Abstract
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.
| Original language | English (US) |
|---|---|
| Journal | Journal of Differential Equations |
| DOIs | |
| State | Accepted/In press - Jan 1 2019 |
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All Science Journal Classification (ASJC) codes
- Analysis
- Applied Mathematics
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A mixed boundary value problem for uxy = f(x,y,u,ux,uy). / Jenssen, Helge Kristian; Kogan, Irina A.
In: Journal of Differential Equations, 01.01.2019.Research output: Contribution to journal › Article
TY - JOUR
T1 - A mixed boundary value problem for uxy = f(x,y,u,ux,uy)
AU - Jenssen, Helge Kristian
AU - Kogan, Irina A.
PY - 2019/1/1
Y1 - 2019/1/1
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
JO - Journal of Differential Equations
JF - Journal of Differential Equations
SN - 0022-0396
ER -