Proceedings of the Royal Society of Edinburgh: Section A Mathematics



Time-periodic solutions to the one-dimensional wave equation with periodic or anti-periodic boundary conditions


Shuguan Ji a1 and Yong Li a1
a1 College of Mathematics and Key Laboratory of Symbolic Computation & Knowledge Engineering of Ministry of Education, Jilin University, Changchun 130012, People's Republic of China (sgji@email.jlu.edu.cn; liyong@mail.jlu.edu.cn)

Article author query
ji s   [Google Scholar] 
li y   [Google Scholar] 
 

Abstract

This paper is devoted to the study of time-periodic solutions to the nonlinear one-dimensional wave equation with $x$-dependent coefficients $u(x)y_{tt}-(u(x)y_{x})_{x}+g(x,t,y)=f(x,t)$ on $(0,\pi)\times\mathbb{R}$ under the periodic boundary conditions $y(0,t)=y(\pi,t)$, $y_x(0,t)=y_{x}(\pi,t)$ or anti-periodic boundary conditions $y(0,t)=-y(\pi,t)$, $y_x(0,t)=-y_x(\pi,t)$. Such a model arises from the forced vibrations of a non-homogeneous string and the propagation of seismic waves in non-isotropic media. Our main concept is that of the ‘weak solution'. For $T$, the rational multiple of $\pi$, we prove some important properties of the weak solution operator. Based on these properties, the existence and regularity of weak solutions are obtained.

(Published Online March 26 2007)
(Received October 21 2005)
(Accepted March 23 2006)