In this paper we study the nonlinear parabolic equation ∂tu = ∆u + a|x| −γ |u| α u, t > 0, x ∈ R N \{0}, N ≥ 1, a ∈ R, α > 0, 0 < γ < min(2, N) and with initial value u(0) = ϕ. We establish local well-posedness in L q (R N) and in C0(R N). In particular, the value q = N α/(2 − γ) plays a critical role. For α > (2 − γ)/N, we show the existence of global self-similar solutions with initial values ϕ(x) = ω(x)|x| −(2−γ)/α , where ω ∈ L ∞ (R N) is homogeneous of degree 0 and ω ∞ is sufficiently small. We then prove that if ϕ(x) ∼ ω(x)|x| −(2−γ)/α for |x| large, then the solution is global and is asymptotic in the L ∞-norm to a self-similar solution of the nonlinear equation. While if ϕ(x) ∼ ω(x)|x| −σ for |x| large with (2 − γ)/α < σ < N, then the solution is global but is asymptotic in the L ∞-norm to e t∆ (ω(x)|x| −σ). The equation with more general potential, ∂tu = ∆u + V (x)|u| α u, V (x)|x| γ ∈ L ∞ (R N), is also studied. In particular, for initial data ϕ(x) ∼ ω(x)|x| −(2−γ)/α , |x| large , we show that the large time behavior is linear if V is compactly supported near the origin, while it is nonlinear if V is compactly supported near infinity.