Solvability conditions for some semi-linear parabolic equations

Consider the Cauchy problem of the semi-linear parabolic equationΔu−∂tu+Vup+w=0,p>1, in RN×(0,T)RN×(0,T), T>0T>0, u(x,0)=u0(x)⩾0u(x,0)=u0(x)⩾0, and u0,V,w∈Lloc1(RN). We establish a necessary condition on the nonlinear potential V=V(x)V=V(x) so that the above has a positive solution for some w=w(x)⩾0w=w(x)⩾0, u0=u0(x)⩾0u0=u0(x)⩾0. An application to the case when V is the important inverse square potential is also given.