English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We consider the semilinear wave equation $V(x) u_{tt} -u_{xx}+q(x)u = \pmf(x,u)$ for three different classes (P1), (P2), (P3) of periodic potentials$V,q$. (P1) consists of periodically extended delta-distributions, (P2) ofperiodic step potentials and (P3) contains certain periodic potentials $V,q\inH^r_{\per}(\R)$ for $r\in [1,3/2)$. Among other assumptions we suppose that$|f(x,s)|\leq c(1+ |s|^p)$ for some $c>0$ and $p>1$. In each class we can findsuitable potentials that give rise to a critical exponent $p^\ast$ such thatfor $p\in (1,p^\ast)$ both in the "+" and the "-" case we can use variationalmethods to prove existence of time-periodic real-valued solutions that arelocalized in the space direction. The potentials are constructed explicitely inclass (P1) and (P2) and are found by a recent result from inverse spectraltheory in class (P3). The critical exponent $p^\ast$ depends on the regularityof $V, q$. Our result builds upon a Fourier expansion of the solution and adetailed analysis of the spectrum of the wave operator. In fact, it turns outthat by a careful choice of the potentials and the spatial and temporalperiods, the spectrum of the wave operator $V(x)\partial_t^2-\partial_x^2+q(x)$(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 asuitable Hilbert space and to apply tools for strongly indefinite variationalproblems.

Real-valued, time-periodic localized weak solutions for a semilinear wave equation with periodic potentials