arXiv 1709.084432017-09math.AP

具有周期位势的半线性wave equation的实值时间周期局部化弱解

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

Andreas Hirsch, Wolfgang Reichel

摘要：

本文考虑半线性波动方程$V(x) u_{tt} -u_{xx}+q(x)u = \pm f(x,u)$对三种不同类型的周期势$V,q$（P1），（P2），（P3）. (P1)由周期性扩展的δ分布组成，（P2）为周期性阶跃电位，（P3）包含$r\in [1,3/2)$的某些周期电位$V,q\in H^r_{\per}(\R)$。在其他假设中，我们假设某些$c>0$和$p>1$的$|f(x,s)|\leq c(1+ |s|^p)$。在每个类中，我们可以找到合适的潜力，产生一个临界指数$p^\ast$，使$p\in (1,p^\ast)$在“+”和“-”的情况下，我们可以使用变分方法来证明存在的时间周期的实值的解决方案，是本地化的空间方向。在（P_1）和（P_2）类中明确地构造了这些势，并在（P_3）类中由逆谱理论的最近结果得到。临界指数$p^\ast$取决于$V, q$的正则性。我们的结果建立在傅立叶展开的解决方案和详细分析的频谱的波算子。事实上，事实证明，通过仔细选择的潜力和空间和时间的周期，频谱的波算子$V(x)\partial_t^2-\partial_x^2+q(x)$（考虑适当的时间周期函数的空间）是有界远离$0$。这使得找到弱解作为临界点的功能在一个合适的希尔伯特空间和应用工具的强不定变分问题。

英文摘要：

We consider the semilinear wave equation $V(x) u_{tt} -u_{xx}+q(x)u = \pm f(x,u)$ for three different classes (P1), (P2), (P3) of periodic potentials $V,q$. (P1) consists of periodically extended delta-distributions, (P2) of periodic step potentials and (P3) contains certain periodic potentials $V,q\in H^r_{\per}(\R)$ 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 find suitable potentials that give rise to a critical exponent $p^\ast$ such that for $p\in (1,p^\ast)$ both in the "+" and the "-" case we can use variational methods to prove existence of time-periodic real-valued solutions that are localized in the space direction. The potentials are constructed explicitely in class (P1) and (P2) and are found by a recent result from inverse spectral theory in class (P3). The critical exponent $p^\ast$ depends on the regularity of $V, q$. Our result builds upon a Fourier 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 potentials and the spatial and temporal periods, 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 a suitable Hilbert space and to apply tools for strongly indefinite variational problems.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

相关论文