  We consider in this note the semilinear heat system $$\partial_t u = \Delta u
+ f(v), \quad \partial_t v = \mu\Delta v + g(u), \quad \mu > 0,$$ where the
nonlinearity has no gradient structure taking of the particular form $$f(v) =
v|v|^{p-1} \quad \text{and}\quad g(u) = u|u|^{q-1} \quad \text{with} \quad p, q
> 1, $$ or
  $$f(v) = e^{pv}\quad \text{and} \quad g(u) = e^{qu} \quad \text{with} \quad
p,q > 0.$$ We exhibit type I blowup solutions for this system and give a
precise description of its blowup profiles. The method relies on two-step
procedure: the reduction of the problem to a finite dimensional one via a
spectral analysis, then solving the finite dimensional problem by a classical
topological argument based on index theory. As a consequence of our technique,
the constructed solutions are stable under a small perturbation of initial
data. The results and the main arguments presented in this note can be found in
our papers [Ann. IHP 2018] and [JDEs 2018].
