We consider the mixed problem for the quasilinear partial functional differential equation with unbounded delay , where is defined by , , and the phase space satisfies suitable axioms. Using the method of bicharacteristics and the fixed-point method we prove a theorem on the local existence and uniqueness of Carathéodory solutions of the mixed problem.
Some methods for the numerical approximation of time-dependent and steady first-order Hamilton-Jacobi equations are reviewed. Most of the discussion focuses on conformal triangular-type meshes, but we show how to extend this to the most general meshes. We review some first-order monotone schemes and also high-order ones specially dedicated to steady problems.
We show that the equation div has, in general, no Lipschitz (respectively ) solution if is (respectively ).
We consider viscosity solutions for first order differential-functional equations. Uniqueness theorems for initial, mixed, and boundary value problems are presented. Our theorems include some results for generalized ("almost everywhere") solutions.
We consider a size structured cell population model where a mother cell gives birth to two daughter cells. We know that the asymptotic behavior of the density of cells is given by the solution to an eigenproblem. The eigenvector gives the asymptotic shape and the eigenvalue gives the exponential growth rate and so the Maltusian parameter. The Maltusian parameter depends on the division rule for the mother cell, i.e., symmetric (the two daughter cells have the same size) or asymmetric. We use a...
In questa conferenza descrivo alcuni recenti sviluppi relativi al problema dell'unicità per l'equazione differenziale ordinaria e per l'equazione di continuità per campi vettoriali debolmente differenziabili. Descrivo infine un'applicazione di questi risultati a un sistema di leggi di conservazione.