TITLE

Time periodic solutions to a nonlinear wave equation with x-dependent coefficients

AUTHOR(S)
Ji, Shuguan
PUB. DATE
June 2008
SOURCE
Calculus of Variations & Partial Differential Equations;Jun2008, Vol. 32 Issue 2, p137
SOURCE TYPE
Academic Journal
DOC. TYPE
Article
ABSTRACT
In this paper, we study the problem of time periodic solutions to the nonlinear wave equation with x-dependent coefficients $$u(x)y_{tt}-(u(x)y_{x})_{x}+|y|^{p-2}y=f(x,t)$$ on $$(0,\pi)\times {\mathbb{R}}$$ under the boundary conditions a 1 y(0, t)+ b 1 y x (0, t) = 0, $$a_2y(\pi, t)+b_2y_{x}(\pi, t)=0$$ ( $$a_i^2+b_i^2\neq0$$ for i = 1, 2) and the periodic conditions y( x, t + T) = y( x, t), y t ( x, t + T) = y t ( x, t). Such a model arises from the forced vibrations of a bounded nonhomogeneous string and the propagation of seismic waves in nonisotropic media. For $$T = 2 \pi/k\,(k\in{\mathbb{N}})$$ , we establish the existence of time periodic solutions in the weak sense by utilizing some important properties of the wave operator with x-dependent coefficients.
ACCESSION #
31342430

 

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