On the Cauchy problem for the fast diffusion equation

For u0,1u0∈Lloc1(RN), the author studies the existence of a kind of weak solution to the Cauchy problemut=div(um−1Du),inRN×(0,T],u(x,0)=u0(x)⩾0,inRN, where m<0 is a constant. The uniqueness and regularity of solutions are also discussed.