The differential rate law relates the rate of reaction to the concentrations of the various species in the system.
Source
So, it's an equation which tells you how fast the reaction will go at any instant if you plug in the concentrations of species in the mix at that instant.
1 liter is dm^3; that is, the volume of a cube with sides of 0.1 meter. If something (call it s) is 'per liter' that is written mathematically as a fraction: s/liter, with the liter on the bottom. An equivalent way to write this is s x L^-1, which equals s x (dm^3)^-1 which equals s x (dm)^-3. Your use of capitals - that is, writing decimeter as DM - is incorrect and if you've seen a liter equated to 'DM^-3' your source has used the capitals improperly. DM means decamolar, not decimeter.