Blow-up of a modified ODEs system arising from the Galerkin approximation of some Navier-Stokes equations

For the third order Galerkin approximation of the Navier-Stokes equations under Navier boundary conditions in a cube we prove global existence and qualitative behaviour of the solution. By modifying properly the signs of the resulting ODEs system and using the test function technique developed by Mitidieri-Pohožaev we prove, instead, finite time blow-up.