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December 2012 Note on lower bounds of energy growth for solutions to wave equations
Shin-ichi Doi, Tatsuo Nishitani, Hideo Ueda
Osaka J. Math. 49(4): 1065-1085 (December 2012).

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

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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.

Information

Published: December 2012
First available in Project Euclid: 19 December 2012

zbMATH: 1273.35069
MathSciNet: MR3007954

Subjects:
Primary: 35L20
Secondary: 35B40 , 35B45

Rights: Copyright © 2012 Osaka University and Osaka City University, Departments of Mathematics

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Vol.49 • No. 4 • December 2012
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