## Differential and Integral Equations

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

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

Davide Catania and Vladimir Georgiev

#### 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