Wave equations associated to Liouville systems and constant mean curvature equations

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.