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DOI: 10.5445/IR/1000074263

Real-valued, time-periodicweak solutions for a semilinear wave equation with periodic δ-potential

Hirsch, Andreas; Reichel, Wolfgang

Abstract:
We consider the semilinear wave equation $V(x)$$u$$_{u}$ − $u$$_{xx}$ = $±|u|$$^{p-1}$$u$ with p ∈ (1, $\frac{5}{3}$) and a periodically extended delta potential $V(x)$ = $α + βδper(x)$. Both the “+” and the “-” case can be treated. We prove the existence of time-periodic real-valued solutions that are localized in the space direction. Our result builds upon a Fourier-Floquet-Bloch expansion of the solution and a detailed analysis of the spectrum of the wave operator. In fact, it turns out that by a careful choice of the parameters α, β and the spatial and temporal periods, the spectrum of the wave operator $V(x)$∂$\frac{2}{t}$ - ∂$\frac{2}{x}$ (considered on suitable space of time-periodic functions) is bounded away from 0. This allows to find weak solutions as critical points of a functional on a suitable Hilbert space and to apply tools for indefinite variational problems.


Zugehörige Institution(en) am KIT Institut für Analysis (IANA)
Sonderforschungsbereich 1173 (SFB 1173)
Publikationstyp Forschungsbericht
Jahr 2017
Sprache Englisch
Identifikator ISSN: 2365-662X
URN: urn:nbn:de:swb:90-742631
KITopen ID: 1000074263
Verlag KIT, Karlsruhe
Umfang 26 S.
Serie CRC 1173 ; 2017/22
Schlagworte semilinear wave equation, breather solutions, time-periodic, variational methods
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