On the mixed problem for some quasilinear hyperbolic system with fully nonlinear boundary condition.

*(English)*Zbl 0689.35055The authors prove the local existence theorem in time of classical solutions to the following mixed problem for second-order systems:

\[ (N)\quad \partial_ t^ 2u(t)-\partial_ i(p^ i(t,D'u(t))+\phi_{\Omega}(t,D'u(t))=f_{\Omega}(t)\quad in\quad (0,T)\times \Omega \] \(\nu_ ip^ i(t,D'u(t))+Q_ r(t,D'u(t))=f_ r(t)\) on \((0,T)\times \Gamma\)

u(0)\(=u_ 0\) and \(\partial_ tu(0)=u_ 1\) in \(\Omega\)

here u denotes an m-vector.

(N) was already treated and the local existence theorem was proved by Y. Shibata and G. Nakamura. But the order of Sobolev spaces in which solutions exist was not best possible. T. Kato treated the mixed problems of the same type as in (N) in his abstract framework. When \(m=1\) and the nonlinear function \(p^ i\), \(Q_{\Omega}\) and \(Q_{\Gamma}\) do not depend on t and \(\partial_ tu\), and \(f_{\Gamma}(t)\in 0\); applying his abstract theory to (N), he gave some improvements of the result due to Y. Shibata regarding the minimal order of the Sobolev spaces in which the solution exists. The purpose of this paper is to give same improvements as the ones given by Kato, where \(m\geq 1\), where the nonlinear functions may depend on t and \(\partial_ tu\) and \(f_{\Gamma}(t)\not\equiv 0\). The approach used in this paper is concrete and elementary, and different from the one used by Kato.

\[ (N)\quad \partial_ t^ 2u(t)-\partial_ i(p^ i(t,D'u(t))+\phi_{\Omega}(t,D'u(t))=f_{\Omega}(t)\quad in\quad (0,T)\times \Omega \] \(\nu_ ip^ i(t,D'u(t))+Q_ r(t,D'u(t))=f_ r(t)\) on \((0,T)\times \Gamma\)

u(0)\(=u_ 0\) and \(\partial_ tu(0)=u_ 1\) in \(\Omega\)

here u denotes an m-vector.

(N) was already treated and the local existence theorem was proved by Y. Shibata and G. Nakamura. But the order of Sobolev spaces in which solutions exist was not best possible. T. Kato treated the mixed problems of the same type as in (N) in his abstract framework. When \(m=1\) and the nonlinear function \(p^ i\), \(Q_{\Omega}\) and \(Q_{\Gamma}\) do not depend on t and \(\partial_ tu\), and \(f_{\Gamma}(t)\in 0\); applying his abstract theory to (N), he gave some improvements of the result due to Y. Shibata regarding the minimal order of the Sobolev spaces in which the solution exists. The purpose of this paper is to give same improvements as the ones given by Kato, where \(m\geq 1\), where the nonlinear functions may depend on t and \(\partial_ tu\) and \(f_{\Gamma}(t)\not\equiv 0\). The approach used in this paper is concrete and elementary, and different from the one used by Kato.

Reviewer: J.Wang

##### MSC:

35L70 | Second-order nonlinear hyperbolic equations |

35L55 | Higher-order hyperbolic systems |

35A07 | Local existence and uniqueness theorems (PDE) (MSC2000) |

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\textit{Y. Shibata} and \textit{M. Kikuchi}, J. Differ. Equations 80, No. 1, 154--197 (1989; Zbl 0689.35055)

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##### References:

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[2] | Kato, T, Abstract differential equations and nonlinear mixed problem, Scuola normale superiore, lezioni fermiane, Pisa, (1985) |

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