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.

Propagation of analyticity for a class of nonlinear hyperbolic equations

Sergio Spagnolo

Abstract

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