| Abstract: | We study the nonlinear Klein-Gordon equation, in one dimension, with a
qudratic term and variable coefficient qubic term. This equation arises
from the asymptotic stability theory of the kink solution.Our main result
is the global existence and decay estimates for this equation.
We discovered a striking new phenomena in this problem: a resonant
interaction between the spacial frequencies of the nonlinear coefficient
and the temporal oscillations of the solution.
A related new effect is the appearance, as times approaches infinity, of a
shock wave singularity at the derivative level of the solution.
The proof requires the development of new normal-form
transformations.
This is a joint work with H. Lindblad, I. Rodnianski and J. Sterbenz. |
