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.

The aim of this work is threefold. First we set up a calculus for partial differential operators with nonsmooth coefficients which is based on the FBI (Fourier-Bros-Iagolnitzer) transform. Then, using this calculus, we prove a weaker version of the Strichartz estimates for second order hyperbolic equations with nonsmooth coefficients. Finally, we apply these new Strichartz estimates to second order nonlinear hyperbolic equations and improve the local theory, i.e. prove local well-posedness for initial...

The energy in a square membrane Ω subject to constant viscous damping on a subset $\omega \subset \Omega$ decays exponentially in time as soon as ω satisfies a geometrical condition known as the “Bardos-Lebeau-Rauch” condition. The rate $\tau \left(\omega \right)$ of this decay satisfies $\tau \left(\omega \right)=2min(-\mu (\omega ),g(\omega \left)\right)$ (see Lebeau [Math. Phys. Stud.19 (1996) 73–109]). Here $\mu \left(\omega \right)$ denotes the spectral abscissa of the damped wave equation operator and $g\left(\omega \right)$ is a number called the geometrical quantity of ω and defined as follows. A ray in Ω is the trajectory generated by the free motion...