  We shed light on a long-standing open question for the semilinear heat
equation $u_t = \Delta u + |u|^{p-1} u$. Namely, without any restriction on the
exponent $p>1$ nor on the smooth domain~$\Omega$, we prove that the critical
$L^q$ norm blows up whenever the solution undergoes {\it type~I~blow-up.}
A~similar property is also obtained for the local critical $L^q$ norm near any
blow-up point.
  In view of recent results of existence of type~II blow-up solutions with
bounded critical $L^q$ norm, which are counter-examples to the open question,
our result seems to be essentially the best possible result in general setting.
This close connection between type I blow-up and critical $L^q$ norm blow-up
appears to be a completely new observation.
  Our proof is rather involved and requires the combination of various
ingredients. It is based on analysis in similarity variables and suitable
rescaling arguments, combined with {\it backward uniqueness and unique
continuation properties} for parabolic equations.
  As a by-product, we obtain the nonexistence of self-similar profiles in the
critical $L^q$ space. Such properties were up to now only known for $ p \le p_S
$ and in radially symmetric case for $ p > p_S $, where $ p_S $ is the Sobolev
exponent.
