**abstract:**
We study the Cauchy problem for the Kirchhoff equation
¥begin{equation}¥label{EQ:Kirchhoff}
¥left¥{
¥begin{aligned}
& ¥partial^{2}_{t} u-¥left(1+¥int_{{¥mathbb{R}}^{n}}

¥nabla u^{2¥,} dx¥right) ¥Delta u=0, &¥quad t>0, ¥quad x¥in ¥mathbb{R}^{n,¥¥
}
& u(0,x)=u_{0}(x), ¥quad ¥partial_{t} u(0,x)=u_{1}(x),
&¥quad x ¥in ¥mathbb{R}^{n.
}
¥end{aligned}¥right.
¥end{equation}
Global existence results for ¥eqref{EQ:Kirchhoff} in an appropriate class are known for initial data $u_{0,} u_{1$
}
that are either small in some Sobolev space or analytic or quasi-analytic (in this case, $u_{0$
}
and $u_{1$} can be arbitrarily large). In this talk, we consider the case of
initial data in the Gevrey class of $L^{2$} type. We inform that if a certain explicit condition
is satisfied, involving the time $T$, the size of $u_{0$} and $u_{1$,} and their regularity, then the
Cauchy problem ¥eqref{EQ:Kirchhoff} has a unique solution on the interval $0, T$ (in an appropriate
class). The proof relies, in particular, on energy estimates for solutions of
the equation
$$
¥partial^{2}_{t} u- c(t) ¥Delta u=0.
$$
This talk is baed on the jopint work with Professor Michael Ruzhansky (Ghent University).

¥begin{thebibliography}{99} ¥bibitem{MR-JAM} T. Matsuyama and M. Ruzhansky, {¥em On the Gevrey well-posedness of the Kirchhoff equation,} J. Anal. Math. {¥bf 137} (2019), 449--468. ¥end{thebibliography}

Mon 10 Feb, 9:20 - 10:10, Aula Dini

Talk

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