Well-posedness, global existence and large time behavior for Hardy-Hénon parabolic equations

The equation with more general potential, ∂tu=Δu+V(x)|u|αu,V(x)|x|γ∈L∞(RN), 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.