Solvability conditions for some semi-linear parabolic equations

Consider the Cauchy problem of the semi-linear parabolic equation in , T>0, u(x,0)=u0(x) 0, and . We establish a necessary condition on the nonlinear potential V=V(x) so that the above has a positive solution for some w=w(x) 0, u0=u0(x) 0. An application to the case when V is the important inverse square potential is also given.