A parabolic Hardy-H\'enon equation with quasilinear degenerate diffusion

Abstract

Local and global well-posedness, along with finite time blow-up, are investigated for the following Hardy-H\'enon equation involving a quasilinear degenerate diffusion and a space-dependent superlinear source featuring a singular potential $$\partial_t u=\Delta u^m+|x|^{\sigma}u^p,\quad t>0,\ x\in\mathbb{R}^N,$$ when $m>1$, $p>1$ and $\sigma\in \big(\max\{-2,-N\},0 \big)$. While the superlinear source induces finite time blow-up when $\sigma=0$, whatever the value of $p>1$, at least for sufficiently large initial conditions, a striking effect of the singular potential $|x|^\sigma$ is the prevention of finite time blow-up for suitably small values of $p$, namely, $1<p\le p_G := [2-\sigma(m-1)]/2$. Such a result, as well as the local existence of solutions for $p>p_G$, 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 $p_G<p\leq p_F:=m+(\sigma+2)/N$, while global existence for small initial data in some critical Lebesgue spaces and blow-up in finite time for initial data with a negative energy are proved for $p>p_F$. 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 $(m,p,\sigma)$ and conditions on $u_0$.