  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 $a(x)\equiv1$, 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.
