Differential and Integral Equations

Blow-up for the semilinear wave equation in the Schwarzschild metric

Davide Catania and Vladimir Georgiev

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Abstract

We study the Cauchy problem for the semilinear wave equation in the Schwarzschild metric ($(3+1)$--dimensional space--time). First, we establish that the problem is locally well posed in $ \mathrm H^\sigma$ for any $\sigma \in [1,p+1)$; then we prove the blow--up of the solution in two cases: a)} $p \in (1,1+\sqrt{2})$ and small initial data supported far away from the black hole, b) $p \in (2,1+\sqrt{2})$ and large data supported near the black hole. In both cases, we also give an estimate from above for the lifespan of the solution.

Article information

Source
Differential Integral Equations, Volume 19, Number 7 (2006), 799-830.

Dates
First available in Project Euclid: 21 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.die/1356050351

Mathematical Reviews number (MathSciNet)
MR2235896

Zentralblatt MATH identifier
1212.35314

Subjects
Primary: 58J45: Hyperbolic equations [See also 35Lxx]
Secondary: 35B40: Asymptotic behavior of solutions 35L70: Nonlinear second-order hyperbolic equations 83C57: Black holes

Citation

Catania, Davide; Georgiev, Vladimir. Blow-up for the semilinear wave equation in the Schwarzschild metric. Differential Integral Equations 19 (2006), no. 7, 799--830. https://projecteuclid.org/euclid.die/1356050351


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