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.

We report on new results concerning the global well-posedness, dissipativity and attractors for the quintic wave equations in bounded domains of ${ℝ}^3$ with damping terms of the form ${(-{\Delta }_x)}^{\theta }{\partial }_{t}u$, where $\theta =0$ or $\theta =1/2$. The main ingredient of the work is the hidden extra regularity of solutions that does not follow from energy estimates. Due to the extra regularity of solutions existence of a smooth attractor then follows from the smoothing property when $\theta =1/2$. For $\theta =0$ existence of smooth attractors is more complicated and follows...

In this paper we will give a brief survey of recent regularity results for Fourier integral operators with complex phases. This will include the case of real phase functions. Applications include hyperbolic partial differential equations as well as non-hyperbolic classes of equations. An application to an oblique derivative problem is also given.

Large time behavior of solutions to the generalized damped wave equation $u_{tt}+A{u}_t+\nu Bu+F(x,t,u,u_t,\nabla u)=0$ for $(x,t)\in {ℝ}^n\times [0,\infty )$ is studied. First, we consider the linear nonhomogeneous equation, i.e. with F = F(x,t) independent of u. We impose conditions on the operators A and B, on F, as well as on the initial data which lead to the selfsimilar large time asymptotics of solutions. Next, this abstract result is applied to the equation where $A{u}_t=u_t$, $Bu=-\Delta u$, and the nonlinear term is either $|u_t{|}^{q-1}{u}_t$ or ${\left|u\right|}^{\alpha -1}u$. In this case, the asymptotic profile of solutions is given...

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.