The sharp lifespan for a system of multiple speed wave equations: Radial case

Marvin Koonce; Jason Metcalfe; Marvin Koonce; Jason Metcalfe

doi:10.3934/cam.2025026

Communications in Analysis and Mechanics

2025, Volume 17, Issue 3: 662-682. doi: 10.3934/cam.2025026

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The sharp lifespan for a system of multiple speed wave equations: Radial case

Marvin Koonce ,
Jason Metcalfe ^,

Department of Mathematics, University of North Carolina, Chapel Hill, NC 27599–3250, USA

Received: 25 January 2025 Revised: 04 June 2025 Accepted: 06 June 2025 Published: 01 July 2025
35L05, 35L71, 35A01

Ohta examined a system of multiple speed wave equations with small initial data and demonstrated a finite time blowup. We show, in the radial case, that the same system exists almost globally with the same lifespan as a lower bound. To do this, we use integrated local energy estimates, $ r^p $ weighted local energy estimates, the Morawetz estimate that results from using the scaling vector field as a multiplier, and mixed-speed ghost weights.
- wave equations,
- local energy estimates,
- multiple speed,
- almost global existence
Citation: Marvin Koonce, Jason Metcalfe. The sharp lifespan for a system of multiple speed wave equations: Radial case[J]. Communications in Analysis and Mechanics, 2025, 17(3): 662-682. doi: 10.3934/cam.2025026

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Abstract

Ohta examined a system of multiple speed wave equations with small initial data and demonstrated a finite time blowup. We show, in the radial case, that the same system exists almost globally with the same lifespan as a lower bound. To do this, we use integrated local energy estimates, $ r^p $ weighted local energy estimates, the Morawetz estimate that results from using the scaling vector field as a multiplier, and mixed-speed ghost weights.

References

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