### Abstract

We study the wave analog of the Liouville equation for constant Gauss curvature on ^{S2}. We show the global existence in time for the sub-critical equation with data in Ḣ1(S2)×L2(S2). For the super-critical and the critical equation we show the global existence in time assuming the data is symmetric with respect to antipodal points on ^{S2}. Use is made of the Moser-Trudinger inequalities for proving this result. For general data without symmetry assumptions and for the critical equation, we establish that if there is a blow-up, then it will occur at a single point of ^{S2}. We also consider the wave analog of the constant mean curvature equation in the Plateau formalism. This is a system. We establish finite time blow up if the energy of the initial data is larger than 8π and with an additional subsidiary assumption. The threshold energy of 8π arises because it is the energy of the basic bubble discovered by Brezis and Coron.

Original language | English (US) |
---|---|

Pages (from-to) | 187-207 |

Number of pages | 21 |

Journal | Advances in Mathematics |

Volume | 235 |

DOIs | |

State | Published - Mar 1 2013 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Mathematics(all)

### Keywords

- Bubbling
- Concentration
- Constant mean curvature equation
- Gauss curvature equation
- Global existence
- Wave equation

### Cite this

*Advances in Mathematics*,

*235*, 187-207. https://doi.org/10.1016/j.aim.2012.11.014

}

*Advances in Mathematics*, vol. 235, pp. 187-207. https://doi.org/10.1016/j.aim.2012.11.014

**Wave equations associated to Liouville systems and constant mean curvature equations.** / Chanillo, Sagun; Yung, Po Lam.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Wave equations associated to Liouville systems and constant mean curvature equations

AU - Chanillo, Sagun

AU - Yung, Po Lam

PY - 2013/3/1

Y1 - 2013/3/1

N2 - We study the wave analog of the Liouville equation for constant Gauss curvature on S2. We show the global existence in time for the sub-critical equation with data in Ḣ1(S2)×L2(S2). For the super-critical and the critical equation we show the global existence in time assuming the data is symmetric with respect to antipodal points on S2. Use is made of the Moser-Trudinger inequalities for proving this result. For general data without symmetry assumptions and for the critical equation, we establish that if there is a blow-up, then it will occur at a single point of S2. We also consider the wave analog of the constant mean curvature equation in the Plateau formalism. This is a system. We establish finite time blow up if the energy of the initial data is larger than 8π and with an additional subsidiary assumption. The threshold energy of 8π arises because it is the energy of the basic bubble discovered by Brezis and Coron.

AB - We study the wave analog of the Liouville equation for constant Gauss curvature on S2. We show the global existence in time for the sub-critical equation with data in Ḣ1(S2)×L2(S2). For the super-critical and the critical equation we show the global existence in time assuming the data is symmetric with respect to antipodal points on S2. Use is made of the Moser-Trudinger inequalities for proving this result. For general data without symmetry assumptions and for the critical equation, we establish that if there is a blow-up, then it will occur at a single point of S2. We also consider the wave analog of the constant mean curvature equation in the Plateau formalism. This is a system. We establish finite time blow up if the energy of the initial data is larger than 8π and with an additional subsidiary assumption. The threshold energy of 8π arises because it is the energy of the basic bubble discovered by Brezis and Coron.

KW - Bubbling

KW - Concentration

KW - Constant mean curvature equation

KW - Gauss curvature equation

KW - Global existence

KW - Wave equation

UR - http://www.scopus.com/inward/record.url?scp=84871750676&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84871750676&partnerID=8YFLogxK

U2 - 10.1016/j.aim.2012.11.014

DO - 10.1016/j.aim.2012.11.014

M3 - Article

AN - SCOPUS:84871750676

VL - 235

SP - 187

EP - 207

JO - Advances in Mathematics

JF - Advances in Mathematics

SN - 0001-8708

ER -