Existence of quasi-periodic solutions with two frequencies of completely resonant, periodically forced, nonlinear wave equations with periodic spatial boundary conditions is established. We consider both the cases the forcing frequency is (Case A) a rational number and (Case B) an irrational number.

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.

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...

We study the “hyperboloidal Cauchy problem” for linear and semi-linear wave equations on Minkowski space-time, with initial data in weighted Sobolev spaces allowing singular behavior at the boundary, or with polyhomogeneous initial data. Specifically, we consider nonlinear symmetric hyperbolic systems of a form which includes scalar fields with a $\lambda {\phi }^p$ nonlinearity, as well as wave maps, with initial data given on a hyperboloid; several of the results proved apply to general space-times admitting conformal...

We show that the set of nonnegative equilibrium-like states, namely, like $(y_d,0)$ of the semilinear vibrating string that can be reached from any non-zero initial state $(y_0,y_1)\in H_0^1(0,1)\times L^2(0,1)$, by varying its axial load and the gain of damping, is dense in the “nonnegative” part of the subspace $L^2(0,1)\times \left\{0\right\}$ of $L^2(0,1)\times H^{-1}(0,1)$. Our main results deal with nonlinear terms which admit at most the linear growth at infinity in $y$ and satisfy certain restriction on their total impact on (0,∞) with respect to the time-variable.

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.

The model order reduction methodology of reduced basis (RB) techniques offers efficient treatment of parametrized partial differential equations (P2DEs) by providing both approximate solution procedures and efficient error estimates. RB-methods have so far mainly been applied to finite element schemes for elliptic and parabolic problems. In the current study we extend the methodology to general linear evolution schemes such as finite volume schemes for parabolic and hyperbolic evolution equations....