J. DITTRICH, P. DUCLOS, N. GONZALEZ
STABILITY AND INSTABILITY OF THE WAVE EQUATION SOLUTIONS IN A PULSATING DOMAIN
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ABSTRACT.  The behavior of energy is studied for the real scalar field satisfying 
d'Alembertequation in a finite space interval $0<x<a(t)$; the endpoint $a(t)$ 
is assumed to move slower than the light and periodically in most parts of the
paper. The boundary conditions are of Dirichlet and Neumann type. We give 
sufficient conditions for the unlimited growth, the boundedness and the 
periodicity of the energy $E$. The case of unbounded energy without infinite 
limit ($0<\liminf_{t\to +\infty}E(t)<\limsup_{t\to +\infty}E(t)=+\infty$) is 
also possible. For the Neumann boundary condition, $E$ may decay to zero as 
the time tends to infinity. If $a$ is periodic, the solution is determined by 
a homeomorphism $\overline{F}$ of the circle related to $a$. The behavior of 
$E$ depends essentially on the number theoretical characteristics of the 
rotation number of $\overline{F}$.