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The semilinear differential equation (1), (2), (3), in $ℝ\times \mathrm{\Omega }$ with $\mathrm{\Omega }\in {ℝ}^N$, (a nonlinear wave equation) is studied. In particular for $\mathrm{\Omega }={ℝ}^3$, the existence is shown of a weak solution $u(t,x)$, periodic with period $T$, non-constant with respect to $t$, and radially symmetric in the spatial variables, that is of the form $u(t,x)=\nu (t,|x|)$. The proof is based on a distributional interpretation for a linear equation corresponding to the given problem, on the Paley-Wiener criterion for the Laplace Transform, and on the alternative method of...

The semilinear differential equation (1), (2), (3), in $ℝ\times \mathrm{\Omega }$ with $\mathrm{\Omega }\in {ℝ}^N$, (a nonlinear wave equation) is studied. In particular for $\mathrm{\Omega }={ℝ}^3$, the existence is shown of a weak solution $u(t,x)$, periodic with period $T$, non-constant with respect to $t$, and radially symmetric in the spatial variables, that is of the form $u(t,x)=\nu (t,|x|)$. The proof is based on a distributional interpretation for a linear equation corresponding to the given problem, on the Paley-Wiener criterion for the Laplace Transform, and on the alternative method of...

Si considera l’equazione non lineare nell'incognita $u(t,x)$ ((1,1) del testo) soddisfatta in un cilindro $G=(o,T)\times \mathrm{\Omega }$ ($\mathrm{\Omega }$ dominio limitato di ${𝐑}^{𝐧}$) con condizioni al contorno tipo Dirichlet o Neumann sulla superficie laterale di $G$ e con relazioni omogenee fra $u$ e $u_t$ sulle basi. Si stabiliscono per la (1) e nel caso di risonanza alcuni teoremi di perturbazione.