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

Abstract

Optimal conditions for initial data leading to non-existence of non-negative solutions to the Cauchy problem for the parabolic Hardy-H{é}non equation $$\partial \_tu=\Delta u^m+|x{|}^{\sigma }{u}^p,(t,x)\in (0,\infty )\times {\mathbb{R}}^N,$$ with $m>0$, $\sigma >0$ and $p>\max \{1,m\}$, are identified. Assuming that the initial condition satisfies $$u\_0\in L^{\infty }({\mathbb{R}}^N),\lim \_|x|\to \infty |x{|}^{\gamma }u\_0(x)=L\in (0,\infty ),u\_0\ge 0,$$ 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 $\gamma$.