We consider the Cauchy problem for a generalized Klein-Gordon-Schrödinger system with Yukawa coupling. We prove the existence of global weak solutions by the compactness method and, through a special choice of the admissible pairs to match two types of equations, we prove the uniqueness of those solutions by an approach similar to the method presented by J. Ginibre and G. Velo for the pure Klein-Gordon equation or pure Schrödinger equation. Though it is very simple in form, the method has an unnatural...

In this paper we study the Cauchy problem for second order strictly hyperbolic operators of the form $Lu:={\sum }_{j,k=0}^n{\partial }_{y_j}\left(a_{j,k}{\partial }_{y_k}u\right)+{\sum }_{j=0}^n\{b_j{\partial }_{y_j}u+{\partial }_{y_j}\left(c_{j}u\right)\}+du=f,$ when the coefficients of the principal part are not Lipschitz continuous, but only “Log-Lipschitz” with respect to all the variables. This class of equation is invariant under changes of variables and therefore suitable for a local analysis. In particular, we show local existence, local uniqueness and finite speed of propagation for the noncharacteristic Cauchy problem. This provides an invariant...

The Boltzmann–Poisson system modeling the electron flow in semiconductors is used to discuss the validity of the Child–Langmuir asymptotics. The scattering kernel is approximated by a simple relaxation time operator. The Child–Langmuir limit gives an approximation of the current-voltage characteristic curves by means of a scaling procedure in which the ballistic velocity is much larger that the thermal one. We discuss the validity of the Child–Langmuir regime by performing detailed numerical comparisons...

The Boltzmann–Poisson system modeling the electron flow in semiconductors is used to discuss the validity of the Child–Langmuir asymptotics. The scattering kernel is approximated by a simple relaxation time operator. The Child–Langmuir limit gives an approximation of the current-voltage characteristic curves by means of a scaling procedure in which the ballistic velocity is much larger that the thermal one. We discuss the validity of the Child–Langmuir regime by performing detailed numerical...

We are concerned with the structure of the operator corresponding to the Lax–Friedrichs method. At first, the phenomenae which may arise by the naive use of the Lax–Friedrichs scheme are analyzed. In particular, it turns out that the correct definition of the method has to include the details of the discretization of the initial condition and the computational domain. Based on the results of the discussion, we give a recipe that ensures that the number of extrema within the discretized version of...

We are concerned with the structure of the operator corresponding to the Lax–Friedrichs method. At first, the phenomenae which may arise by the naive use of the Lax–Friedrichs scheme are analyzed. In particular, it turns out that the correct definition of the method has to include the details of the discretization of the initial condition and the computational domain. Based on the results of the discussion, we give a recipe that ensures that the number of extrema within the discretized version...

Extending our previous work, we show that the Cauchy problem for wave equations with critical exponential nonlinearities in 2 space dimensions is globally well-posed for arbitrary smooth initial data.

In questa nota viene introdotto un nuovo metodo per ottenere espressioni esplicite dell'energia della soluzione dell'equazione iperbolica $${\left(\frac{\partial }{\partial t}\right)}^{m}u+\sum _{|\nu |+j\le m;j\le m-1}{a}_{\nu ,j}(t){\left(\frac{\partial }{\partial x}\right)}^{\nu }{\left(\frac{\partial }{\partial t}\right)}^{j}u=0.$$ Stimando opportunamente queste espressioni si ottengono nuovi risultati di buona positura negli spazi di Gevrey per l'equazione $(\cdot )$ quando questa è debolmente iperbolica.

We consider the following Darboux problem for the functional differential equation $\partial ²u/\partial x\partial y(x,y)=f(x,y,u_{(x,y)},\partial u/\partial x(x,y),\partial u/\partial y(x,y))$ a.e. in [0,a]×[0,b], u(x,y) = ψ(x,y) on [-a₀,a]×[-b₀,b]$0,a\left]\times \right(0,b],$where the function $u_{(x,y)}:[-a₀,0]\times [-b₀,0]\to {ℝ}^k$ is defined by $u_{(x,y)}(s,t)=u(s+x,t+y)$ for (s,t) ∈ [-a₀,0]×[-b₀,0]. We prove a theorem on existence of the Carathéodory solutions of the above problem.