In this paper, we study the quasineutral limit of the isothermal Euler-Poisson equation for ions, in a domain with boundary. This is a follow-up to our previous work [5], devoted to no-penetration as well as subsonic outflow boundary conditions. We focus here on the case of supersonic outflow velocities. The structure of the boundary layers and the stabilization mechanism are different.

The aim of this talk is to present some recent existence results about quasi-periodic solutions for PDEs like nonlinear wave and Schrödinger equations in ${𝕋}^d$, $d\ge 2$, and the $1$-$d$ derivative wave equation. The proofs are based on both Nash-Moser implicit function theorems and KAM theory.

We prove the existence of quasi-periodic solutions for Schrödinger equations with a multiplicative potential on ${𝕋}^d,d\ge 1$, finitely differentiable nonlinearities, and tangential frequencies constrained along a pre-assigned direction. The solutions have only Sobolev regularity both in time and space. If the nonlinearity and the potential are $C^{\infty }$ then the solutions are $C^{\infty }$. The proofs are based on an improved Nash-Moser iterative scheme, which assumes the weakest tame estimates for the inverse linearized operators...

It is proved that a function can be estimated in the norm with a higher degree of summability if it satisfies some integral relations similar to the reverse Hölder inequalities (quasireverse Hölder inequalities). As an example, we apply this result to derive an a priori estimate of the Hölder norm for a solution of strongly nonlinear elliptic system.

Dopo aver introdotto la nozione di quasi-simmetrizzatore per sistemi del prim'ordine debolmente iperbolici, si dimostra che ad ogni sistema di tipo Sylvester, cioè proveniente da un'equazione scalare di ordine superiore, si può associare in modo regolare un quasi-simmetrizzatore. Come applicazione di questo risultato si prova che, per qualunque sistema semi-lineare $N\times N$ debolmente iperbolico, le soluzioni Gevrey in x di ordine $s<N/\left(N-1\right)$ restano analitiche non appena lo siano all'istante iniziale.

On considère un opérateur $L$ défini par$$Lu=\sum _{i=1}^n{D}_i{\mathbf{P}}_{i,p}\left(u\right)-\sum _{k=1}^N{D}_t[{\alpha }_{p,k}{u}_k]$$où $u$ est une application de $\bar{\Omega }\times [0,T]$ dans ${\mathbf{R}}^N$ ($\Omega$ ouvert quelconque de $R^n$), ${\mathbf{P}}_{i,p}\left(u\right)$$(1\le i\le n...$

This paper concerns the study of the numerical approximation for the following boundary value problem: $$\left\{\begin{array}{cccc}{u}_t(x,t)-u_{xx}(x,t)=-u^{-p}(x,t),\hfill & 0<x<1,t>0,u_x(0,t)=0,\hfill & u(1,t)=1,t>0,u(x,0)=u_0\left(x\right)>0,& 0\le x\le 1,\end{array}\right.$$ where $p>0$. We obtain some conditions under which the solution of a semidiscrete form of the above problem quenches in a finite time and estimate its semidiscrete quenching time. We also establish the convergence of the semidiscrete quenching time. Finally, we give some numerical experiments to illustrate our analysis.

In this paper, we consider the following initial-boundary value problem $$\left\}\begin{array}{c}{u}_{tt}(x,t)=\epsilon Lu(x,t)+f\left(u(x,t)\right)\text{in}\Omega \times (0,T),\hfill \\ u(x,t)=0\text{on}\partial \Omega \times (0,T),\hfill \\ u(x,0)=0\text{in}\Omega ,\hfill \\ u_t(x,0)=0\text{in}\Omega ,\hfill \end{array}\right.$$ where $\Omega$ is a bounded domain in ${ℝ}^N$ with smooth boundary $\partial \Omega$, $L$ is an elliptic operator, $\epsilon$ is a positive parameter, $f\left(s\right)$ is a positive, increasing, convex function for $s\in (-\infty ,b)$, ${lim}_{s\to b}f\left(s\right)=\infty$ and ${\int }_0^b\frac{ds}{f\left(s\right)}<\infty$ with $b=const>0$. Under some assumptions, we show that the solution of the above problem quenches in a finite time and its quenching time goes to that of the solution of the following differential equation $$\left\}\begin{array}{cc}{\alpha }^{\text{'}\text{'}}\left(t\right)=f\left(\alpha \left(t\right)\right),\hfill & t>0,\hfill \\ \alpha \left(0\right)=0,{\alpha }^{\text{'}}\left(0\right)=0,\hfill \end{array}\right.$$ as $\epsilon$ goes to zero. We also show that the above result remains...