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arXiv · 2611.08662

Blow-Up and Life-Span for Nonlinear Hardy-Hénon Parabolic Equations

Byrame Ben Slimene·Slim Tayachi
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Abstract

We consider the nonlinear Hardy-Hénon parabolic equation $ u_t=Δu+|x|^l|u|^αu,$ $t>0,$ $x\in\mathbb{R}^N$ or $x\in\mathbb{R}^N\setminus\{0\}$, with $N\ge1$, $l>-\min\{2,N\}$, $α>0$, with nonnegative initial datum $u_0$. We first establish new criteria for finite-time blow-up. We then derive optimal upper bounds for the life-span $T_{\max}(λφ)$ of solutions arising from the initial datum $u_0=λφ$. These estimates establish a precise dependence of $T_{\max}(λφ)$ on both the initial profile $φ$ and the parameter $λ$ in the small and large-amplitude regimes.

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Byrame Ben Slimene, Slim Tayachi. 2026-11-05. Blow-Up and Life-Span for Nonlinear Hardy-Hénon Parabolic Equations. https://arxiv.org/abs/2611.08662

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