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Lower regularity
solutions of the biharmonic Schrödinger equation in a quarter
plane
Roberto de A. Capistrano-Filho, Márcio Cavalcante and Fernando A. Gallego |
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Vol. 309 (2020), No. 1, 35–70
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DOI: 10.2140/pjm.2020.309.35
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Abstract |
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We deal with the initial-boundary value problem of the biharmonic cubic nonlinear Schrödinger equation in a quarter plane with inhomogeneous Dirichlet–Neumann boundary data. We prove local well-posedness in the low regularity Sobolev spaces by introducing Duhamel boundary forcing operator associated to the linear equation in order to construct solutions in the whole line. With this in hand, the energy and nonlinear estimates allow us to apply the Fourier restriction method, introduced by J. Bourgain, to obtain our main result. Additionally, we discuss adaptations of this approach for the biharmonic cubic nonlinear Schrödinger equation on star graphs. |
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Keywords
biharmonic Schrödinger equation, initial-boundary value
problem, local well-posedness, quarter plane
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Mathematical Subject Classification 2010
Primary: 35A07, 35C15, 35G15, 35G30, 35Q55
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Milestones
Received: 23 December 2018
Revised: 29 December 2019
Accepted: 25 July 2020
Published: 26 December 2020
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Authors |
| Roberto de A. Capistrano-Filho | |
| Universidade Federal de
Pernambuco Recife, PE Brazil |
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| Márcio Cavalcante | |
| Universidade Federal de Alagoas
(UFAL) Maceió, AL Brazil |
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| Fernando A. Gallego | |
| Universidad Nacional de Colombia,
Sede Manizales Manizales Colombia |
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