Grow-up for a quasilinear heat equation with a localized reaction

We study the behaviour of global solutions to the quasilinear heat equation with a reaction localized $$u_t=(u^m{)}_{xx}+a(x)u^p,$$ $m,p>0$ and $a(x)$ being the characteristic function of an interval. we prove that there exists $p_0=\max \{1,\frac{m+1}{2}\}$ such that all global solution are bounded if $p>p_0$, while for $p\le p_0$ all the solution are global and unbounded. In the last case, we prove that if $p<m$ the grow-up rate is different to the one obtained when $a(x)\equiv 1$, while if $p>m$ the grow-up rate coincides with that rate, but only inside the support of $a$; outside the interval the rate is smaller.