Abstract
In this note we study lower bounds of energy growth for solutions to wave equations which are compact in space perturbations of the wave equation $\partial_{t}^{2}u - \Delta u = 0$. Assuming that there exists a null bicharacteristic $(x(t),\xi(t))$, parametrized by the time $t$, such that $x(t)$ remains inside a ball and $\xi(t)$ outside a ball for $t \geq 0$ we prove that the solution operator $R(t)$ is bounded from below by constant times $\sqrt{\lvert\xi(t)\rvert/\lvert\xi(0)\rvert}$ in the operator norm. We apply this result to examples constructed by the same idea as in Colombini and Rauch [1] and show that there exist compact in space perturbations which cause $\exp(ct^{\alpha})$ growth of the energy for any given $0\leq \alpha \leq 1$.
Citation
Shin-ichi Doi. Tatsuo Nishitani. Hideo Ueda. "Note on lower bounds of energy growth for solutions to wave equations." Osaka J. Math. 49 (4) 1065 - 1085, December 2012.
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