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

Published online by Cambridge University Press:  26 March 2007

Shuguan Ji  and
Yong Li
Shuguan Ji
Affiliation:
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)
Yong Li
Affiliation:
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)
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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)ytt – (u(x)yx)x + g(x,t,y) = f(x,t) on (0,π) × ℝ under the periodic boundary conditions y(0,t) = y(π,t), yx(0,t) = yx(π,t) or anti-periodic boundary conditions y(0, t) = –y(π,t), yx[0,t) = – yx(π,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 π, we prove some important properties of the weak solution operator. Based on these properties, the existence and regularity of weak solutions are obtained.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 137 , Issue 2 , 2007 , pp. 349 - 371
Copyright
Copyright © Royal Society of Edinburgh 2007

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