## Abstract

Consider a single hyperbolic PDE u_{xy}=f(x,y,u,u_{x},u_{y}), with locally prescribed data: u along a non-characteristic curve M and u_{x} along a non-characteristic curve N. We assume that M and N are graphs of one-to-one functions, intersecting only at the origin, and located in the first quadrant of the (x,y)-plane. It is known that if M is located above N, then there is a unique local solution, obtainable by successive approximation. We show that in the opposite case, when M lies below N, the uniqueness can fail in the following strong sense: for the same boundary data, there are two solutions that differ at points arbitrarily close to the origin. In the latter case, we also establish existence of a local solution (under a Lipschitz condition on the function f). The construction, via Picard iteration, makes use of a careful choice of additional u-data which are updated in each iteration step.

Original language | English (US) |
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Pages (from-to) | 7535-7560 |

Number of pages | 26 |

Journal | Journal of Differential Equations |

Volume | 268 |

Issue number | 12 |

DOIs | |

State | Published - Jun 5 2020 |

## All Science Journal Classification (ASJC) codes

- Analysis
- Applied Mathematics