Singularity and blow-up estimates via Liouville-type theorems for Hardy–Hénon parabolic equations

Abstract

Abstract. We consider the Hardy-Hénon parabolic equation ut − ∆u = |x| a |u| p−1 u with p > 1 and a ∈ R. We establish the space-time singularity and decay estimates, and Liouvilletype theorems for radial and nonradial solutions. As applications, we study universal and a priori bound of global solutions as well as the blow-up estimates for the corresponding initial boundary value problem.

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

“…7) follows now by the Hölder inequality.Proof of Part (ii). Letv(t) = e t∆ ω(•)| • | − 2−γα We will show the following:u(t) − v(t) p ≤ Ct −β(p)−δ , ∀ p ∈ [r, ∞].We haveu(t) − v(t) = e t∆ (ϕ − Φ) + t (t−σ)∆ V (.…”

“…We are unaware of any previous lower blow-up estimates. On the hand, the following upper blow-up estimate has been established by [1, Theorems 1.2 and 1.3] and [7,Theorem 1.6] in the case q = ∞ (with various restriction on α):…”

Optimal singularities of initial data for solvability of the Hardy parabolic equation

“…They also obtained a global-in-time solution under u 0 (x) ≤ c|x| −(2−γ)/(p−1) if p > p γ and c > 0 small. For related results, see [1,3,7,8,9,10,13,14,15,18,19]. Subsequently, the first author and Sierż ' ega [12] examined necessary conditions of initial data for the existence of solutions including a fractional case.…”

Singularity and Decay Estimates for a Degenerate Parabolic Equation

“…In the special case θ � a � 0, equation (1) has been widely studied by many authors. Here, the space-time singularity and decay estimates of positive solutions of equation (1) have been established by Peter et al in [2] for the case θ � a � 0 and by Phan [4] for the case θ � 0. Motivated by the aforementioned work, we establish the space-time singularity and decay estimates of solutions of equation (1).…”