2017/09/25 by Andreas Hirsch, Hirsch, Andreas, Wolfgang Reichel +1 · 1 citation
Computer Science · Mathematics · #49J40 #Advanced Mathematical Modeling in Engineering #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Primary: 35L71 #Secondary: 35B10 #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.1709.08443
openalex publication_date 2017/09/25 · openalex created_date 2022/10/06 · openalex updated_date 2026/07/28
We consider the semilinear wave equation V(x) utt -uxx+q(x)u = \±\nf(x,u) for three different classes (P1), (P2), (P3) of periodic potentials\nV,q. (P1) consists of periodically extended delta-distributions, (P2) of\nperiodic step potentials and (P3) contains certain periodic potentials V,q\∈\nHr per( R) for r\∈ [1,3/2). Among other assumptions we suppose that\n|f(x,s)|\≤ c(1+ |s|p) for some c>0 and p>1. In each class we can find\nsuitable potentials that give rise to a critical exponent p^\∗ such that\nfor p\∈ (1,p^\∗) both in the "+" and the "-" case we can use variational\nmethods to prove existence of time-periodic real-valued solutions that are\nlocalized in the space direction. The potentials are constructed explicitely in\nclass (P1) and (P2) and are found by a recent result from inverse spectral\ntheory in class (P3). The critical exponent p^\∗ depends on the regularity\nof V, q. Our result builds upon a Fourier expansion of the solution and a\ndetailed analysis of the spectrum of the wave operator. In fact, it turns out\nthat by a careful choice of the potentials and the spatial and temporal\nperiods, the spectrum of the wave operator V(x)\∂t2-\∂x2+q(x)\n(considered on suitable space of time-periodic functions) is bounded away from\n0. This allows to find weak solutions as critical points of a functional on a\nsuitable Hilbert space and to apply tools for strongly indefinite variational\nproblems.\n