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

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We study the behaviour of nonnegative global solutions to the quasilinear heat equation with a reaction localized in a ball $$\begin{aligned} u_t={\varDelta } u^m+a(x)u^p, \end{aligned}$$ for $$m>0$$ , $$0<p\le \max \{1,m\}$$ , $$a(x)=\mathbb {1}_{B_L}(x)$$ , $$0<L<\infty $$ and $$N\ge 2$$ . We study when the solutions 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 $$L=\infty $$ .