  We consider a degenerate chemotaxis model with two-species and two-stimuli in
dimension $d \geq 3$. Under the hypothesis of integrable initial data with
finite second moment and energy, we show local-in-time existence for any mass
of free-energy solutions, namely weak solutions with some free energy
estimates. We exhibit that the qualitative behavior of solutions is decided by
a set of critical values: there is a critical value of a parameter pair in the
system of equations for which there is a global-in-time energy solution and
there exist blowing-up free-energy solutions under a criticality condition is
violated for the parameter pair.

Analysis of PDEs