TY - JOUR

T1 - A mixed boundary value problem for uxy = f(x,y,u,ux,uy)

AU - Jenssen, Helge Kristian

AU - Kogan, Irina A.

N1 - Funding Information:
H.K. Jenssen was partially supported by NSF grants DMS-1311353 and DMS-1813283. I.A. Kogan was partially supported by NSF DMS-1311743.
Funding Information:
H.K. Jenssen was partially supported by NSF grants DMS-1311353 and DMS-1813283 . I.A. Kogan was partially supported by NSF DMS-1311743 .
Publisher Copyright:
© 2019 Elsevier Inc.

PY - 2020/6/5

Y1 - 2020/6/5

N2 - Consider a single hyperbolic PDE uxy=f(x,y,u,ux,uy), with locally prescribed data: u along a non-characteristic 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.

AB - Consider a single hyperbolic PDE uxy=f(x,y,u,ux,uy), with locally prescribed data: u along a non-characteristic 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.

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U2 - 10.1016/j.jde.2019.11.063

DO - 10.1016/j.jde.2019.11.063

M3 - Article

AN - SCOPUS:85076254291

VL - 268

SP - 7535

EP - 7560

JO - Journal of Differential Equations

JF - Journal of Differential Equations

SN - 0022-0396

IS - 12

ER -