Grow-up for a quasilinear heat equation with a localized reaction in higher dimensions

We study the behaviour of nonnegative solutions to the quasilinear heat equation with a reaction localized in a ball $$u_t=\mathrm{\Delta }{u}^m+a(x)u^p,$$ for $m>0$, $0<p\le \max \{1,m\}$, $a(x)=\text{mathds}{1}_{B_L}(x)$, $0<L<\mathrm{\infty }$ and $N\ge 2$. We study when solutions, which are global in time, are bounded or unbounded. In particular we show that the precise value of the length $L$ plays a crucial role in the critical case $p=m$ for $N\ge 3$. We also obtain the asymptotic behaviour of unbounded solutions and prove that the grow-up rate is different in most of the cases to the one obtained when $L=\mathrm{\infty }$.