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 $u_t-\Delta u=|x|^{-\gamma}\,u^q$, $0<q<1$, $\gamma>0$ in the whole space $\mathbb{R}^N$, and for non-negative initial data $u_0\in C_0(\mathbb{R}^N)$.