A Mixed Boundary Value Problem For U(Xy) = F (X , Y, U, U(X), U(Y))

Consider a single hyperbolic PDE u(xy) = f(x, y, u, u(x) , u(y)), with locally prescribed data: u along a noncharacteristic curve M and ux 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. (C) 2019 Elsevier Inc. All rights reserved.