Initial value problems and time-periodic solutions for a nonlinear wave equation

1957 ◽  
Vol 10 (3) ◽  
pp. 331-356 ◽  
Author(s):  
F. A. Ficken ◽  
B. A. Fleishman
Keyword(s):  
Wave Equation ◽  
Periodic Solutions ◽  
Nonlinear Wave ◽  
Nonlinear Wave Equation ◽  
Initial Value Problems ◽  
Initial Value ◽  
Time Periodic Solutions ◽  
Time Periodic

Time-periodic solutions to a nonlinear wave equation with periodic or anti-periodic boundary conditions

2008 ◽  
Vol 465 (2103) ◽  
pp. 895-913 ◽  
Author(s):  
Shuguan Ji
Keyword(s):  
Wave Equation ◽  
Boundary Conditions ◽  
Periodic Solutions ◽  
Nonlinear Wave ◽  
Nonlinear Wave Equation ◽  
Periodic Boundary ◽  
Periodic Boundary Conditions ◽  
Main Concept ◽  
Time Periodic Solutions ◽  
Time Periodic

This paper is concerned with the existence of time-periodic solutions to the nonlinear wave equation with x -dependent coefficients u ( x ) y tt − ( u ( x ) y x ) x + au ( x ) y +| y | p −2 y = f ( x ,  t ) on (0,  π )× under the periodic or anti-periodic boundary conditions y (0, t )=± y ( π ,  t ), y x (0,  t )=± y x ( π ,  t ) and the time-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 non-homogeneous string and the propagation of seismic waves in non-isotropic media. A main concept is the notion ‘weak solution’ to be given in §2. For T =2 π / k ( k ∈ ), we establish the existence of time-periodic solutions in the weak sense by investigating some important properties of the wave operator with x -dependent coefficients.

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