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

Abstract

In this paper we study the nonlinear parabolic equation ∂tu=Δu+a|x|−γ|u|αu,t>0,x∈RN∖{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 Lq(RN) and in C0(RN). In particular, the value q=Nα/(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∞(RN) 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 etΔ(ω(x)|x|−σ).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.