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 -