there exists a considerable degree of freedom to prescribe the evolution of point singularities.