Inserted: 1 feb 2018
Last Updated: 1 feb 2018
We are concerned with wave equations associated to some Liouville-type problems on compact surfaces, focusing on sinh-Gordon equation and general Toda systems. Our aim is on one side to develop the analysis for wave equations associated to the latter problems and second, to substantially refine the analysis initiated in 11 concerning the mean field equation. In particular, by exploiting the variational analysis recently derived for Liouville-type problems we prove global existence in time for the sub-critical case and we give general blow up criteria for the super-critical and critical case. The strategy is mainly based on fixed point arguments and improved versions of the Moser-Trudinger inequality.
Keywords: wave equation, Moser-Trudinger inequality, Mean field equation, Sinh-Gordon equation, Toda system, Global existence, Blow up criteria, Liouville-type equation