Sharp non-existence threshold for a parabolic Hardy-H{\'e}non equation with quasilinear diffusion

TL;DR

This paper establishes the precise threshold conditions under which solutions to a parabolic Hardy-Hénon equation with quasilinear diffusion do not exist, revealing an optimal boundary based on initial data decay rates.

Contribution

It identifies the exact non-existence threshold for solutions to a nonlinear PDE with Hardy-Hénon type nonlinearity, extending understanding of solution behavior in such equations.

Findings

Non-existence occurs when initial data decay rate is below a specific threshold.

The threshold for non-existence is proven to be optimal.

Existence of self-similar solutions confirms the sharpness of the threshold.

Abstract

Optimal conditions for initial data leading to non-existence of non-negative solutions to the Cauchy problem for the parabolic Hardy-H{\'e}non equation $$ \partial\_tu=\Delta u^m+|x|^{\sigma}u^p, \quad (t,x)\in(0,\infty)\times\mathbb{R}^N, $$ with $m>0$, $\sigma>0$ and $p>\max\{1,m\}$, are identified. Assuming that the initial condition satisfies $$ u\_0\in L^{\infty}(\mathbb{R}^N), \quad \lim\limits\_{|x|\to\infty}|x|^{\gamma}u\_0(x)=L\in(0,\infty), \quad u\_0\geq0, $$ it is shown that non-existence of solution occurs for $$ \gamma<\frac{\sigma+2}{p-m} - \frac{2\max{\{p-p\_G,0\}}}{(p-1)(p-m)} $$ with $$ p\_G:=1+\frac{\sigma(1-m)}{2}. $$ The above threshold for non-existence is optimal, in view of the existence of self-similar solutions for the limiting value of $\gamma$.