Periodic solutions to one-dimensional wave equation with x-dependent coefficients

By S. Ji, Yong Li. 2006

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Abstract In this paper, we study the nonlinear one-dimensional periodic wave equation with x-dependent coefficients u ( x ) y t t − ( u ( x ) y x ) x + g ( x , t , y ) = f ( x , t ) on ( 0 , π ) × R under the boundary conditions a 1 y ( 0 , t ) + b 1 y x ( 0 , t ) = 0 , a 2 y ( π , t ) + b 2 y x ( π , t ) = 0 ( a i 2 + b i 2 ≠ 0 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 nonhomogeneous string and the propagation of seismic waves in nonisotropic media. A main concept is the notion “weak solution” to be given in Section 2. For T is 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.

Subfields: Control and Systems Engineering, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/34afa2debe68b99fd3448f61c4e108f1bcfa0711