We consider the mixed problem for the hyperbolic partial differential-functional equation of the first order $$D_{x}z(x,y)=f(x,y,z_{(x,y)},D_{y}z(x,y)),$$ where $z_{(x,y)}[-\tau ,0]\times [0,h]\to ℝ$ is a function defined by $z_{(x,y)}(t,s)=z(x+t,y+s)$, $(t,s)\in [-\tau ,0]\times [0,h]$. Using the method of bicharacteristics and the method of successive approximations for a certain integral-functional system we prove, under suitable assumptions, a theorem of the local existence of generalized solutions of this problem.

We consider the mixed problem for the quasilinear partial functional differential equation with unbounded delay $D_{t}z(t,x)={\sum }_{i=1}^n{f}_i(t,x,z_{(t,x)})D_{x_i}z(t,x)+h(t,x,z_{(t,x)})$, where $z_{(t,x)}\in X{̶}_0$ is defined by $z_{(t,x)}(\tau ,s)=z(t+\tau ,x+s)$, $(\tau ,s)\in (-\infty ,0]\times [0,r]$, and the phase space $X{̶}_0$ 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.

Several realistic situations in vehicular traffic that give rise to queues can be modeled through conservation laws with boundary and unilateral constraints on the flux. This paper provides a rigorous analytical framework for these descriptions, comprising stability with respect to the initial data, to the boundary inflow and to the constraint. We present a framework to rigorously state optimal management problems and prove the existence of the corresponding optimal controls. Specific cases are...

Several realistic situations in vehicular traffic that give rise to queues can be modeled through conservation laws with boundary and unilateral constraints on the flux. This paper provides a rigorous analytical framework for these descriptions, comprising stability with respect to the initial data, to the boundary inflow and to the constraint. We present a framework to rigorously state optimal management problems and prove the existence of the corresponding optimal controls. Specific cases...

We consider a body immersed in a perfect gas and moving under the action of a constant force. Body and gas are in thermal equilibrium. We assume a stochastic interaction body/medium: when a particle of the medium hits the body, it is absorbed and immediately re-emitted with a Maxwellian distribution. This system gives rise to a microscopic model of friction. We study the approach of the body velocity V(t) to the limiting velocity $V_{\infty }$ and prove that, under suitable smallness assumptions, the approach...

The paper deals with the 2-D system of gas dynamics without pressure which was introduced in 1970 by Ua. Zeldovich to describe the formation of largescale structure of the Universe. Such system occurs to be an intermediate object between the systems of ordinary differential equations and hyperbolic systems of PDE. The main its feature is the arising of singularities: discontinuities for velocity and d-functions of various types for density. The rigorous notion of generalized solutions in terms of...

We obtain a precise decay estimate of the energy of the solutions to the initial boundary value problem for the wave equation with nonlinear internal and boundary feedbacks. We show that a judicious choice of the feedbacks leads to fast energy decay.

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 study the statistical properties of the solutions of the Kadomstev-Petviashvili equations (KP-I and KP-II) on the torus when the initial datum is a random variable. We give ourselves a random variable $u_0$ with values in the Sobolev space $H^s$ with $s$ big enough such that its Fourier coefficients are independent from each other. We assume that the laws of these Fourier coefficients are invariant under multiplication by $e^{i\theta }$ for all $\theta \in ℝ$. We investigate about the persistence of the decorrelation between the...

The paper deals with initial-boundary value problem for generalized solutions of single quasilinear nonautonomous conservation law. For the case so-called "processes with aggravation" the localization property and inner boundedness are studied. Also in case when boundary function tends to zero as t ⇒ +∞ the localization effect is regarded.

B. Rzepecki in [5] examined the Darboux problem for the hyperbolic equation $z_{xy}=f(x,y,z,z_{xy})$ on the quarter-plane x ≥ 0, y ≥ 0 via a fixed point theorem of B.N. Sadovskii [6]. The aim of this paper is to study the Picard problem for the hyperbolic equation $z_{xy}=f(x,y,z,z_x,z_{xy})$ using a method developed by A. Ambrosetti [1], K. Goebel and W. Rzymowski [2] and B. Rzepecki [5].

This short report is a review on recent results of S. Caprino, C. Marchioro, E. Miot and the author on the initial value problem associated to the evolution of a continuous distribution of charges (plasma) in presence of a finite number of point charges.