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Wellposedness for a $(1+1)$-dimensional wave equation with quasilinear boundary conditions

Ohrem, Sebastian 1; Reichel, Wolfgang 1; Schnaubelt, Roland 1
1 Institut für Analysis (IANA), Karlsruher Institut für Technologie (KIT)

Abstract:

We consider the linear wave equation $V(x)u_{tt}(x, t)−u_{xx}(x, t)=0$ on $[0,\infty)\times[0,\infty)$ with initial conditions and a nonlinear Neumann boundary condition $u_x (0, t) = (f (u_t (0, t)))_t$ at $x = 0$. This problem is an exact reduction of a nonlinear Maxwell problem in electrodynamics. In the case where $f : \mathbb{R} \to \mathbb{R}$ is an increasing homeomorphism we study global existence, uniqueness and wellposedness of the initial value problem by the method of characteristics and fixed point methods. We also prove conservation of energy and momentum and discuss why there is no wellposedness in the case where f is a decreasing homeomorphism. Finally we show that previously known time-periodic, spatially localized solutions (breathers) of the wave equation with the nonlinear Neumann boundary condition at $x = 0$ have enough regularity to solve the initial value problem with their own initial data.


Volltext §
DOI: 10.5445/IR/1000151943
Veröffentlicht am 28.10.2022
Cover der Publikation
Zugehörige Institution(en) am KIT Institut für Analysis (IANA)
Sonderforschungsbereich 1173 (SFB 1173)
Publikationstyp Forschungsbericht/Preprint
Publikationsmonat/-jahr 10.2022
Sprache Englisch
Identifikator ISSN: 2365-662X
KITopen-ID: 1000151943
Verlag Karlsruher Institut für Technologie (KIT)
Umfang 31 S.
Serie CRC 1173 Preprint ; 2022/54
Projektinformation SFB 1173/2 (DFG, DFG KOORD, SFB 1173/2 2019)
Externe Relationen Siehe auch
Schlagwörter Maxwell equations, wave equation, nonlinear boundary condition, wellposedness
Nachgewiesen in arXiv
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