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A Local Energy Estimate for Damped Wave Equations (2023)

Undergraduates: Xiao-Ming Porter, Yizhou Gu

Faculty Advisor: Jason Metcalfe
Department: Mathematics

Abstract. We develop a weighted local energy estimate to be used for damped waves. The estimate introduces linear damping to build on an estimate by Metcalfe & Rhoads [2], and Metcalfe & Stewart [4], which combines the r^p-weight by Dafermos & Rodnianski [2] with the ghost-weight by Alinhac [1]. The damped estimate demonstrates an improved lower bound compared to the undamped case. We will introduce the backgrounds and state the theorem for the energy estimate.

[1] Alinhac, S.: The null condition for quasilinear wave equations in two space dimensions I. Invent. Math. 145(3), 597–618 (2001)
[2] Dafermos, M., Rodnianski, I.: A new physical-space approach to decay for the wave equation with applications to black hole spacetimes. In: XVIth International Congress on Mathematical Physics, pp. 421–432. World Science Publication, Hackensack, NJ (2010)
[3] Metcalfe, J., Rhoads, T.: Long-time existence for systems of quasilinear wave equations. arXiv preprint arXiv:2203.08599 (2022)
[4] Metcalfe, J., Stewart, A.: On a system of weakly null semilinear wave equations. Anal. Math. Phys. 12, Paper No. 125. arXiv:2204.06665 (2022)

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