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Consider a single hyperbolic PDE $u_{xy}=f(x,y,u,u_x,u_y)$, with locallyprescribed data: $u$ along a non-characteristic curve $M$ and $u_x$ along anon-characteristic curve $N$. We assume that $M$ and $N$ are graphs ofone-to-one functions, intersecting only at the origin, and located in the firstquadrant 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 canfail in the following strong sense: for the same boundary data, there are twosolutions that differ at points arbitrarily close to the origin. In the lattercase, we also establish existence of a local solution (under a Lipschitzcondition on the function $f$). The construction, via Picard iteration, makesuse of a careful choice of additional $u$-data which are updated in eachiteration step.

A mixed boundary value problem for $u_{xy}=f(x,y,u,u_x,u_y)$