# Propagation of analyticity for a class of nonlinear hyperbolic equations

### Sergio Spagnolo

Università di Pisa, Italy

## Abstract

We consider the semilinear hyperbolic equations of the form

$\partial_{t}^mu \,+, a_1(t)\,\partial_{t}^{m-1}\partial_{x} u \,+, \cdots \,+, a_m(t)\,\partial_{x}^mu \,=\, f(u)$

with $f(u)$ entire analytic, where the characteristic roots satisfy

$\lambda_i^2(t),+,\lambda_j^2(t) \le M( \lambda_i(t)-\lambda_j(t))^2 \,, \quad \hbox{} \ \ i\neq j.$

We prove that, if the $\,a_h(t)$'s are analytic functions, al the solutions bounded in $\mathcal {C}^\infty$ enjoy the propagation of analyticity; while, if the $a_h(t)$'s are $\mathcal {C}^\infty$ functions, such a property holds for those solutions which are bounded in some Gevrey class.

## Cite this article

Sergio Spagnolo, Propagation of analyticity for a class of nonlinear hyperbolic equations. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 22 (2011), no. 2, pp. 135–149

DOI 10.4171/RLM/591