New dissipated energies for the thin fluid film equation

The thin fluid film evolution h_t = -(hⁿh _xxx)_x is known to conserve the fluid volume ∫ h dx and to dissipate the "energies" ∫ h^1.5-n dx and ∫ h_x² dx. We extend this last result by showing the energy ∫ h^ph_x² dx is dissipated for some values of p < 0, when 1/2 < n < 3. For example when n = 1, the Hele-Shaw equation h_t = -(hh_xxx)_x dissipates ∫ h ^-1/2h_x² dx.