An inhomogeneous nonlinear parabolic equation with concave nonlinearity

We establish both the existence and uniqueness of non-negative global solutions for the nonlinear heat equation $u_t-\mathrm{\Delta }u=\mathrm{\mid }x{\mathrm{\mid }}^{-\gamma }{u}^q$, $0<q<1$, $\gamma >0$ in the whole space ${\mathbb{R}}^N$, and for non-negative initial data $u_0\in C_0({\mathbb{R}}^N)$.