A parabolic Hardy-Hénon equation with quasilinear degenerate diffusion

Abstract

Local and global well-posedness, along with finite time blow-up, are investigated for the following Hardy-Hénon equation involving a quasilinear degenerate diffusion and a space-dependent superlinear source featuring a singular potential $$\partial_t u=Δu^m+|x|^σu^p,\quad t>0,\ x\in\mathbb{R}^N,$$ when $m>1$, $p>1$ and $σ\in \big(\max\{-2,-N\},0 \big)$. While the superlinear source induces finite time blow-up when $σ=0$, whatever the value of $p>1$, at least for sufficiently large initial conditions, a striking effect of the singular potential $|x|^σ$ is the prevention of finite time blow-up for suitably small values of $p$, namely, $1 p_G$, is obtained by employing the Caffarelli-Kohn-Nirenberg inequalities. Another interesting feature is that uniqueness and comparison principle hold true for generic non-negative initial conditions when $p>p_G$, but their validity is restricted to initial conditions which are positive in a neighborhood of $x=0$ when $p\in (1,p_G)$, a range in which non-uniqueness holds true without this positivity condition. Finite time blow-up of any non-trivial, non-negative solution is established when $p_G p_F$. Optimal temporal growth rates are also derived for global solutions when $p\in (1,p_G]$. All the results are sharp with respect to the exponents $(m,p,σ)$ and conditions on $u_0$.