### Abstract

The notion of systems with integration by parts is introduced.With this notion, the spatial operator of the transport equation and the spatial operator of the wave or heat equations on graphs can be defined.Th e graphs, which we consider, can consist of arbitrarily many edges and vertices.The respective adjoints of the operators on those graphs can be calculated and skew-selfadjoint operators can be classified via boundary values.Using the work of R.Picard (Math. Meth.App.Sci.32: 1768–1803 [2009]), we can therefore show well-posedness results for the respective evolutionary problems.

Original language | English |
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Title of host publication | Spectral Theory, Mathematical System Theory, Evolution Equations, Differential and Difference Equations |

Subtitle of host publication | 21st International Workshop on Operator Theory and Applications, Berlin, July 2010 |

Editors | Wolfgang Arendt, Joseph A. Ball, Jussi Behrndt, Karl-Heinz Förster, Volker Mehrmann, Carsten Trunk |

Place of Publication | Heidelberg |

Pages | 653-666 |

Number of pages | 14 |

DOIs | |

Publication status | Published - 16 Jun 2012 |

### Publication series

Name | Operator Theory: Advances and Applications |
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Publisher | Springer |

Volume | 221 |

### Keywords

- evolutionary equations
- (skew)-self-adjoint operators
- spatial operator

## Cite this

Waurick, M., & Kaliske, M. (2012). On the well-posedness of evolutionary equations on infinite graphs. In W. Arendt, J. A. Ball, J. Behrndt, K-H. Förster, V. Mehrmann, & C. Trunk (Eds.),

*Spectral Theory, Mathematical System Theory, Evolution Equations, Differential and Difference Equations: 21st International Workshop on Operator Theory and Applications, Berlin, July 2010*(pp. 653-666). (Operator Theory: Advances and Applications ; Vol. 221).. https://doi.org/10.1007/978-3-0348-0297-0