Long time existence of solutions to the Kirchhoff equation

speaker: Tokio Matsuyama (Chuo University (Tokyo, Japan))

abstract: We study the Cauchy problem for the Kirchhoff equation ¥begin{equation}¥label{EQ:Kirchhoff} ¥left¥{ ¥begin{aligned} & ¥partial²_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)=u₀(x), ¥quad ¥partial_t u(0,x)=u₁(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 $$ ¥partial²_t u- c(t) ¥Delta u=0. $$ 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}

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

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