We prove that the initial value problem for the semi-linear Schrödinger and wave equations is well-posed in the Besov space ${\dot{B}}_2^{\frac{n}{2}-\frac{2}{p},\infty }\left({𝐑}^n\right)$, when the nonlinearity is of type $u^p$, for $p\in 𝐍$. This allows us to obtain self-similar solutions, as well as to recover previously known results for the solutions under weaker smallness assumptions on the data.

Small data scattering for nonlinear Schrödinger equations (NLS), nonlinear wave equations (NLW), nonlinear Klein-Gordon equations (NLKG) with power type nonlinearities is studied in the scheme of Sobolev spaces on the whole space ${ℝ}^n$ with order $s\<n/2$. The assumptions on the nonlinearities are described in terms of power behavior $p_1$ at zero and $p_2$ at infinity such as $1+4/n\le p_1\le p_2\le 1+4/(n-2s)$ for NLS and NLKG, and $1+4/(n-1)\le p_1\le p_2\le 1+4/(n-2s)$ for NLW.

In this paper, the system consisting of two nonlinear equations is studied. The former is hyperbolic with a dissipative term and the latter is elliptic. In a special case, the system reduces to the approximate model for the damped transversal vibrations of a string proposed by G. F. Carrier and R. Narasimha. Taking advantage of accelerated convergence methods, the existence of at least one time-periodic solution is stated on condition that the right-hand side of the system is sufficiently small.

Dans ce travail, on s’intéresse à l’existence globale de solutions classiques et au sens de Shatah-Struwe de l’équation des ondes critique à coefficients variables en dimension $d$ d’espace$${\square }_{A}u+{\left|u\right|}^{4/(d-2)}u={\partial }_t^{2}u-\mathrm{div}(A\left(x\right)·{\nabla }_{x}u)+{\left|u\right|}^{4/(d-2)}u=0,{ℝ}_t\times {ℝ}_x^d,$$où $A$ est une fonction régulière à valeurs dans les matrices $d\times d$ définies positives, valant l’identité en dehors d’un compact fixe.

Presentiamo nuovi risultati di esistenza e molteplicità di soluzioni periodiche di piccola ampiezza per equazioni alle derivate parziali Hamiltoniane. Otteniamo soluzioni periodiche di equazioni «completamente risonanti» aventi nonlinearità generali grazie ad una riduzione di tipo Lyapunov-Schmidt variazionale ed usando argomenti di min-max. Per equazioni «non risonanti» dimostriamo l'esistenza di soluzioni periodiche di tipo Birkhoff-Lewis, mediante un'opportuna forma normale di Birkhoff e realizzando...

We consider three types of semilinear second order PDEs on a cylindrical domain $\Omega \times (0,\infty )$, where $\Omega$ is a bounded domain in ${ℝ}^N$, $N\ge 2$. Among these, two are evolution problems of parabolic and hyperbolic types, in which the unbounded direction of $\Omega \times (0,\infty )$ is reserved for time $t$, the third type is an elliptic equation with a singled out unbounded variable $t$. We discuss the asymptotic behavior, as $t\to \infty$, of solutions which are defined and bounded on $\Omega \times (0,\infty )$.